research article dynamics underlying the gaussian ...tions of an sed harmonic oscillator are a...

Hindawi Publishing CorporationJournal of Computational Methods in PhysicsVolume 2013 Article ID 308538 19 pageshttpdxdoiorg1011552013308538

Research ArticleDynamics Underlying the Gaussian Distribution of the ClassicalHarmonic Oscillator in Zero-Point Radiation

Wayne Cheng-Wei Huang and Herman Batelaan

Department of Physics and Astronomy University of Nebraska-Lincoln Lincoln NE 68588 USA

Correspondence should be addressed to Wayne Cheng-Wei Huang waynehhuskersunledu

Received 12 April 2013 Accepted 29 July 2013

Academic Editor Marta B Rosales

Copyright copy 2013 W C-W Huang and H Batelaan This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Stochastic electrodynamics (SED) predicts aGaussian probability distribution for a classical harmonic oscillator in the vacuumfieldThis probability distribution is identical to that of the ground state quantum harmonic oscillator Thus the Heisenberg minimumuncertainty relation is recovered in SED To understand the dynamics that give rise to the uncertainty relation and the Gaussianprobability distribution we perform a numerical simulation and follow the motion of the oscillator The dynamical informationobtained through the simulation provides insight to the connection between the classic double-peak probability distribution and theGaussian probability distribution Amain objective for SED research is to establish to what extent the results of quantummechanicscan be obtainedThe present simulationmethod can be applied to other physical systems and it may assist in evaluating the validityrange of SED

1 Introduction

According to quantum electrodynamics the vacuum is not atranquil place A background electromagnetic field called theelectromagnetic vacuum field is always present independentof any external electromagnetic source [1] The first experi-mental evidence of the vacuum field dates back to 1947 whenLamb and his student Retherford found an unexpected shiftin the hydrogen fine structure spectrum [2 3] The physicalexistence of the vacuum field has inspired an interestingmodification to the classical mechanics known as stochas-tic electrodynamics (SED) [4] As a variation of classicalelectrodynamics SED adds a background electromagneticvacuum field to the classical mechanics The vacuum fieldas formulated in SED has no adjustable parameters exceptthat each field mode has a random initial phase and the fieldstrength is set by the Planck constant ℎ With the aid ofthis background field SED is able to reproduce a numberof results that were originally thought to be pure quantumeffects [1 4ndash8]

Despite that the classical mechanics and SED are boththeories that give trajectories of particles the probabilitydistributions of the harmonic oscillator in both theories arevery different In a study of the harmonic oscillator Boyer

showed that themoments ⟨119909119899⟩ of an SED harmonic oscillatorare identical to those of the ground state quantum harmonicoscillator [9] As a consequence the Heisenberg minimumuncertainty relation is satisfied and the probability distribu-tions of an SED harmonic oscillator are a Gaussian identicalto that of the ground state quantum harmonic oscillatorWhile in classical mechanics it is most likely to find anoscillator at the two turning points of the trajectory hencethe double-peak probability distribution the SED Gaussianprobability distribution has the maximum in the center (seeFigure 1) How do the dynamics differ in the two classicaltheories so that the probability distributions become sodifferent

Although the analytical solution of an SED harmonicoscillator was given a long time ago it is not straightforwardto see the dynamical properties of a single particle fromthe complicated solution Additionally many results in SEDsuch as probability distribution are obtained from (ensemble)phase averaging Thus they cannot be used for the interpre-tation of a single particlersquos dynamical behavior in time (somemay consider applying the central limit theorem and treatthe positions in the time sequence as independent randomvariables However for an SED system this cannot workbecause the correlation between the motion at two points

2 Journal of Computational Methods in Physics

Vacuum field SED

Radiationdamping

X

X

2

1

0

minus1

minus2649647645

Time (s)

times10minus8

times10minus13

Posit

ion

(m)

2

1

0

minus1

minus2649647645

Time (s)

times10minus8

times10minus13

Posit

ion

(m)

2

1

0

minus1

minus2

times10minus8

Posit

ion

(m)

CMCM

CM

Probability density

SED (theory)

0 5 10

2

1

0

minus1

minus2

times10minus8

Posit

ion

(m)

Probability density0 5 10

times107

times107

Figure 1 A comparison between the harmonic oscillators with and without the vacuum field Top without any external force except for thevacuum field the SED harmonic oscillator undergoes a motion that results in a Gaussian probability distributionThis motion is investigatedwith our simulation Bottom in the absence of the vacuum field or any external drive a harmonic oscillator that is initially displaced fromequilibrium performs a simple harmonic oscillation with constant oscillation amplitude The resulting probability distribution has peaks atthe two turning points

in time persists beyond many cycles of oscillation) unlessthe system is proved to be ergodic As an analytical proof ofthe ergodicity is very difficult we take a numerical approachto study the particlersquos dynamical behavior Besides what isalready known from the analytical solution our numericalstudies construct the probability distribution from a singleparticlersquos trajectory We investigate the relation betweensuch a probability distribution and the particlersquos dynamicalbehavior Ultimately we want to know the underlying mech-anism that turns the classic double-peak distribution into theGaussian distribution

While most works in the field of SED are analyticalnumerical studies are rare [10ndash12] The advantage of numer-ical simulation is that it may be extended to other physicalsystems with relative ease and is flexible in testing differentassumptions and approximations For example in most SEDanalyses the effect of the Lorentz force due to the vacuumfield is often neglected as the first-order approximation [5]However when the field gradient is nonzero the Lorentzforce from the magnetic field can work with the electricfield to give a nonzero drift to a charged particle known asthe ponderomotive force Therefore the Lorentz force of thevacuum field may play a significant role in SED In fact it isshown in the literature that the magnetic part of the vacuumfield is responsible for the self-ionization of the atoms andthe acquisition of an energy 1020 eV in a few nanoseconds bypart of a free electron in vacuum [13 14] Using numericalsimulation one can easily examine SED beyond the first-order approximation (in this work we limit ourselves to

physical parameters where we expect the Lorentz force effectsto be minimal as this work is compared to analytical resultswhere this approximation is made) and the Lorentz forceeffects in SED can be investigated

The major challenge for the numerical simulation is toproperly account for the vacuum field modes A represen-tative sampling of the modes is thus the key for successfulsimulations In this study one of our goals is to use a simplephysical system namely the simple harmonic oscillator tobenchmark our numericalmethod of vacuummode selectionso that it can be used to test the validity range of SED asdiscussed in the following

Over the decades SED has been criticized for severaldrawbacks [15] Authors like Cavalleri argued that SEDcan neither explain electron slit-diffraction nor derive thenonlinear Schrodinger equation moreover SED impliesbroad radiation and absorption spectra for rarefied gases Inthe case of the quartic anharmonic oscillator Pesquera andClaverie showed that SEDdisagrees with quantummechanics[16] Additionally the results claimed in [17ndash19] were shownto be wrong due to improper relativistic approximation [20]While these theoretical analyses are documented in detail itmay be useful to use numerical simulation as an independentcheck in order to establish the validity range of SED which isone of the objectives of our work

Meanwhile a modified theory called stochastic electro-dynamics with spin (SEDS) was recently proposed [15] AsSEDS are an extension of SED with a model of electron spinmotion it is argued that the introduction of electron spin

Journal of Computational Methods in Physics 3

can eliminate several drawbacks of SED Among all it isclaimed that SEDS allows the derivation of the complete andeven the generalized Schrodinger equation [21ndash24] Also itis claimed that SEDS can explain electron slit-diffraction andthe sharp spectral lines of the rarefied gases [25] In view ofthe fact that SEDS is a modified theory from SED and mayextend its validity range it is probably important as a next stepto apply numerical simulation to such a model as a ldquoquasi-experimentrdquo and test some of its claims Models that includeconstraints can be incorporated in our numericalmodel withfor example Lagrange multipliers

The organization of this paper is the following Firstin Section 2 Boyerrsquos results on the SED harmonic oscillatorare briefly reviewed Based on these results the probabilitydistribution for the oscillator is derived Second in Section 3details for simulating the vacuumfield and the SEDharmonicoscillator are documented Third in Section 4 the trajectoryof the SED harmonic oscillator is solved numerically andthe constructed probability distribution is compared to theanalytical probability distribution Two sampling methodsare used in constructing the probability distribution from thesimulated trajectories The first method is ldquosequential sam-plingrdquo which is suitable for studying the relation between thedynamics of the SED harmonic oscillator and its probabilitydistribution The second approach is ldquoensemble samplingrdquowhich lends itself well to parallel computing and is convenientfor statistical interpretations Lastly in Section 6 we discusssome potential applications of the numerical simulation instudying other quantum phenomena Numerical studies havean advantage over the analytical solutions in that they canbe adopted to a range of physical systems and we hope thatthe current method of simulation may also assist in assessingSEDrsquos validity range

2 Theory of Stochastic Electrodynamics

21 Brief Review of Boyerrsquos Work In his 1975 papers [4 9]Boyer calculated the statistical features of an SED harmonicoscillator and the Heisenberg minimum uncertainty relationis shown to be satisfied for such an oscillator The vacuumfield used in Boyerrsquos work arises from the homogeneoussolution of Maxwellrsquos equations which is assumed to bezero in classical electrodynamics [4] In an unbounded (free)space the vacuum field has an integral form (a detailedaccount of the vacuum field in unbounded space is given inAppendix A1) [26]

Evac (r 119905) =2

sum

120582=1

int1198893

119896120576 (k 120582)120578 (k 120582)2

times (119886 (k 120582) 119890119894(ksdotrminus120596119905)

+119886lowast

(k 120582) 119890minus119894(ksdotrminus120596119905))

(1)

120578 (k 120582) equiv radic ℎ120596

812058731205980

119886 (k 120582) equiv 119890119894120579(k120582)

(2)

where 120596 = 119888|k| and 120579(k 120582) is the random phase uniformlydistributed in [0 2120587] The integral is to be taken over all k-space The two unit vectors 120576(k 1) and 120576(k 2) describe apolarization basis in a plane that is perpendicular to the wavevector k

120576 (k 120582) sdot k = 0 (3)

Furthermore the polarization basis vectors are chosen to bemutually orthogonal

120576 (k 1) sdot 120576 (k 2) = 0 (4)

To investigate the dynamics of the SED harmonic oscillatorBoyer used the dipole approximation

k sdot r ≪ 1 (5)

to remove the spatial dependence in the vacuum field (1)Therefore the equation ofmotion for an SEDharmonic oscil-lator used in Boyerrsquos analysis is

119898 = minus1198981205962

0119909 + 119898Γ + 119902119864vac119909 (119905) (6)

where Γ equiv (21199022

31198981198883

)(141205871205980) is the radiation damping

parameter119898 is the mass 119902 is the charge and1205960is the natural

frequency The 119909-component of the vacuum field in (6) is

119864vac119909 (119905) =2

sum

120582=1

int1198893

119896120576119909(k 120582)

120578 (k 120582)2

times (119886 (k 120582) 119890minus119894120596119905 + 119886lowast (k 120582) 119890119894120596119905)

(7)

and the steady-state solution is obtained as

119909 (119905) =119902

119898

2

sum

120582=1

int1198893

119896120576119909(k 120582)

120578 (k 120582)2

times (119886 (k 120582)119862 (k 120582)

119890minus119894120596119905

+119886lowast

(k 120582)119862lowast (k 120582)

119890119894120596119905

)

(8)

where 119862(k 120582) equiv (minus1205962

+ 1205962

0) minus 119894Γ120596

3 Additionally using thecondition of sharp resonance

Γ1205960≪ 1 (9)

Boyer further calculated the standard deviation of positionandmomentum from (8) by averaging over the randomphase120579 [9]

120590119909= radic⟨1199092⟩

120579minus ⟨119909⟩2

120579

= radicℎ

21198981205960

(10)

120590119901= radic⟨1199012⟩

120579minus ⟨119901⟩

2

120579= radic

ℎ1198981205960

2 (11)

where the phase averaging ⟨ ⟩120579represents the ensemble aver-

age over many realizations In each realization the random


phase 120579 of the vacuum field is different The above resultsatisfies the Heisenberg minimum uncertainty relation

120590119909120590119901=ℎ

2 (12)

From an energy argument Boyer showed that this uncer-tainty relation can also be derived from a delicate balancebetween the energy gain from the vacuum field and theenergy loss through radiation damping [4]

22 Probability Distribution Given the knowledge of themoments ⟨119909119899⟩

120579 the Fourier coefficients119865

120579(119896) of the probabil-

ity distribution119875120579(119909) can be determined by Taylor expanding

119890minus119894119896119909 in powers of 119909119899

119865120579(119896) = int

+infin

minusinfin

119890minus119894119896119909

119875120579(119909) 119889119909

=

infin

sum

119899=0

(minus119894119896)119899

119899int

+infin

minusinfin

119909119899

119875120579(119909) 119889119909

=

infin

sum

119899=0

(minus119894119896)119899

119899⟨119909119899

⟩120579

(13)

Using (8) and the relation from Boyerrsquos paper [9]

⟨119890plusmn119894(120579(k120582)+120579(k1015840 1205821015840))

⟩120579

= 0

⟨119890plusmn119894(120579(k120582)minus120579(k1015840 1205821015840))

⟩120579

= 12057512058210158401205821205753

(k1015840 minus k) (14)

the moments ⟨119909119899⟩120579can be evaluated

⟨1199092119898+1

⟩120579

= 0

⟨1199092119898

⟩120579

=(2119898)

1198982119898(

ℎ

21198981205960

)

119898

(15)

where119898 is a natural number Consequently only even-powerterms are contributing in (13) and the Fourier coefficients119865120579(119896) can be determined

119865120579(119896) =

infin

sum

119898=0

(minus119894119896)2119898

(2119898)⟨1199092119898

⟩

=

infin

sum

119898=0

1

119898(minusℎ1198962

41198981205960

)

119898

= exp(minus ℎ

41198981205960

1198962

)

(16)

Therefore although not explicitly given it is implied byBoyerrsquos work [9] that the probability distribution of the SEDharmonic oscillator is

119875120579(119909) =

1

2120587int

+infin

minusinfin

119890119894119896119909

119865120579(119896) 119889119896

= radic1198981205960

120587ℎexp(minus

1198981205960

ℎ1199092

)

(17)

which is identical to the probability distribution of the quan-tumharmonic oscillator in the ground state (this result is con-sistent with the phase space probability distribution given inMarshallrsquos work [27])

3 Methods of Numerical Simulation

31 Vacuum Field in Bounded Space While the vacuum fieldin unbounded space is not subject to any boundary conditionand thus every wave vector k is allowed [4] the fieldconfined in a space of volume 119881 with zero value boundarycondition has a discrete spectrum and a summation overinfinitely many countable wave vectors k is required [1] Ina simulation it is convenient to write the vacuum field in thesummation form

Evac = sumk120582radicℎ120596

1205980119881cos (k sdot r minus 120596119905 + 120579k120582) 120576k120582 (18)

where 119886k120582 equiv 119890119894120579k120582 120596 = 119888|k| 120579k120582 is the random phase

uniformly distributed in [0 2120587] and 119881 is the volume of thebounded space A derivation of the summation form of thevacuum field in bounded space is given in Appendix A2

Since the range of the allowed wave vectors k is over thewhole k-space we choose to sample only the wave vectors kwhose frequencies are within the finite range [120596

0minus Δ2 120596

0+

Δ2] Such sampling is valid as long as the chosen frequencyrange Δ completely covers the characteristic resonance widthΓ1205962

0of the harmonic oscillator

Γ1205962

0≪ Δ (19)

On the other hand the distribution of the allowed wavevectors k depends on the specific shape of the bounded spaceIn a cubic space of volume 119881 the allowed wave vectorsk are uniformly distributed at cubic grid points and thecorresponding vacuum field is

Evac =2

sum

120582=1

sum

(119896119909 119896119910119896119911)

radicℎ120596


The sampling density is uniform and has a simple relationwith the space volume 119881

120588k =119881

(2120587)3 (21)

Nevertheless such uniform cubic sampling is not convenientfor describing a frequency spectrum and it requires a largenumber of sampled modes to reach numerical convergenceIn order to sample only the wave vectors k in the resonanceregion spherical coordinates are used For the sampling to beuniform each sampled wave vector k must occupy the samesize of finite discrete volume element Δ3119896 = 1198962 sin 120579Δ119896Δ120579Δ120601To sample for modes in the resonance region with eachmodeoccupying the same volume size we use a set of specifically


chosen numbers (120581119894119895119899 120599119894119895119899 120593119894119895119899) to sample the wave vectors k

For 119894 = 1 119873120581 119895 = 1 119873

120599 and 119899 = 1 119873

120593

120581119894119895119899=(1205960minus Δ2)

3

31198883+ (119894 minus 1) Δ120581

120599119894119895119899= minus1 + (119895 minus 1) Δ120599

120593119894119895119899= 119877(0)

119894119895+ (119899 minus 1) Δ120593

(22)

where 119877(0)119894119895

is a random number uniformly distributed in[0 2120587] and the stepsizes are constant

Δ120581 =[(1205960+ Δ2)

3

31198883

minus (1205960minus Δ2)

3

31198883

]

119873120581minus 1

(23)

Δ120599 =2

119873120599minus 1

Δ120593 =2120587

119873120593

(24)

The random number 119877(0)119894119895

is used here to make the samplingmore efficient A set of numbers (120581

119894119895119899 120599119894119895119899 120593119894119895119899) is then used

for assigning the spherical coordinates to each sampled wavevector k

k119894119895119899= (

119896119909

119896119910

119896119911

) = (

119896119894119895119899

sin (120579119894119895119899) cos (120601

119894119895119899)

119896119894119895119899

sin (120579119894119895119899) sin (120601

119894119895119899)

119896119894119895119899

cos (120579119894119895119899)

) (25)

where

119896119894119895119899= (3120581

119894119895119899)13

120579119894119895119899= cosminus1 (120599

119894119895119899)

120601119894119895119899= 120593119894119895119899

(26)

Therefore each sampled wave vector k119894119895119899

is in the resonanceregion and occupies the same size of finite discrete volumeelement

Δ3

119896 = 1198962 sin 120579Δ119896Δ120579Δ120601 = Δ120581Δ120599Δ120593 = constant (27)

where the differential equalities 119889120599 = sin 120579 119889120579 119889120581 = 1198962

119889119896and 119889120593 = 119889120601 are used The differential limit is reached when119873120581 119873120599 and 119873

120593are sufficiently large The numerical result

converges to the analytical solution in this limit Under theuniform spherical sampling method (described by (22) (25)and (26)) the expression for the vacuum field (18) becomes

Evac =2

sum

120582=1

sum

(120581120599120593)

radicℎ120596


where k = k119894119895119899 It is worth noting that when the total

number of wave vectors119873k becomes very large both uniformspherical and cubic sampling converge to each other because

they both effectively sample all the wave vectors k in k-space In the limit of large sampling number 119873k rarr infinthe two sampling methods are equivalent (the relation 120588k =1Δ3

119896 = 119873k119881k = 119881(2120587)3 implies that the limit of large

sampling number (ie 119873k rarr infin) is equivalent to the limitof unbounded space (ie119881 rarr infin) At this limit the volumeelement becomes differential (denoted as 1198893119896) and is freefrom any specific shape associated with the space boundaryTherefore all sampling methods for the allowed wave vectorsk become equivalent and the summation approaches theintegralThis is consistent with the fact that no volume factor119881 is involved in the vacuum field integral as shown in (1))and (21) can be used for both sampling methods to calculatethe volume factor 119881 in (20) and (28)

119881 = (2120587)3

120588k = (2120587)3119873k119881k (29)

where

119881k =4120587

3(1205960+ Δ2

119888)

3

minus4120587

3(1205960minus Δ2

119888)

3

(30)

In the simulation the summation indices in (28) can berewritten as

Evac =2

sum

120582=1

119873120581

sum

119894=1

119873120599

sum

119895=1

119873120593

sum

119899=1

radicℎ120596


where the multiple sums indicate a numerical nested loopand the wave vector k = k

119894119895119899is chosen according to the

uniform spherical sampling method To achieve numericalconvergence 119873

120599and 119873

120593need to be sufficiently large so

that the wave vector k at a fixed frequency may be sampledisotropically In addition a large119873

120581is required for represen-

tative samplings in frequency As a result 119873k = 119873120581119873120599119873120593

needs to be very largewhen using uniform spherical samplingfor numerical simulation To improve the efficiency of thecomputer simulation we sample k at one random angle(120579119894 120601119894) for each frequency Namely for 119894 = 1 119873

120596

k119894= (

119896119909

119896119910

119896119911

) = (

119896119894sin 120579119894cos120601119894

119896119894sin 120579119894sin120601119894

119896119894cos 120579119894

) (32)

where

119896119894= (3120581119894)13

120579119894= cosminus1 (120599

119894)

120601119894= 120593119894

(33)

120581119894=(1205960minus Δ2)

3

31198883+ (119894 minus 1) Δ120581

120599119894= 119877(1)

119894

120593119894= 119877(2)

119894

(34)


The stepsize Δ120581 is specified in (23) 120599 = 119877(1)

119894is a random

number uniformly distributed in [minus1 1] and 120593 = 119877(2)

119894is

another random number uniformly distributed in [0 2120587] Asthe number of sampled frequencies 119873

120596becomes sufficiently

large the random angles (120579119894 120601119894) will approach the angles

specified in uniform spherical sampling (22)In the limit of Δ120596

0≪ 1 the above sampling method

(described by (32) (33) and (34)) and the uniform sphericalsampling method both approach a uniform sampling on aspherical surface at the radius 119903

119896= 1205960119888 In this limit the

addition of the condition in (19) leads to the possible choicesfor the frequency range Δ

Γ1205960≪

Δ

1205960

≪ 1 (35)

Within this range (35) the expression for the vacuum field in(18) becomes

Evac =2

sum

120582=1

119873120596

sum

119894=1

radicℎ120596


where k = k119894 For large 119873k = 119873

120596 the volume factor 119881 is

calculated using (29)Finally for a complete specification of the vacuum field

(36) the polarizations 120576k120582 need to be chosen From (29)we notice that large 119873k gives large 119881 Since for large 119881the vacuum field is not affected by the space boundarythere should be no preferential polarization direction andthe polarizations should be isotropically distributed Theconstruction for isotropically distributed polarizations isdiscussed in detail in Appendix B Here we give the resultthat satisfies the property of isotropy and the properties ofpolarization (described by (3) and (4))

120576k1198941= (

1205761119909

1205761119910

1205761119911

)

= (

cos 120579119894cos120601119894cos120594119894minus sin120601

119894sin120594119894

cos 120579119894sin120601119894cos120594119894+ cos120601

119894sin120594119894

minus sin 120579119894cos120594119894

)

120576k1198942= (

1205762119909

1205762119910

1205762119911

)

= (

minus cos 120579119894cos120601119894sin120594119894minus sin120601

119894cos120594119894

minus cos 120579119894sin120601119894sin120594119894+ cos120601

119894cos120594119894

sin 120579119894sin120594119894

)

(37)

where120594 is a randomnumber uniformly distributed in [0 2120587]With the wave vectors k (described by (32) (33) and (34))and the polarizations 120576k120582 (described by (37)) the endpointsof the sampled vacuum field vector are plotted on a unitsphere as shown in Figure 2 which illustrates the isotropyof the distribution

In summary the vacuum field mode (k 120582) in (36) canbe sampled by a set of four numbers (120581

119894 120599119894 120593119894 120594119894) which are

specified in (32) (33) (34) and (37) The only assumption

1

05

0

minus05

minus1

1050

minus05minus1

1

050

minus05minus1

y

z

x

Figure 2The isotropic distribution of the polarization field vectors120576k1 (number of sampled frequencies number 119873

120596= 3 times 10

3) Theendpoints of the polarization field vectors 120576k1 are plotted for a ran-dom sampling of modes (k 1) according to the methods describedin (32) (33) and (34)

used in determining these numbers is (35) which is equiv-alent to the sharp resonance condition (9) used in Boyerrsquosanalysis

32 Equation of Motion in Numerical Simulation In theunbounded (free) space the equation of motion in Boyerrsquosanalysis is

119898 = minus1198981205962

0119909 + 119898Γ + 119902119864vac119909 (119905) (38)

where the dipole approximation k sdot r ≪ 1 (5) is used In thebounded space the equation ofmotion remains the same butthe vacuum field formulated has the summation form (36)


1205980119881

1

2(119886k120582119890

minus119894120596119905

+ 119886lowast

k120582119890119894120596119905

) 120576k120582 (39)

where 119886k120582 equiv 119890119894120579k120582 The steady-state solution to (38) in the

bounded space can be found following Boyerrsquos approach

119909 (119905) =119902

119898sum

k120582radicℎ120596

1205980119881

1

2(119886k120582119862k120582

119890minus119894120596119905

+119886lowast

k120582119862lowastk120582

119890119894120596119905

) 120576k120582119909 (40)

where119862k120582 equiv (minus1205962

+1205962

0)minus119894Γ120596

3 While this analytical solutioncan be evaluated using ourmethod of vacuummode selection((32) (33) (34) and (37)) our goal with the numericalsimulation is to first reproduce Boyerrsquos analytical results (8)and then later extend the methods to other physical systemsOne major obstacle for the numerical approach is the third-order derivative in the radiation damping term119898Γ To cir-cumvent this problemwe follow the perturbative approach in


[28ndash30] According to classical electrodynamics the equationof motion for an electron with radiation damping is

119898 = 119865 + 119898Γ (41)

where 119865 is the force and Γ equiv (21198902

31198981198883

)(141205871205980) is the

radiation damping coefficient Under the assumption119898Γ ≪119865 the zero-order equation of motion is

119898 ≃ 119865 (42)

The justification for the assumption 119898Γ ≪ 119865 is that apoint particle description of the electron is used in classicalelectrodynamics [28 30] Using (42) the radiation dampingterm may be estimated by

119898Γ ≃ Γ (43)

which can be iterated back to the original equation (41) andget a perturbative expansion

119898 ≃ 119865 + Γ (44)

Thus in this approximated equation of motion we replace thethird derivative of position119909 by the first derivative of the force119865 Applying (44) to (38) the equation of motion becomes

119898 ≃ minus1198981205962

0119909 minus 119898Γ120596

2

0 + 119902119864vac119909 (119905) + 119902Γvac119909 (119905) (45)

The order of magnitude for each term on the right-hand sideis

119874(1198981205962

0119909) = 119898120596

2

01199090 (46)

119874 (119902119864vac119909) = 1198901198640 (47)

119874(119898Γ1205962

0) = (Γ120596

0)1198981205962

01199090 (48)

119874(119902Γvac119909) = (Γ1205960) 1199021198640 (49)

where 1199090and 119864

0are the order of magnitude for the particle

motion119909 and the vacuumfield119864vac119909The order ofmagnitudefor the time scale of particle motion is given by 1120596

0because

of the sharp resonance condition (9) In order to comparethe two radiation damping terms ((48) and (49)) we use arandom walk model to estimate 119909

0and 119864

0 For a fixed time

119905 = 1199050 the order ofmagnitude for119864vac119909(1199050) and119909(1199050) (see (39)

and (40)) is equal to 1198640and 119909

0 Written as complex numbers

the mathematical form of 119864vac119909(1199050) and 119909(1199050) is analogousto a two-dimensional random walk on the complex planewith random variable Θ

k120582 where k 120582 denotes a set ofmodes (k 120582) Averaging over k 120582 the order of magnitudefor 119864vac119909(1199050) and 119909(1199050) can be estimated by the root-mean-squared distance of119864vac119909(1199050) and 119909(1199050) In a two-dimensionalrandom walk model [31] the root-mean-squared distance119863rms is given by

119863rms = radic119873119904 sdot Δ119904 (50)

where 119873119904is the number of steps taken and Δ119904 is a typical

stepsize for 119863(119864)rms Δ119904 = (12)radic(ℎ12059601205980119881) and for 119863(119909)rms

Δ119904 = (12)((119902119898Γ1205963

0)radic(ℎ120596

01205980119881)) Hence the order of

magnitude 1198640and 119909

0 may be estimated as (using 119881 =

(2120587)3

119873120596119881k and 119881k ≃ 4120587120596

2

0(Γ1205962

0)1198883 ((29) and (30)) the

value of 1199090in (51) can be estimated as 119909

0≃ radic3120587radicℎ2119898120596

0

which is consistent with Boyerrsquos calculation for the standarddeviation of position in (10))

1198640≃ 119863(119864)

rms = radic2119873120596 sdot1

2radicℎ1205960

1205980119881

1199090≃ 119863(119909)


2(

119902

119898Γ12059630

radicℎ1205960

1205980119881)

(51)

The order of magnitude for the two radiation damping termsis evaluated accordingly

119874(119898Γ1205962

0) ≃ 119902radic

119873120596

2sdot radic

ℎ1205960

1205980119881

119874 (119902Γvac119909) ≃ (Γ1205960) 119902radic119873120596

2sdot radic

ℎ1205960

1205980119881

(52)

Using the sharp resonance condition Γ1205960≪ 1 (9) we approx-

imate the equation of motion (45) to its leading order

119898 ≃ minus1198981205962

0119909 minus 119898Γ120596

2

0 + 119902119864vac119909 (119905) (53)

As an additional note given the estimation of1198640and1199090in

(51) the three force terms in (53) have the following relation

119874(1198981205962

0119909) ≫ 119874(119898Γ120596

2

0) ≃ 119874 (119902119864vac119909) (54)

Thus the linear restoring force11989812059620119909 is the dominating drive

for an SEDharmonic oscillator while the vacuumfield 119902119864vac119909and radiation damping 119898Γ120596

2

0 act as perturbations The

balance between the vacuum field and the radiation dampingconstrains the oscillation amplitude to fluctuate in the vicinityof 1199090Finally as we have established an approximated equation

of motion (53) for numerical simulation the total integrationtime 120591int (ie how long the simulation is set to run) needs tobe specified Upon inspection two important time scales areidentified from the analytical solution of (53)

119909 (119905) = 119890minusΓ1205962

01199052

(119860119890119894120596119877119905

+ 119860lowast

119890minus119894120596119877119905

)

+119902

119898sum

k120582radicℎ120596

1205980119881

1

2(119886k120582119861k120582

119890minus119894120596119905

+119886lowast

k120582119861lowastk120582

119890119894120596119905

) 120576k120582119909(55)

where 119860 is a coefficient determined by the initial conditionsand

120596119877equiv 1205960

radic1 minus (Γ1205960

2)

2

119861k120582 equiv (minus1205962

+ 1205962

0) minus 119894 (Γ120596

2

0) 120596

119886k120582 equiv 119890119894120579k120582

(56)


The first term in (55) represents the transient motion andthe second term represents the steady-state motion Thecharacteristic time for the transient motion is

120591tran =2

Γ12059620

(57)

Thus the simulation should run beyond 120591tran if one is inter-ested in the steady-state motion As the steady-state solutionis a finite discrete sum of periodic functions it would havea nonphysical repetition time 120591rep The choice of 120591int shouldsatisfy 120591int le 120591rep to avoid repetitive solutions A detaileddiscussion about 120591rep can be found in Appendix C Here wegive a choice of 120591int

120591int =2120587

Δ120596 (58)

where Δ120596 is the frequency gap and it can be estimated using(23) (33) and (34)

Δ120596 ≃119888(31205810)13

3

Δ120581

1205810

(59)

where 1205810equiv (13)(120596

0119888)3

To summarize (53) is the approximated equation ofmotion to be used in numerical simulationThe vacuum field119864vac119909 in (53) is given by (39)The specifications of the vacuumfield modes (k 120582) polarizations 120576k120582 and other relevant vari-ables can be found in Section 31 To approximate (38) by (53)two conditions need to be used namely the dipole approxi-mation (5) and the sharp resonance condition (9)Theparam-eters 119902 119898 and 120596

0simulation should be chosen to satisfy

these two conditions as these two conditions are also used inBoyerrsquos analysis Lastly the integration time 120591int for the sim-ulation is chosen to be within the range 120591trans ≪ 120591int le 120591repwhere 120591tran and 120591rep are given in (57) and (58) respectively

4 Simulation Results

In Section 2 it was shown that the probability distributionfor an SED harmonic oscillator is a Gaussian In Section 3 wedevelop themethods for a numerical simulation to investigatethe dynamics of the SED harmonic oscillator and how it givesrise to the Gaussian probability distribution In this sectionthe results of the simulation are presented and the relationbetween the trajectory and the probability distribution isdiscussed

To construct the probability distribution from a particlersquostrajectory two sampling methods are used The first methodis sequential sampling and the second method is ensemblesampling In sequential sampling the position or velocity isrecorded in a time sequence from a single particlersquos trajectorywhile in ensemble sampling the same is recorded only atthe end of the simulation from an ensemble of particletrajectoriesThe recorded positions or velocities are collectedin histogram and then converted to a probability distributionfor comparison to the analytical result (17) Whereas thesequential sampling illustrates the relationship between the

2

10

minus1

minus1

minus215 16 17 18 19

0040020minus002minus004 Va

cuum

fiel

d (V

m)

002

0

minus002

Vacu

um fi

eld

(Vm

)

times10minus8

times10minus8times10minus12

times10minus12

ParticleField

1

0

17 171 172 173Time (s)

Time (s)

Posit

ion

(m)

Posit

ion

(m) 120591(X)

mod

)(E120591mod

Figure 3 A comparison between particle trajectory and the tem-poral evolution of the vacuum field Top the vacuum field (red) iscompared to the trajectory of the SED harmonic oscillator (black)Bottom a magnified section of the trajectory shows that there is nofixed phase or amplitude relation between the particle trajectory andthe instantaneous driving field The modulation time for the field isalso shown to be longer than that for the motion of the harmonicoscillator

buildup of probability distribution and the dynamics of par-ticle trajectory the ensemble sampling is convenient for sta-tistical interpretation In addition the ensemble sampling issuitable for parallel computing which can be used to improvethe computation efficiency

41 Particle Trajectory and the Probability Distribution Bysolving (53) numerically the steady-state trajectory for theSED harmonic oscillator is obtained and shown in Figure 3For a comparison the temporal evolution of the vacuum field(39) is also includedThe ordinary differential equation (53) issolved using the adaptive 5th order Cash-Karp Runge-Kuttamethod [32] and the integration stepsize is set as small asone-twentieth of the natural period (120)(2120587120596

0) to avoid

numerical aliasing The charge 119902 mass 119898 natural frequency1205960 and vacuum field frequency range Δ are chosen to be

119902 = 119890

119898 = 10minus4

119898119890

1205960= 1016 rads

Δ = 220 times Γ1205962

0

(60)

where 119890 is the electron charge 119898119890is the electron mass and

Γ1205962

0is the resonance width of the harmonic oscillator The

choice of 119898 = 10minus4

119898119890is made to bring the modulation time

and the natural period of the harmonic oscillator closer toeach other In other words the equation ofmotion (53) coverstime scales at two extremes and the choice of mass 119898 =

10minus4

119898119890brings these two scales closer so that the integration


time ismanageable without losing the physical characteristicsof the problem

Here we would like to highlight some interesting featuresof the simulated trajectory (Figure 3) First there appears tobe no fixed phase or amplitude relation between the particletrajectory and the instantaneous driving field Second therate of amplitude modulation in the particle trajectory isslower than that in the driving field To gain insights intothese dynamical behaviors we study the steady-state solutionof (55) in the Green function form [33]

119909 (119905)

=119902

119898120596119877

int

119905

minusinfin

119864vac119909 (1199051015840

) 119890minusΓ1205962

0(119905minus1199051015840)2 sin (120596

119877(119905 minus 1199051015840

)) 1198891199051015840

(61)

where 120596119877equiv 1205960radic1 minus (Γ120596

02)2 The solution indicates that the

effect of the driving field119864vac119909(1199051015840

) at any given time 1199051015840 lasts fora time period of 1Γ1205962

0beyond 1199051015840 In other words the particle

motion 119909(119905) at time 119905 is affected by the vacuum field 119864vac119909(1199051015840

)

from all the previous moments (1199051015840 le 119905) As the vacuum fieldfluctuates in time the fields at two points in time only becomeuncorrelated when the time separation is much longer thanone coherence time (the coherence time of the vacuumfield is120591coh = 2120587|120596 minus 1205960|max see (64))This property of the vacuumfield reflects on the particle trajectory and it explains why theparticle trajectory has no fixed phase or amplitude relationwith the instantaneous driving field Another implication of(61) is that it takes a characteristic time 1Γ1205962

0for the particle

to dissipate the energy gained from the instantaneous drivingfield Thus even if the field already changes its amplitude itwould still take a while for the particle to followThis explainswhy the amplitude modulation in the particle trajectory isslower compared to that in the driving field (however incase of slow field modulation when the field bandwidth isshorter than the resonance width of the harmonic oscillatorthemodulation time of the field and the particle trajectory arethe same)

The sequential sampling of a simulated trajectory givesthe probability distributions in Figure 4 While Boyerrsquos resultis obtained through ensemble (phase) averaging the Gaus-sian probability distribution shown here is constructed froma single trajectory and is identical to the probability distribu-tion of a ground state quantum harmonic oscillator

To understand how the trajectory gives rise to a Gaussianprobability distribution we investigate the particle dynamicsat two time scales At short time scale the particle oscillatesin a harmonic motion The oscillation amplitude is constantand the period is 119879 = 2120587120596

0 Such an oscillation makes

a classical double-peak probability distribution At largetime scale the oscillation amplitude modulates As a resultdifferent parts of the trajectory have double-peak probabilitydistributions associated with different oscillation amplitudeswhich add to make the final probability distribution a Gaus-sian distribution as shown in Figure 5 To verify this idea weattempt to reconstruct the Gaussian probability distribution

6

4

2

0minus4 minus2 0 2 4

Position (m)

Prob

abili

ty d

ensit

y

times107

6

4

2


Momentum (kgmiddotms)

Prob

abili

ty d

ensit

yQMSED

QMSED

times1025

times10minus26

2

1

0

minus1

minus212 18 24

Posit

ion

(m)

Time (s)times10minus12

times10minus26

2

1

0

minus1

minus2178 1781

Posit

ion

(m)


times10minus26

times10minus8

1782

Figure 4 The probability distribution constructed from a singleparticlersquos trajectory (number of sampled frequencies119873

120596= 2 times 10

4)Left the position and momentum probability distributions for theSED harmonic oscillator (black dot) and the ground state quantumharmonic oscillator (red and blue lines) are compared Right anillustration of the sequential sampling shows how positions arerecorded at a regular time sequence (red cross) Note that at smalltime scale the oscillation amplitude is constant but at large timescale it modulates

from the double-peak probability distributions at differentsections of the trajectory We approach this problem bynumerically sampling the oscillation amplitudes at a fixedtime-step To determine the appropriate sampling time-stepwe inspect the steady-state solution (55) in its complex form

119909 (119905) =119902

119898sum

k120582radicℎ120596

1205980119881

119886k120582119861k120582

119890minus119894120596119905

120576k120582119909 (62)

Since the frequency components can be sampled symmetri-cally around120596

0 we factorize 119909(119905) into an amplitude term119860(119905)

and an oscillation term 119890minus1198941205960119905

119909 (119905) = 119860 (119905) 119890minus1198941205960119905

119860 (119905) = sum

k120582(120576k120582119909

119902

119898radicℎ120596

1205980119881

119886k120582119861k120582

)119890minus119894(120596minus120596

0)119905

(63)

The complex components 119890minus119894(120596minus1205960)119905 rotate in the complexplane at different rates 120596 minus 120596

0 At any given time the con-

figuration of these components determines the magnitude of119860(119905) as shown in Figure 6 As time elapses the configurationevolves and the amplitude 119860(119905) changes with time When theelapsed time Δ119905 is much shorter than the shortest rotating


2

1

0

minus1

minus205 1 15 2 25 3

Posit

ion

(m)

times10minus8

times10minus12

Prob

abili

ty d

ensit

y 6

4

2

0minus3 0 3

times107 15

10

5

0minus3 0 3

times1078

6

4

2

0minus3 0 3

times107

Time (s)

Position (10minus8 m) Position (10minus8 m) Position (10minus8 m)

Figure 5 Contributions of different oscillation amplitudes in thefinal probability distribution Top several sections (red) of a steady-state trajectory (black) are shown A section is limited to the dura-tion of the characteristicmodulation timeThe oscillation amplitudechanges significantly beyond the characteristic modulation timeso different sections of the trajectory obtain different oscillationamplitudes Bottom the probability distributions of each section ofthe trajectory are shown As the oscillation amplitude is approx-imately constant in each section the corresponding probabilitydistribution (red bar) is close to the classic double-peak distributionThe probability distributions in different sections of the trajectorycontribute to different areas of the final probability distribution(black dashed line) The final probability distribution is constructedfrom the steady-state trajectory

period 2120587|120596 minus 1205960|max the change in the amplitude 119860(119905) is

negligible

119860 (119905 + Δ119905) ≃ 119860 (119905) for Δ119905 ≪ 120591coh (64)

where 120591coh = 2120587|120596 minus 1205960|max Here we denote this shortest

rotating period as coherence time (the coherence time 120591cohas defined here is equivalent to the temporal width ofthe first-order correlation function (autocorrelation) As theautocorrelation of the simulated trajectory is the Fouriertransform of the spectrum according to Wiener-Khinchintheorem it has a temporal width the same as the coherencetime 120591coh calculated here) 120591coh For our problem at hand it isclear that the sampling of oscillation amplitudes should use atime-step greater than 120591coh

A representative sampling of the oscillation amplitudeswith each sampled amplitude separated by 3120591coh is shownin Figure 7 In this figure the histogram of the sampledoscillation amplitudes shows that the occurrence of large orsmall amplitudes is rare Most of the sampled amplitudeshave a medium value This is because the occurrence ofextreme values requires complete alignment ormisalignmentof the complex components in 119860(119905) For most of the timethe complex components are in partial alignment and thusgive a medium value of119860(119905) Interestingly the averaged value

3

15

0

minus15

minus33150minus15minus3

Real

Imag

inar

y

A(t0)

Δ

A(t0 + Δt)

Figure 6 A schematic illustration of the oscillation amplitude asa sum of different frequency components in the complex planeAt a particular time 119905 = 119905

0 an oscillation amplitude 119860(119905

0) (blue

solid arrow) is formed by a group of frequency components (bluedashed arrow) which rotate in the complex plane at different ratesAfter a time Δ119905 ≪ 120591coh the angles of the frequency components(red dashed arrow) change only a little Therefore the oscillationamplitude119860(119905

0+Δ119905) (red solid arrow) does not changemuch within

the coherence time

of 119860(119905) is close to the oscillation amplitude 1199090as predicted

by the random walk model (51) Using the amplitude dis-tribution given in Figure 7 a probability distribution canbe constructed by adding up the double-peak probabilitydistributions

119875 (119909) = sum

119860

119875119860(119909) = int119875

119860(119909) 119891 (119860) 119889119860 (65)

where 119860 is oscillation amplitude 119891(119860) is the amplitude dis-tribution and 119875

119860(119909) is the corresponding double-peak prob-

ability distribution This constructed probability distributionis a Gaussian and is identical to the simulation result shownin Figure 4 The reconstruction of the Gaussian probabilitydistribution indicates the transitioning from double-peakdistribution to theGaussian distribution due to the amplitudemodulation driven by the vacuum field

42 Phase Averaging and Ensemble Sampling In many SEDanalyses [4ndash7 9 34] the procedure of random phase aver-aging is often used to obtain the statistical properties of thephysical system A proper comparison between numericalsimulation and analysis should thus be based on ensemblesampling In each realization of ensemble sampling the par-ticle is prepared with identical initial conditions but the vac-uum field differs in its initial random phase 120579k120582 The differ-ence in the initial random phase 120579k120582 corresponds to the dif-ferent physical realizations in randomphase averaging At theend of the simulation physical quantities such as position andmomentum are recorded from an ensemble of trajectories

The ensemble sampling of the simulation gives the proba-bility distributions in Figure 8 The position and momentum


5

0

minus51 3 5 7 9

times10minus8

times10minus11

Posit

ion

(m)

Time (s)

2

0

minus2461 462 463 464

times10minus8

times10minus11

Posit

ion

(m)

Time (s)

10

5

0minus4 minus3 minus2 minus1 0 1 2 3 4

times107

times10minus8Position (m)

Prob

abili

ty d

ensit

y

AmplitudeReconstructionSimulation

(rare)Average amplitude

(frequent)Large amplitude

(rare)

= 15 times 107

Small amplitude

3120591coh

Δs

Figure 7 Sampled oscillation amplitude and the reconstructedGaussian probability distribution Top the oscillation amplitudes(red dot) are sampled from a steady-state trajectory (black) witha sampling time-step equal to 3120591coh where 120591coh is the coherencetime Middle a magnified section of the trajectory shows that thesampling time-step (red dot) is 3120591coh Bottom using the amplitudedistribution (red bar) a probability distribution (blue line) is con-structed and shown to agreewith the simulatedGaussian probabilitydistribution (black dot) The simulated probability distribution(black dot) is offset by Δ

119904= 15 times 10

7 for better visualizationThis result confirms that the underlyingmechanism for theGaussianprobability distribution is the addition of a series of double-peakprobability distributions according to the amplitude distributiongiven by the vacuum field It is also worth noting that the mostfrequent oscillation amplitude in the amplitude distribution is at thehalf-maximum of the position distribution (black dot)

distributions satisfy the Heisenberg minimum uncertainty aspredicted by Boyerrsquos analysis In addition Boyer proposed amechanism for the minimum uncertainty using an energy-balance argument Namely he calculated the energy gainfrom the vacuum field and the energy loss through radiationdamping and he found that the delicate balance results in theminimumuncertainty relation [4]We confirm this balancingmechanism by turning off the radiation damping in thesimulation and see that theminimumuncertainty relation nolonger holds (see Figure 9)

Unlike sequential sampling ensemble sampling has theadvantage that the recorded data are fully uncorrelated Asa result the integration time does not need to be very longcompared to the coherence time 120591coh However since onlyone data point is recorded from each trajectory a simulationwith ensemble sampling actually takes longer time than withsequential sampling For example a typical simulation runwith sequential sampling takes 23 hours to finish (for numberof sampled frequencies 119873

120596= 2 times 10

4) but with ensemble

6

4

2


times107

times10minus8

Prob

abili

ty d

ensit

y

QMSED

120590x ≃ 761 times 10minus9

6

4

2


times1025

times10minus26

Prob

abili

ty d

ensit

y


QMSED

120590p ≃ 693 times 10minus27 5

0

minus50 2 4Po

sitio

n (m

)

Position (m)

Time (s)

times10minus8

times10minus12

Trial-15

0

minus50 2 4

Posit

ion

(m)

Time (s)

times10minus8

times10minus12

Trial-25

0

minus50 2 4

Posit

ion

(m)

Time (s)

times10minus8

times10minus12

Trial-3

Figure 8 The probability distribution constructed from an ensem-ble sampling (number of particles119873

119901= 2 times 10

5 number of sampledfrequencies 119873

120596= 2 times 10

3) Left the position and momentumprobability distributions are shown for the SED harmonic oscillator(black dot) and the ground state quantum harmonic oscillator(red and blue line) The ensemble sampling corresponds to theprocedure of random phase averaging ((10) and (11)) so the width ofthe probability distributions should satisfy Heisenbergrsquos minimumuncertainty relation (120590

119909120590119901= ℎ2) as predicted by the analysis Right

an illustration of the ensemble sampling shows how positions (redcross) are recorded from an ensemble of trajectories (black line)

sampling it takes 61 hours (for number of particles 119873119901=

2 times 105 and number of sampled frequencies119873

120596= 5 times 10

2)A remedy to this problem is to use parallel computing for thesimulation The parallelization scheme (the parallelizationof the simulation program is developed and benchmarkedwith assistance from the University of Nebraska HollandComputing Center The program is written in Fortran andparallelized using message passing interface (MPI) [32 35]The compiler used in this work is the GNU CompilerCollection (GCC) gcc-441 and the MPI wrapper used isopenmpi-143) for our simulation with ensemble sam-pling is straightforward since each trajectory is independentexcept for the random initial phases 120579k120582 To reduce theamount of interprocessor communication and computationoverhead each processor is assigned an equal amount ofworkTheparallelized program is benchmarked and shows aninverse relation between the computation time and the num-ber of processors (see Figure 10) As the computation speedupis defined as 119878

119901= 1198791119879119901 where 119879

1is the single processor

computation time and 119879119901is the multiprocessor computation

time the inverse relation shown in Figure 10 indicates idealperformance of linear speedup As an additional note theparallelized code is advantageous for testing the numericalconvergence of the simulation In Figure 11 the convergence


4

2

0

minus2

minus40 02 04 06 08 1 12 14

Time (s)

Posit

ion

(m)

No dampingWith damping

times10minus8

times10minus12

4

2

0

minus2

minus40 02 04 06 08 1 12 14

Time (s)

Posit

ion

(m)


times10minus12

times10minus26

Figure 9 Radiation damping and Heisenbergrsquos minimum uncer-tainty relation Under the balance between vacuum field driving andthe radiation damping the trajectory of the SEDharmonic oscillator(black) satisfies the Heisenberg minimum uncertainty relationWhen the radiation damping is turned off in the simulation (red andblue) the minimum uncertainty relation no longer holds althoughthe range of the particlersquos motion is still bounded by the harmonicpotential

10 30 50 70 90

10

20

30

40

Number of processors

Com

puta

tion

time (

hr)

Np = 100 k = 05 kNp = 100 k = 1 kNp = 200 k = 05 k

N120596

N120596

N120596

Figure 10 The inverse relation between the computation time andthe number of processorsThe data (black open square blue and reddot) follow an inverse relation 119910 = 119862119909 (black blue and red dashedlines)The parameter119862 is determined by the product of119873

119901(number

of particles) and119873120596(number of sampled frequencies)

0 200 400 600 800 10000

14

12

34

1

Np = 200 k

Ener

gy (ℏ

1205960)

N120596

Figure 11 The ensemble-averaged energy of the SED harmonicoscillator and its convergence as a function of sampled frequencynumber119873

120596 The number of particles used in the simulation is119873

119901=

2 times 105 and the sampled frequency range is Δ = 220 times Γ120596

2

0

The energy converges as the number of sampled frequencies 119873120596

increases At119873120596= 5 times 10

2 (blue cross) the deviation between theenergy of the SED harmonic oscillator and the ground state energyof the quantum oscillator is 1

of the ensemble-averaged energy as a function of sampledfrequency number is shownAs highlighted in the figure only119873120596= 5 times 10

2 sampled modes need to be used for thesimulation to agree with the analytical resultThe fact that119873

120596

is low indicates that our method of vacuummode selection isefficient

5 Conclusions

The analytical probability distribution of an SED harmonicoscillator is obtained in Section 2 The details of our numer-ical methods including vacuum mode selection are docu-mented in Section 3 Agreement is found between the simu-lation and the analytical results as both sequential samplingand ensemble sampling give the same probability distributionas the analytical result (see Figures 4 and 8) Numericalconvergence is reached with a low number of sampledvacuum field mode (119873

120596= 5 times 10

2) which is an indicationthat our method of vacuum mode selection ((32) (33) (34)and (37)) is effective in achieving a representative sampling

As the probability distribution constructed from a sin-gle trajectory is a Gaussian and satisfies the Heisenbergminimum uncertainty relation we investigate the relationbetween the Gaussian probability distribution and the par-ticlersquos dynamical properties As a result the amplitude modu-lation of the SED harmonic oscillator at the time scale of 120591cohis found to be the cause for the transitioning from the double-peak probability distribution to the Gaussian probabilitydistribution (see Figures 5 7 and 12)


Vacuum field

SEDRadiationdamping

X

X

2

1

0

minus1

minus2

Time (s)

times10minus8

times10minus12

Posit

ion

(m)

2

1

0

minus1

minus2649647645

Time (s)

times10minus8

times10minus13

Posit

ion

(m)

2

1

0

minus1

minus2

times10minus8

Posit

ion

(m)

CMCM

Probability density

SED (theory)

0 5 10

2

1

0

minus1

minus2

times10minus8

Posit

ion

(m)


SED (simulation)

2 3 4 5 6 7

times107

times107

Figure 12 A comparison between the harmonic oscillators with and without the vacuum field Top compared to Figure 1 it is clear from thisfigure that the SED harmonic oscillator undergoes an oscillatory motion with modulating oscillation amplitude The oscillation amplitudemodulates at the time scale of coherence time 120591coh and is responsible for the resultingGaussian probability distribution Bottom in the absenceof the vacuum field or any external drive a harmonic oscillator that is initially displaced from equilibrium performs a simple harmonicoscillation with constant oscillation amplitude The resulting probability distribution has peaks at the two turning points

6 Discussions Application of Simulation toOther Physical Systems

In quantum mechanics the harmonic oscillator has excitedcoherent and squeezed states A natural extension of ourcurrent work is to search for the SED correspondence of suchstates Currently we are investigating how a Gaussian pulsewith different harmonics of 120596

0will affect the SED harmonic

oscillator Can the SEDharmonic oscillator support a discreteexcitation spectrum and if so how does it compare withthe prediction from quantum mechanics Such a study isinteresting in the broader view of Milonnirsquos comment thatSED is unable to account for the discrete energy levels ofinteracting atoms [1] and also Boyerrsquos comment that at presentthe line spectra of atoms are still unexplained in SED [36]

The methods of our numerical simulation may be appli-cable to study other quantum systems that are related to theharmonic oscillator such as a charged particle in a uniformmagnetic field and the anharmonic oscillator [5 34] For thefirst example classically a particle in a uniformmagnetic fieldperforms cyclotron motion Such a system can be viewed asa two-dimensional oscillator having the natural frequencyset by the Larmor frequency On the other hand a quantummechanical calculation for the same system reveals Landauquantization The quantum orbitals of cyclotron motion arediscrete and degenerate Such a system presents a challengeto SED For the second example a harmonic potential can be

modified to include anharmonic terms of various strengthsHeisenberg considered such a system a critical test in the earlydevelopment of quantummechanics [37 38] We think that astudy of the anharmonic oscillator is thus a natural extensionof our current study and may serve as a test for SED

Lastly over the last decades there has been a sustainedinterest to explain the origin of electron spin and the mecha-nism behind the electron double-slit diffractionwith SED [1539ndash41] Several attempts were made to construct a dynamicalmodel that accounts for electron spin In 1982 de la Penacalculated the phase averaged mechanical angular momen-tum of a three-dimensional harmonic oscillator The resultdeviates from the electron spin magnitude by a factor of 2[39] One year later Sachidanandam derived the intrinsicspin one-half of a free electron in a uniform magnetic field[40] Whereas Sachidanandamrsquos calculation is based on thephase averaged canonical angular momentum his result isconsistent with Boyerrsquos earlier work where Landau diamag-netism is derived via the phase averaged mechanical angularmomentum of an electron in a uniform magnetic field [5]Although these results are intriguing the most importantaspect of spin the spin quantization has not been shownIf passed through a Stern-Gerlach magnet will the electronsin the SED description split into two groups of trajectories(electron Stern-Gerlach effect is an interesting but controver-sial topic in its own right Whereas Bohr and Pauli assertedthat an electron beam cannot be separated by spin based


Figure 13 Schematic illustration of a vacuum field based mecha-nism for electron double-slit diffraction Several authors have pro-posed different SED mechanisms that explain the electron double-slit diffraction The central idea is that the vacuum field in one slitis affected by the presence of the other slit As the vacuum fieldperturbs the electronrsquos motion an electron passing through only oneslit can demonstrate a dynamical behavior that reflects the presenceof both slits Such a mechanism may reconcile the superpositionprinciple with the concept of particle trajectory

on the concept of classical trajectories [42] Batelaan et al[43 44] and Dehmelt argue that one can do so with cer-tain Stern-Gerlach-like devices [45 46]) At this point thedynamics become delicate and rather complex To furtherinvestigate such a model of spin a numerical simulation maybe helpful

On the other hand over the years claims have beenmade that SED can predict double-slit electron diffraction[4 15 41] In order to explain the experimentally observedelectron double-slit diffraction (see [47 48] for a movieof the single electron buildup of a double-slit diffractionpattern) [49ndash51] different mechanisms motivated by SEDwere proposed [15 41] but no concrete calculation has beengiven except for a detailed account of the slit-diffractedvacuumfield [52] In 1999 Kracklauer suggested that particlessteered by the modulating waves of the SED vacuum fieldshould display a diffraction pattern when passing througha slit since the vacuum field itself is diffracted [41] Inrecent years another diffraction mechanism is proposedby Cavalleri et al in relation to a postulated electron spinmotion [15] Despite these efforts Boyer points out in a recentreview article that at present there is still no concrete SEDcalculation on the double-slit diffraction [36] Boyer suggeststhat as the correlation function of the vacuum field nearthe slits is modified by the slit boundary the motion of theelectron near the slits should be influenced as well Can thescattering of the vacuum field be the physical mechanismbehind the electron double-slit diffraction (see Figure 13) AsHeisenbergrsquos uncertainty relation is a central feature in allmatter diffraction phenomena any proposed mechanism forelectron double-slit diffraction must be able to account forHeisenbergrsquos uncertainty relation In the physical system of

the harmonic oscillator SED demonstrates a mechanism thatgives rise to the Heisenberg minimum uncertainty We hopethat the current simulation method may help in providing adetailed investigation on the proposed SED mechanisms forthe electron slit-diffraction

Appendices

A The Vacuum Field in Unbounded andBounded Space

In ldquounboundedrdquo space the modes are continuous and thefield is expressed in terms of an integral In ldquoboundedrdquo spacethe modes are discrete and the field is expressed in termsof a summation In both cases the expression for the fieldamplitude needs to be obtained (see Appendices A1 andA2)The integral expression helps comparison with analyticalcalculations in previous papers [4ndash7 9] while the summationexpression is what we use in our numerical work

A1 Unbounded Space The homogeneous solution ofMaxwellrsquos equations in unbounded space is equivalent to thesolution for a wave equation

E (r 119905)

=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)

times (119860 (k 120582) 119890119894(ksdotrminus120596119905) + 119860lowast (k 120582) 119890minus119894(ksdotrminus120596119905))

B (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896 (k times 120576 (k 120582))

times (119860 (k 120582) 119890119894(ksdotrminus120596119905) + 119860lowast (k 120582) 119890minus119894(ksdotrminus120596119905)) (A1)

where 119860(k 120582) is the undetermined field amplitude for themode (k 120582) and has the unit of electric field (Vm) k isdefined as the unit vector of k and the two vectors 120576(k 1)and 120576(k 2) describe an orthonormal polarization basis in aplane that is perpendicular to the wave vector k

Without loss of generality a random phase 119890119894120579(k120582) canbe factored out from the field amplitude 119860(k 120582) =

119860(k 120582)119890119894120579(k120582)

E (r 119905)

=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)

times (119860 (k 120582) 119890119894(ksdotrminus120596119905)119890119894120579(k120582)

+119860lowast

(k 120582) 119890minus119894(ksdotrminus120596119905)119890minus119894120579(k120582))


B (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896120585 (k 120582)


+119860lowast


(A2)

where 120585(k 120582) equiv k times 120576(k 120582) The field amplitude 119860(k 120582) canbe determined through the phase averaged energy density

⟨119906⟩120579=1205980

2⟨|E|2⟩

120579

+1

21205830

⟨|B|2⟩120579

(A3)

where 1205980is the vacuum permittivity 120583

0is the vacuum

permeability and 120579 is the random phase in (A2) To evaluatethe phase averaged energy density ⟨119906⟩

120579 we first calculate |E|2

and |B|2 using (A2)

|E|2 (r 119905)

= E (r 119905)Elowast (r 119905)

=1

4sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120576 (k 120582) sdot 120576 (k1015840 1205821015840)) 119891

|B|2 (r 119905)

= B (r 119905)Blowast (r 119905)

=1

41198882sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120585 (k 120582) sdot 120585 (k1015840 1205821015840)) 119891

(A4)

where 119891 equiv 119891(k 120582 k1015840 1205821015840 r 119905)

119891 (k 120582 k1015840 1205821015840 r 119905)

= 119860 (k 120582) 119860lowast (k1015840 1205821015840) 119890119894(ksdotrminus120596119905+120579(k120582))

times 119890minus119894(k1015840 sdotrminus1205961015840119905+120579(k1015840 1205821015840))

+ 119860 (k 120582) 119860 (k1015840 1205821015840) 119890119894(ksdotrminus120596119905+120579(k120582))

times 119890119894(k1015840 sdotrminus1205961015840119905+120579(k1015840 1205821015840))

+ 119860lowast

(k 120582) 119860lowast (k1015840 1205821015840) 119890minus119894(ksdotrminus120596119905+120579(k120582))


+ 119860lowast

(k 120582) 119860 (k1015840 1205821015840) 119890minus119894(ksdotrminus120596119905+120579(k120582))


(A5)

The random phase average can be calculated with the follow-ing relation [9]

⟨119890plusmn119894(120579(k120582)+120579(k1015840 1205821015840))

⟩120579

= 0

⟨119890plusmn119894(120579(k120582)minus120579(k1015840 1205821015840))

⟩120579

= 12057512058210158401205821205753

(k1015840 minus k) (A6)

Applying (A6) to (A5) we obtain

⟨119891 (k 120582 k1015840 1205821015840 r 119905)⟩120579

= 119860 (k 120582) 119860lowast (k1015840 1205821015840) 119890119894(ksdotrminus120596119905)

times 119890minus119894(k1015840 sdotrminus1205961015840119905)

12057512058210158401205821205753

(k1015840 minus k)

+ 119860lowast

(k 120582) 119860 (k1015840 1205821015840) 119890minus119894(ksdotrminus120596119905)

times 119890119894(k1015840 sdotrminus1205961015840119905)

12057512058210158401205821205753

(k1015840 minus k)

(A7)

Consequently ⟨|E|2⟩120579and ⟨|B|2⟩

120579can be evaluated using

(A4) and (A7)

⟨|E|2⟩120579

=1

4sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120576 (k 120582) sdot 120576 (k1015840 1205821015840)) ⟨119891⟩120579

=1

2

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2

⟨|B|2⟩120579

=1

41198882sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120585 (k 120582) sdot 120585 (k1015840 1205821015840)) ⟨119891⟩120579

=1

21198882

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2

(A8)

The above calculation leads to a relation between the fieldamplitude119860(k 120582) and the phase averaged energy density ⟨119906⟩

120579

in unbounded space

⟨119906⟩120579=1205980

2

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2 (A9)

Now if we postulate that the vacuum energy is ℎ1205962 for eachmode (k 120582) then in a bounded cubic space of volume 119881 thevacuum energy density is

119906vac =1

119881sum

k120582

ℎ120596

2

=1

119881sum

k120582

ℎ120596

2(119871119909

2120587Δ119896119909)(

119871119910

2120587Δ119896119910)(

119871119911

2120587Δ119896119911)

=

2

sum

120582=1

1

(2120587)3sum

kΔ3

119896ℎ120596

2

(A10)


In the limit of unbounded space (ie 119881 rarr infin) the volumeelement Δ3119896 becomes differential (ie Δ3119896 rarr 119889

3

119896) and thevacuum energy density becomes

119906vac =2

sum

120582=1

1

(2120587)3int1198893

119896ℎ120596

2 (A11)

Comparing this result with (A9) we find

1003816100381610038161003816119860vac (k 120582)10038161003816100381610038162

=ℎ120596

(2120587)3

1205980

(A12)

Assuming 119860vac(k 120582) is a positive real number the vacuumfield amplitude in unbounded space is determined

119860vac (k 120582) = radicℎ120596

812058731205980

(A13)

Therefore the vacuum field in unbounded space is found tobeEvac (r 119905)

=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)radic ℎ120596

812058731205980

times (119890119894(ksdotrminus120596119905)

119890119894120579(k120582)

+ 119890minus119894(ksdotrminus120596119905)

119890minus119894120579(k120582)

)

Bvac (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896120585 (k 120582)radic ℎ120596

812058731205980


119890119894120579(k120582)


119890minus119894120579(k120582)

)

(A14)

which is (1)

A2 Bounded Space The solution of homogeneousMaxwellrsquosequations in bounded space has the summation form

E (r 119905) = 1

2sum

k120582(119860k120582119890

119894(ksdotrminus120596119905)+ 119860lowast

k120582119890minus119894(ksdotrminus120596119905)

) 120576k120582

B (r 119905) = 1

2119888sum

k120582(119860k120582119890



) 120585k120582

(A15)

where 120585k120582 = k times 120576k120582 119860k120582 is the undetermined fieldamplitude for themode (k 120582) and has the unit of electric field(Vm) k is defined as the unit vector of k and the two vec-tors 120576k1 and 120576k2 describe an orthonormal polarization basisin a plane that is perpendicular to the wave vector k

Using the relation (if the twomodes are not identical (iek1015840 = k or 1205821015840 = 120582) then 119890119894120579k120582 and 119890119894120579k1015840 1205821015840 are independent whichleads to the factorization ⟨119890119894(120579k120582plusmn120579k1015840 1205821015840 )⟩

120579= ⟨119890119894120579k120582⟩120579⟨119890plusmn119894120579k1015840 1205821015840 ⟩

120579=

0)

⟨119890plusmn119894(120579k120582+120579k1015840 1205821015840 )⟩

120579

= 0

⟨119890plusmn119894(120579k120582minus120579k1015840 1205821015840 )⟩

120579

= 1205751205821015840120582120575k1015840 k

(A16)

we can follow the same argument in Appendix A1 and obtainthe phase averaged energy density in bounded space

⟨119906⟩120579=1205980

2sum

k120582

1003816100381610038161003816119860k12058210038161003816100381610038162

(A17)

where 119860k120582 = 119860k120582119890119894120579k120582 Again if we postulate that the

vacuum energy is ℎ1205962 for each mode (k 120582) then in abounded space of volume 119881 the vacuum energy density is

119906vac =1

119881sum

k120582

ℎ120596

2 (A18)

Comparing (A17) and (A18) the vacuum field amplitude inbounded space is determined

119860vack120582 = radicℎ120596

1205980119881 (A19)

Therefore the RED vacuum field in bounded space is

Evac (r 119905) =1

2sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120576k120582

Bvac (r 119905) =1

2119888sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120585k120582

(A20)

B Isotropic Polarization Vectors

A wave vector chosen along the 119911-axis

k = (119909

119910

119911

) = (

0

0

119896

) (B1)

has an orthonormal polarization basis in the 119909119910-plane

k1 = (cos120594sin1205940

) k2 = (minus sin120594cos1205940

) (B2)

where the random angle 120594 is uniformly distributed in [0 2120587]To obtain the wave vector k in (32) k can be first rotatedcounterclockwise about the 119910-axis by an angle 120579 and thencounterclockwise about the 119911-axis by an angle 120601 The corre-sponding rotation matrix is described by

= (119911)

120601(119910)

120579

= (

cos120601 minus sin120601 0

sin120601 cos120601 0

0 0 1

)(

cos 120579 0 sin 1205790 1 0

minus sin 120579 0 cos 120579)

= (

cos 120579 cos120601 minus sin120601 cos120601 sin 120579cos 120579 sin120601 cos120601 sin120601 sin 120579minus sin 120579 0 cos 120579

)

(B3)


2

1

0

minus1

minus2

0 3 6 9 12Time (s)

Fiel

d (V

m)

120591rep

120591env

Figure 14 Repetition time of a beat wave A beat wave (black solidline) ismade of two frequency components (red and blue solid lines)The oscillation periods of the two frequency components are 119879

1=

15 and 1198792= 2 thus the repetition time is 120591rep = [119879

1 1198792]LCM = 6

Note that the periodicity of the envelope (black dashed line) is 120591env =12 which is different from the repetition time 120591rep = 6

and k is obtained accordingly

k = k = (119896 sin 120579 cos120601119896 sin 120579 sin120601119896 cos 120579

) (B4)

In the same manner we can rotate k1 and k2 with the rota-tion matrix and obtain an isotropically distributed (afterthe rotation the uniformly distributed circle will span into auniformly distributed spherical surface) polarization basis asdescribed in (37)

120576k1 = k1 = (

cos 120579 cos120601 cos120594 minus sin120601 sin120594cos 120579 sin120601 cos120594 + cos120601 sin120594

minus sin 120579 cos120594)

120576k2 = k2 = (

minus cos 120579 cos120601 sin120594 minus sin120601 cos120594minus cos 120579 sin120601 sin120594 + cos120601 cos120594

sin 120579 sin120594)

(B5)

C Repetitive Time

A field composed of finite discrete frequencies

119864 (119905) =

119873

sum

119896=1

119864119896cos (120596

119896119905) (C1)

repeats itself at the least common multiple (LCM) of all theperiods of its frequency components

119864 (119905 + 120591rep) = 119864 (119905) (C2)

120591rep = [1198791 1198792 119879119873]LCM (C3)

where 119879119896= 2120587120596

119896 An example of a two-frequency beat wave

is given in Figure 14 Given the frequency spectrum of 119864(119905)one can draw a relation between the repetition time 120591rep andthe greatest common divider (GCD) of the frequencies

120591rep =2120587

(1205961 1205962 120596

119873)GCD

(C4)

The derivation of the relation in (C4) is the following Firstwe factorize the sum of all the frequencies into two terms

1205961+ 1205962+ sdot sdot sdot + 120596

119873

=2120587

1198791

+2120587

1198792

+ sdot sdot sdot +2120587

119879119873

=2120587

[1198791 1198792 119879

119873]LCM

(1198991+ 1198992+ sdot sdot sdot + 119899

119873)

(C5)

where 119899119896are positive integers

119899119896=[1198791 1198792 119879

119873]LCM

119879119896

(C6)

Now it is true that (1198991 1198992 119899

119873)GCD = 1 otherwise it would

lead to a contradiction to (C6) Therefore one can concludethat

Δ120596gcd equiv (1205961 1205962 120596119873)GCD

=2120587

[1198791 1198792 119879

119873]LCM

(C7)

From (C3) and (C7) the relation in (C4)

120591rep =2120587

Δ120596gcd(C8)

is drawn Since the simulation should only be carried outthrough an integration time 120591int le 120591rep to avoid repetitivesolutions in our case the choice of the integration time (58)

120591int =2120587

Δ120596 (C9)

whereΔ120596 is the smallest frequency gap (Δ120596 le Δ120596gcd) sufficesour purpose The frequency gap as a function of 120581 can beestimated using (33)

Δ120596 (120581) = 120596 (120581) minus 120596 (120581 minus Δ120581)

= 119888(3120581)13

minus 119888(3120581)13

(1 minusΔ120581

120581)

13

(C10)

Applying the sharp resonance condition (9) to (23) and (34)it can be further shown that Δ120581 is much smaller than 120581

0and

120581 ≃ 1205810

Δ120581

1205810

= (3

119873k minus 1)Δ

1205960

≪ 1

120581 = 1205810+ 119874(

Δ

1205960

)

(C11)


where 1205810= (13)(120596

0119888)3 Therefore the size of Δ120596(120581) is

approximately fixed within the sampled frequency range Δand the smallest frequency gap Δ120596 can be approximated as

Δ120596 ≃119888(31205810)13

3

Δ120581

1205810

(C12)

Acknowledgments

This work was completed utilizing the Holland ComputingCenter of the University of Nebraska In particular theauthors thank Dr Adam Caprez Charles Nugent PragaAngu and Stephen Mkandawire for their work on paralleliz-ing the SED simulation code Huang thanks Dr Bradley AShadwick Roger Bach and Dr Steven R Dunbar for stimu-lating discussions This work is supported by NSF Grant no0969506

References

[1] PWMilonniTheQuantumVacuum An Introduction to Quan-tum Electrodynamics Academic Press Boston 1994

[2] W E Lamb Jr and R C Retherford ldquoFine structure of thehydrogen atom by a microwave methodrdquo Physical Review vol72 no 3 pp 241ndash243 1947

[3] H A Bethe ldquoThe electromagnetic shift of energy levelsrdquo Physi-cal Review vol 72 no 4 pp 339ndash341 1947

[4] T H Boyer ldquoRandom electrodynamics the theory of classicalelectrodynamics with classical electromagnetic zero-point radi-ationrdquo Physical Review D vol 11 no 4 pp 790ndash808 1975

[5] T H Boyer ldquoDiamagnetism of a free particle in classicalelectron theory with classical electromagnetic zero-point radi-ationrdquo Physical Review A vol 21 no 1 pp 66ndash72 1980

[6] T H Boyer ldquoThermal effects of acceleration through randomclassical radiationrdquo Physical Review D vol 21 no 8 pp 2137ndash2148 1980

[7] T H Boyer ldquoRetarded van der waals forces at all distancesderived from classical electrodynamics with classical electro-magnetic zero-point radiationrdquo Physical Review A vol 7 no 6pp 1832ndash1840 1973

[8] H E Puthoff ldquoGround state of hydrogen as a zero-point-fluctuation-determined staterdquo Physical Review D vol 35 no 10pp 3266ndash3269 1987

[9] T H Boyer ldquoGeneral connection between random electrody-namics and quantum electrodynamics for free electromagneticfields and for dipole oscillator systemsrdquo Physical Review D vol11 no 4 pp 809ndash830 1975

[10] D C Cole and Y Zou ldquoQuantum mechanical ground stateof hydrogen obtained from classical electrodynamicsrdquo PhysicsLetters A vol 317 no 1-2 pp 14ndash20 2003

[11] D C Cole and Y Zou ldquoSimulation study of aspects of the classi-cal hydrogen atom interacting with electromagnetic radiationelliptical orbitsrdquo Journal of Scientific Computing vol 20 no 3pp 379ndash404 2004

[12] D C Cole and Y Zou ldquoSimulation study of aspects of the classi-cal hydrogen atom interacting with electromagnetic radiationcircular orbitsrdquo Journal of Scientific Computing vol 20 no 1 pp43ndash68 2004

[13] T W Marshall and P Claverie ldquoStochastic electrodynamics ofnonlinear systems I Partial in a central field of forcerdquo Journalof Mathematical Physics vol 21 no 7 pp 1819ndash1825 1979

[14] A Rueda and G Cavalleri ldquoZitterbewegung in stochastic elec-trodynamics and implications on a zero-point field accelerationmechanismrdquo Il Nuovo Cimento C vol 6 no 3 pp 239ndash2601983

[15] G Cavalleri F Barbero G Bertazzi et al ldquoA quantitativeassessment of stochastic electrodynamics with spin (SEDS)physical principles and novel applicationsrdquo Frontiers of Physicsin China vol 5 no 1 pp 107ndash122 2010

[16] L Pesquera and P Claverie ldquoThe quartic anharmonic oscillatorin stochastic electrodynamicsrdquo Journal of Mathematical Physicsvol 23 no 7 pp 1315ndash1322 1981

[17] B Haisch A Rueda and H E Puthoff ldquoInertia as a zero-point-field Lorentz forcerdquo Physical Review A vol 49 no 2 pp 678ndash694 1994

[18] D C Cole A Rueda and Danley ldquoStochastic nonrelativisticapproach to gravity as originating from vacuum zero-point fieldvan der Waals forcesrdquo Physical Review A vol 63 no 5 ArticleID 054101 2 pages 2001

[19] A Rueda and B Haisch ldquoGravity and the quantum vacuuminertia hypothesisrdquo Annalen der Physik vol 14 no 8 pp 479ndash498 2005

[20] Y S Levin ldquoInertia as a zero-point-field force critical analysisof the Haisch-Rueda-Puthoff inertia theoryrdquo Physical Review Avol 79 no 1 Article ID 012114 14 pages 2009

[21] G Cavalleri ldquoSchrodingerrsquos equation as a consequence ofzitterbewegungrdquo Lettere Al Nuovo Cimento vol 43 no 6 pp285ndash291 1985

[22] G Cavalleri and G Spavieri ldquoSchrodinger equation for manydistinguishable particles as a consequence of a microscopicstochastic steady-statemotionrdquo Il Nuovo Cimento B vol 95 no2 pp 194ndash204 1986

[23] G Cavalleri and G Mauri ldquoIntegral expansion often reducingto the density-gradient expansion extended to non-Markovstochastic processes consequent non-Markovian stochasticequation whose leading terms coincide with SchrodingersrdquoPhysical Review B vol 41 no 10 pp 6751ndash6758 1990

[24] G Cavalleri and A Zecca ldquoInterpretation of a Schrodinger-like equation derived from a non-Markovian processrdquo PhysicalReview B vol 43 no 4 pp 3223ndash3227 1991

[25] A Zecca and G Cavalleri ldquoGaussian wave packets passingthrough a slit a comparison between the predictions of theSchrodinger QM and of stochastic electrodynamics with spinrdquoNuovo Cimento della Societa Italiana di Fisica B vol 112 no 1pp 1ndash9 1997

[26] M Ibison andBHaisch ldquoQuantum and classical statistics of theelectromagnetic zero-point fieldrdquo Physical Review A vol 54 no4 pp 2737ndash2744 1996

[27] T W Marshall ldquoRandom electrodynamicsrdquo Proceedings of theRoyal Society A vol 276 no 1367 pp 475ndash491 1963

[28] L D Landau and E M Lifshitz The Classical Theory of FieldsPergamon Press New York NY USA 4th edition 1987

[29] J D Jackson Classical Electrodynamics Wiley New York NYUSA 3rd edition 1998

[30] E Poisson ldquoAn introduction to the Lorentz-Dirac equationrdquohttparxivorgabsgr-qc9912045

[31] G WoanThe Cambridge Handbook of Physics Formulas Cam-bridge University Press New York NY USA 2003

[32] WH Press S A TeukolskyW T Vetterling and B P FlanneryNumerical Recipes in Fortran 77TheArt of Scientific ComputingCambridge University Press New York NY USA 2nd edition1996


[33] S T Thornton and J B Marion Classical Dynamics of Particlesand Systems BrooksCole Belmont Calif USA 5th edition2004

[34] T H Boyer ldquoEquilibrium of random classical electromagneticradiation in the presence of a nonrelativistic nonlinear electricdipole oscillatorrdquo Physical Review D vol 13 no 10 pp 2832ndash2845 1976

[35] P S Pacheco Parallel Programming with MPI Morgan Kauf-mann San Francisco Calif USA 1997

[36] T H Boyer ldquoAny classical description of nature requires classi-cal electromagnetic zero-point radiationrdquo American Journal ofPhysics vol 79 no 11 pp 1163ndash1167 2011

[37] K Gottfried ldquoP A M Dirac and the discovery of quantummechanicsrdquo American Journal of Physics vol 79 no 3 pp 261ndash266 2011

[38] I J R Aitchison D A MacManus and T M Snyder ldquoUnder-standing Heisenbergrsquos ldquomagicalrdquo paper of July 1925 a new lookat the calculational detailsrdquoAmerican Journal of Physics vol 72no 11 pp 1370ndash1379 2004

[39] L de la Pena and A Jauregui ldquoThe spin of the electron accord-ing to stochastic electrodynamicsrdquo Foundations of Physics vol12 no 5 pp 441ndash465 1982

[40] S Sachidanandam ldquoA derivation of intrinsic spin one-half fromrandom electrodynamicsrdquo Physics Letters A vol 97 no 8 pp323ndash324 1983

[41] A F Kracklauer ldquoPilot wave steerage a mechanism and testrdquoFoundations of Physics Letters vol 12 no 5 pp 441ndash453 1999

[42] W Pauli ldquoLes theories quantities du magnetisive lrsquoelectronmagnetique (The theory of magnetic quantities the magneticelectron)rdquo in Proceedings of the 6th Solvay Conference pp 183ndash186 Gauthier-Villars Brussels Belgium 1930

[43] H Batelaan T J Gay and J J Schwendiman ldquoStern-gerlacheffect for electron beamsrdquo Physical Review Letters vol 79 no23 pp 4517ndash4521 1997

[44] G A Gallup H Batelaan and T J Gay ldquoQuantum-mechanicalanalysis of a longitudinal Stern-Gerlach effectrdquo Physical ReviewLetters vol 86 no 20 pp 4508ndash4511 2001

[45] H Dehmelt ldquoExperiments on the structure of an individualelementary particlerdquo Science vol 247 no 4942 pp 539ndash5451990

[46] H Dehmelt ldquoNew continuous Stern-Gerlach effect and a hintof ldquotherdquo elementary particlerdquo Zeitschrift fur Physik D vol 10 no2-3 pp 127ndash134 1988

[47] R Bach and H Batelaan The Challenge of Quantum Realityproduced by D Pope a film made by Perimeter Institute forTheoretical Physics 2009

[48] R Bach D Pope S H Liou and H Batelaan ldquoControlleddouble-slit electron diffractionrdquo New Journal of Physics vol 15Article ID 033018 2013

[49] C Jonsson ldquoElektroneninterferenzen an mehreren kunstlichhergestellten Feinspaltenrdquo Zeitschrift fur Physik vol 161 no 4pp 454ndash474 1961

[50] C Jonsson ldquoElectron diffraction at multiple slitsrdquo AmericanJournal of Physics vol 42 no 1 pp 4ndash11 1974

[51] S Frabboni C Frigeri G C Gazzadi and G Pozzi ldquoTwo andthree slit electron interference and diffraction experimentsrdquoAmerican Journal of Physics vol 79 no 6 pp 615ndash618 2011

[52] J Avendano and L de La Pena ldquoReordering of the ridge patternsof a stochastic electromagnetic field by diffraction due to anideal slitrdquo Physical Review E vol 72 no 6 Article ID 0666052005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


Vacuum field SED

Radiationdamping

X

X

2

1

0

minus1

minus2649647645

Time (s)

times10minus8

times10minus13

Posit

ion

(m)

2

1

0

minus1

minus2649647645

Time (s)

times10minus8

times10minus13

Posit

ion

(m)

2

1

0

minus1

minus2

times10minus8

Posit

ion

(m)

CMCM

CM

Probability density

SED (theory)

0 5 10

2

1

0

minus1

minus2

times10minus8

Posit

ion

(m)


times107

times107

Figure 1 A comparison between the harmonic oscillators with and without the vacuum field Top without any external force except for thevacuum field the SED harmonic oscillator undergoes a motion that results in a Gaussian probability distributionThis motion is investigatedwith our simulation Bottom in the absence of the vacuum field or any external drive a harmonic oscillator that is initially displaced fromequilibrium performs a simple harmonic oscillation with constant oscillation amplitude The resulting probability distribution has peaks atthe two turning points

in time persists beyond many cycles of oscillation) unlessthe system is proved to be ergodic As an analytical proof ofthe ergodicity is very difficult we take a numerical approachto study the particlersquos dynamical behavior Besides what isalready known from the analytical solution our numericalstudies construct the probability distribution from a singleparticlersquos trajectory We investigate the relation betweensuch a probability distribution and the particlersquos dynamicalbehavior Ultimately we want to know the underlying mech-anism that turns the classic double-peak distribution into theGaussian distribution

While most works in the field of SED are analyticalnumerical studies are rare [10ndash12] The advantage of numer-ical simulation is that it may be extended to other physicalsystems with relative ease and is flexible in testing differentassumptions and approximations For example in most SEDanalyses the effect of the Lorentz force due to the vacuumfield is often neglected as the first-order approximation [5]However when the field gradient is nonzero the Lorentzforce from the magnetic field can work with the electricfield to give a nonzero drift to a charged particle known asthe ponderomotive force Therefore the Lorentz force of thevacuum field may play a significant role in SED In fact it isshown in the literature that the magnetic part of the vacuumfield is responsible for the self-ionization of the atoms andthe acquisition of an energy 1020 eV in a few nanoseconds bypart of a free electron in vacuum [13 14] Using numericalsimulation one can easily examine SED beyond the first-order approximation (in this work we limit ourselves to

physical parameters where we expect the Lorentz force effectsto be minimal as this work is compared to analytical resultswhere this approximation is made) and the Lorentz forceeffects in SED can be investigated

The major challenge for the numerical simulation is toproperly account for the vacuum field modes A represen-tative sampling of the modes is thus the key for successfulsimulations In this study one of our goals is to use a simplephysical system namely the simple harmonic oscillator tobenchmark our numericalmethod of vacuummode selectionso that it can be used to test the validity range of SED asdiscussed in the following

Over the decades SED has been criticized for severaldrawbacks [15] Authors like Cavalleri argued that SEDcan neither explain electron slit-diffraction nor derive thenonlinear Schrodinger equation moreover SED impliesbroad radiation and absorption spectra for rarefied gases Inthe case of the quartic anharmonic oscillator Pesquera andClaverie showed that SEDdisagrees with quantummechanics[16] Additionally the results claimed in [17ndash19] were shownto be wrong due to improper relativistic approximation [20]While these theoretical analyses are documented in detail itmay be useful to use numerical simulation as an independentcheck in order to establish the validity range of SED which isone of the objectives of our work

Meanwhile a modified theory called stochastic electro-dynamics with spin (SEDS) was recently proposed [15] AsSEDS are an extension of SED with a model of electron spinmotion it is argued that the introduction of electron spin






Evac (r 119905) =2

sum

120582=1

int1198893

119896120576 (k 120582)120578 (k 120582)2


+119886lowast


(1)

120578 (k 120582) equiv radic ℎ120596

812058731205980

119886 (k 120582) equiv 119890119894120579(k120582)

(2)


120576 (k 120582) sdot k = 0 (3)


120576 (k 1) sdot 120576 (k 2) = 0 (4)


k sdot r ≪ 1 (5)


119898 = minus1198981205962

0119909 + 119898Γ + 119902119864vac119909 (119905) (6)


31198981198883




119864vac119909 (119905) =2

sum

120582=1

int1198893

119896120576119909(k 120582)

120578 (k 120582)2


(7)


119909 (119905) =119902

119898

2

sum

120582=1

int1198893

119896120576119909(k 120582)

120578 (k 120582)2

times (119886 (k 120582)119862 (k 120582)

119890minus119894120596119905

+119886lowast

(k 120582)119862lowast (k 120582)

119890119894120596119905

)

(8)


+ 1205962

0) minus 119894Γ120596


Γ1205960≪ 1 (9)


120590119909= radic⟨1199092⟩

120579minus ⟨119909⟩2

120579

= radicℎ

21198981205960

(10)

120590119901= radic⟨1199012⟩

120579minus ⟨119901⟩

2

120579= radic

ℎ1198981205960

2 (11)





120590119909120590119901=ℎ

2 (12)







119865120579(119896) = int

+infin

minusinfin

119890minus119894119896119909

119875120579(119909) 119889119909

=

infin

sum

119899=0

(minus119894119896)119899

119899int

+infin

minusinfin

119909119899

119875120579(119909) 119889119909

=

infin

sum

119899=0

(minus119894119896)119899

119899⟨119909119899

⟩120579

(13)


⟨119890plusmn119894(120579(k120582)+120579(k1015840 1205821015840))

⟩120579

= 0

⟨119890plusmn119894(120579(k120582)minus120579(k1015840 1205821015840))

⟩120579

= 12057512058210158401205821205753

(k1015840 minus k) (14)


⟨1199092119898+1

⟩120579

= 0

⟨1199092119898

⟩120579

=(2119898)

1198982119898(

ℎ

21198981205960

)

119898

(15)


119865120579(119896) =

infin

sum

119898=0

(minus119894119896)2119898

(2119898)⟨1199092119898

⟩

=

infin

sum

119898=0

1

119898(minusℎ1198962

41198981205960

)

119898

= exp(minus ℎ

41198981205960

1198962

)

(16)


119875120579(119909) =

1

2120587int

+infin

minusinfin

119890119894119896119909

119865120579(119896) 119889119896

= radic1198981205960

120587ℎexp(minus

1198981205960

ℎ1199092

)

(17)









0minus Δ2 120596

0+



Γ1205962

0≪ Δ (19)


Evac =2

sum

120582=1

sum

(119896119909 119896119910119896119911)

radicℎ120596



120588k =119881

(2120587)3 (21)




For 119894 = 1 119873120581 119895 = 1 119873

120599 and 119899 = 1 119873

120593

120581119894119895119899=(1205960minus Δ2)

3

31198883+ (119894 minus 1) Δ120581

120599119894119895119899= minus1 + (119895 minus 1) Δ120599

120593119894119895119899= 119877(0)

119894119895+ (119899 minus 1) Δ120593

(22)

where 119877(0)119894119895


Δ120581 =[(1205960+ Δ2)

3

31198883


3

31198883

]

119873120581minus 1

(23)

Δ120599 =2

119873120599minus 1

Δ120593 =2120587

119873120593

(24)



119894119895119899 120599119894119895119899 120593119894119895119899) is then used


k119894119895119899= (

119896119909

119896119910

119896119911

) = (

119896119894119895119899

sin (120579119894119895119899) cos (120601

119894119895119899)

119896119894119895119899

sin (120579119894119895119899) sin (120601

119894119895119899)

119896119894119895119899

cos (120579119894119895119899)

) (25)

where

119896119894119895119899= (3120581

119894119895119899)13

120579119894119895119899= cosminus1 (120599

119894119895119899)

120601119894119895119899= 120593119894119895119899

(26)



Δ3






Evac =2

sum

120582=1

sum

(120581120599120593)

radicℎ120596







119881 = (2120587)3

120588k = (2120587)3119873k119881k (29)

where

119881k =4120587

3(1205960+ Δ2

119888)

3

minus4120587

3(1205960minus Δ2

119888)

3

(30)


Evac =2

sum

120582=1

119873120581

sum

119894=1

119873120599

sum

119895=1

119873120593

sum

119899=1

radicℎ120596





120599and 119873






120596

k119894= (

119896119909

119896119910

119896119911

) = (

119896119894sin 120579119894cos120601119894

119896119894sin 120579119894sin120601119894

119896119894cos 120579119894

) (32)

where

119896119894= (3120581119894)13

120579119894= cosminus1 (120599

119894)

120601119894= 120593119894

(33)

120581119894=(1205960minus Δ2)

3

31198883+ (119894 minus 1) Δ120581

120599119894= 119877(1)

119894

120593119894= 119877(2)

119894

(34)



119894is a random


119894is







119896= 1205960119888 In this limit the


Γ1205960≪

Δ

1205960

≪ 1 (35)


Evac =2

sum

120582=1

119873120596

sum

119894=1

radicℎ120596






120576k1198941= (

1205761119909

1205761119910

1205761119911

)

= (


119894sin120594119894

cos 120579119894sin120601119894cos120594119894+ cos120601

119894sin120594119894

minus sin 120579119894cos120594119894

)

120576k1198942= (

1205762119909

1205762119910

1205762119911

)

= (


119894cos120594119894


119894cos120594119894

sin 120579119894sin120594119894

)

(37)



119894 120599119894 120593119894 120594119894) which are


1

05

0

minus05

minus1

1050

minus05minus1

1

050

minus05minus1

y

z

x


120596= 3 times 10




119898 = minus1198981205962

0119909 + 119898Γ + 119902119864vac119909 (119905) (38)



1205980119881

1

2(119886k120582119890

minus119894120596119905

+ 119886lowast

k120582119890119894120596119905

) 120576k120582 (39)



119909 (119905) =119902

119898sum

k120582radicℎ120596

1205980119881

1

2(119886k120582119862k120582

119890minus119894120596119905

+119886lowast

k120582119862lowastk120582

119890119894120596119905

) 120576k120582119909 (40)


+1205962

0)minus119894Γ120596




119898 = 119865 + 119898Γ (41)


31198981198883

)(141205871205980) is the


119898 ≃ 119865 (42)


119898Γ ≃ Γ (43)


119898 ≃ 119865 + Γ (44)


119898 ≃ minus1198981205962

0119909 minus 119898Γ120596

2

0 + 119902119864vac119909 (119905) + 119902Γvac119909 (119905) (45)


119874(1198981205962

0119909) = 119898120596

2

01199090 (46)

119874 (119902119864vac119909) = 1198901198640 (47)

119874(119898Γ1205962

0) = (Γ120596

0)1198981205962

01199090 (48)

119874(119902Γvac119909) = (Γ1205960) 1199021198640 (49)

where 1199090and 119864



0because


0and 119864

0 For a fixed time






119863rms = radic119873119904 sdot Δ119904 (50)



Δ119904 = (12)((119902119898Γ1205963

0)radic(ℎ120596




(2120587)3

119873120596119881k and 119881k ≃ 4120587120596

2

0(Γ1205962

0)1198883 ((29) and (30)) the


0≃ radic3120587radicℎ2119898120596

0


1198640≃ 119863(119864)


2radicℎ1205960

1205980119881

1199090≃ 119863(119909)


2(

119902

119898Γ12059630

radicℎ1205960

1205980119881)

(51)


119874(119898Γ1205962

0) ≃ 119902radic

119873120596

2sdot radic

ℎ1205960

1205980119881

119874 (119902Γvac119909) ≃ (Γ1205960) 119902radic119873120596

2sdot radic

ℎ1205960

1205980119881

(52)



119898 ≃ minus1198981205962

0119909 minus 119898Γ120596

2

0 + 119902119864vac119909 (119905) (53)



119874(1198981205962

0119909) ≫ 119874(119898Γ120596

2

0) ≃ 119874 (119902119864vac119909) (54)



2




119909 (119905) = 119890minusΓ1205962

01199052

(119860119890119894120596119877119905

+ 119860lowast

119890minus119894120596119877119905

)

+119902

119898sum

k120582radicℎ120596

1205980119881

1

2(119886k120582119861k120582

119890minus119894120596119905

+119886lowast

k120582119861lowastk120582

119890119894120596119905

) 120576k120582119909(55)


120596119877equiv 1205960


2)

2

119861k120582 equiv (minus1205962

+ 1205962

0) minus 119894 (Γ120596

2

0) 120596

119886k120582 equiv 119890119894120579k120582

(56)



120591tran =2

Γ12059620

(57)


120591int =2120587

Δ120596 (58)


Δ120596 ≃119888(31205810)13

3

Δ120581

1205810

(59)

where 1205810equiv (13)(120596

0119888)3







2

10

minus1

minus1

minus215 16 17 18 19


cuum

fiel

d (V

m)

002

0

minus002

Vacu

um fi

eld

(Vm

)

times10minus8


times10minus12

ParticleField

1

0

17 171 172 173Time (s)

Time (s)

Posit

ion

(m)

Posit

ion

(m) 120591(X)

mod

)(E120591mod




0) to avoid


119902 = 119890

119898 = 10minus4

119898119890

1205960= 1016 rads

Δ = 220 times Γ1205962

0

(60)


Γ1205962





10minus4





119909 (119905)

=119902

119898120596119877

int

119905

minusinfin

119864vac119909 (1199051015840

) 119890minusΓ1205962

0(119905minus1199051015840)2 sin (120596

119877(119905 minus 1199051015840

)) 1198891199051015840

(61)







)


0for the particle






6

4

2


Position (m)

Prob

abili

ty d

ensit

y

times107

6

4

2



Prob

abili

ty d

ensit

yQMSED

QMSED

times1025

times10minus26

2

1

0

minus1

minus212 18 24

Posit

ion

(m)


times10minus26

2

1

0

minus1

minus2178 1781

Posit

ion

(m)


times10minus26

times10minus8

1782


120596= 2 times 10



119909 (119905) =119902

119898sum

k120582radicℎ120596

1205980119881

119886k120582119861k120582

119890minus119894120596119905

120576k120582119909 (62)




119909 (119905) = 119860 (119905) 119890minus1198941205960119905

119860 (119905) = sum

k120582(120576k120582119909

119902

119898radicℎ120596

1205980119881

119886k120582119861k120582

)119890minus119894(120596minus120596

0)119905

(63)





2

1

0

minus1

minus205 1 15 2 25 3

Posit

ion

(m)

times10minus8

times10minus12

Prob

abili

ty d

ensit

y 6

4

2

0minus3 0 3

times107 15

10

5

0minus3 0 3

times1078

6

4

2

0minus3 0 3

times107

Time (s)




negligible

119860 (119905 + Δ119905) ≃ 119860 (119905) for Δ119905 ≪ 120591coh (64)




3

15

0

minus15


Real

Imag

inar

y

A(t0)

Δ

A(t0 + Δt)



0) (blue



the coherence time



119875 (119909) = sum

119860

119875119860(119909) = int119875

119860(119909) 119891 (119860) 119889119860 (65)







5

0

minus51 3 5 7 9

times10minus8

times10minus11

Posit

ion

(m)

Time (s)

2

0

minus2461 462 463 464

times10minus8

times10minus11

Posit

ion

(m)

Time (s)

10

5


times107


Prob

abili

ty d

ensit

y




(rare)

= 15 times 107

Small amplitude

3120591coh

Δs


119904= 15 times 10




120596= 2 times 10


6

4

2


times107

times10minus8

Prob

abili

ty d

ensit

y

QMSED

120590x ≃ 761 times 10minus9

6

4

2


times1025

times10minus26

Prob

abili

ty d

ensit

y


QMSED

120590p ≃ 693 times 10minus27 5

0

minus50 2 4Po

sitio

n (m

)

Position (m)

Time (s)

times10minus8

times10minus12

Trial-15

0

minus50 2 4

Posit

ion

(m)

Time (s)

times10minus8

times10minus12

Trial-25

0

minus50 2 4

Posit

ion

(m)

Time (s)

times10minus8

times10minus12

Trial-3


119901= 2 times 10


120596= 2 times 10






120596= 5 times 10


119901= 1198791119879119901 where 119879





4

2

0

minus2

minus40 02 04 06 08 1 12 14

Time (s)

Posit

ion

(m)


times10minus8

times10minus12

4

2

0

minus2

minus40 02 04 06 08 1 12 14

Time (s)

Posit

ion

(m)


times10minus12

times10minus26


10 30 50 70 90

10

20

30

40


Com

puta

tion

time (

hr)

Np = 100 k = 05 kNp = 100 k = 1 kNp = 200 k = 05 k

N120596

N120596

N120596


119901(number


0 200 400 600 800 10000

14

12

34

1

Np = 200 k

Ener

gy (ℏ

1205960)

N120596



119901=


2

0






120596


5 Conclusions


120596= 5 times 10




Vacuum field

SEDRadiationdamping

X

X

2

1

0

minus1

minus2

Time (s)

times10minus8

times10minus12

Posit

ion

(m)

2

1

0

minus1

minus2649647645

Time (s)

times10minus8

times10minus13

Posit

ion

(m)

2

1

0

minus1

minus2

times10minus8

Posit

ion

(m)

CMCM

Probability density

SED (theory)

0 5 10

2

1

0

minus1

minus2

times10minus8

Posit

ion

(m)


SED (simulation)

2 3 4 5 6 7

times107

times107














Appendices




E (r 119905)

=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)


B (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896 (k times 120576 (k 120582))




119860(k 120582)119890119894120579(k120582)

E (r 119905)

=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)


+119860lowast



B (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896120585 (k 120582)


+119860lowast


(A2)


⟨119906⟩120579=1205980

2⟨|E|2⟩

120579

+1

21205830

⟨|B|2⟩120579

(A3)


0is the vacuum



and |B|2 using (A2)

|E|2 (r 119905)

= E (r 119905)Elowast (r 119905)

=1

4sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120576 (k 120582) sdot 120576 (k1015840 1205821015840)) 119891

|B|2 (r 119905)

= B (r 119905)Blowast (r 119905)

=1

41198882sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120585 (k 120582) sdot 120585 (k1015840 1205821015840)) 119891

(A4)

where 119891 equiv 119891(k 120582 k1015840 1205821015840 r 119905)

119891 (k 120582 k1015840 1205821015840 r 119905)



+ 119860 (k 120582) 119860 (k1015840 1205821015840) 119890119894(ksdotrminus120596119905+120579(k120582))


+ 119860lowast



+ 119860lowast



(A5)


⟨119890plusmn119894(120579(k120582)+120579(k1015840 1205821015840))

⟩120579

= 0

⟨119890plusmn119894(120579(k120582)minus120579(k1015840 1205821015840))

⟩120579

= 12057512058210158401205821205753

(k1015840 minus k) (A6)


⟨119891 (k 120582 k1015840 1205821015840 r 119905)⟩120579



12057512058210158401205821205753

(k1015840 minus k)

+ 119860lowast



12057512058210158401205821205753

(k1015840 minus k)

(A7)



(A4) and (A7)

⟨|E|2⟩120579

=1

4sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120576 (k 120582) sdot 120576 (k1015840 1205821015840)) ⟨119891⟩120579

=1

2

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2

⟨|B|2⟩120579

=1

41198882sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120585 (k 120582) sdot 120585 (k1015840 1205821015840)) ⟨119891⟩120579

=1

21198882

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2

(A8)


120579

in unbounded space

⟨119906⟩120579=1205980

2

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2 (A9)


119906vac =1

119881sum

k120582

ℎ120596

2

=1

119881sum

k120582

ℎ120596

2(119871119909

2120587Δ119896119909)(

119871119910

2120587Δ119896119910)(

119871119911

2120587Δ119896119911)

=

2

sum

120582=1

1

(2120587)3sum

kΔ3

119896ℎ120596

2

(A10)



3


119906vac =2

sum

120582=1

1

(2120587)3int1198893

119896ℎ120596

2 (A11)


1003816100381610038161003816119860vac (k 120582)10038161003816100381610038162

=ℎ120596

(2120587)3

1205980

(A12)


119860vac (k 120582) = radicℎ120596

812058731205980

(A13)


=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)radic ℎ120596

812058731205980


119890119894120579(k120582)


119890minus119894120579(k120582)

)

Bvac (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896120585 (k 120582)radic ℎ120596

812058731205980


119890119894120579(k120582)


119890minus119894120579(k120582)

)

(A14)

which is (1)


E (r 119905) = 1

2sum

k120582(119860k120582119890



) 120576k120582

B (r 119905) = 1

2119888sum

k120582(119860k120582119890



) 120585k120582

(A15)



120579= ⟨119890119894120579k120582⟩120579⟨119890plusmn119894120579k1015840 1205821015840 ⟩

120579=

0)

⟨119890plusmn119894(120579k120582+120579k1015840 1205821015840 )⟩

120579

= 0

⟨119890plusmn119894(120579k120582minus120579k1015840 1205821015840 )⟩

120579

= 1205751205821015840120582120575k1015840 k

(A16)


⟨119906⟩120579=1205980

2sum

k120582

1003816100381610038161003816119860k12058210038161003816100381610038162

(A17)



119906vac =1

119881sum

k120582

ℎ120596

2 (A18)


119860vack120582 = radicℎ120596

1205980119881 (A19)


Evac (r 119905) =1

2sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120576k120582

Bvac (r 119905) =1

2119888sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120585k120582

(A20)



k = (119909

119910

119911

) = (

0

0

119896

) (B1)


k1 = (cos120594sin1205940

) k2 = (minus sin120594cos1205940

) (B2)


= (119911)

120601(119910)

120579

= (


sin120601 cos120601 0

0 0 1

)(

cos 120579 0 sin 1205790 1 0

minus sin 120579 0 cos 120579)

= (


)

(B3)


2

1

0

minus1

minus2

0 3 6 9 12Time (s)

Fiel

d (V

m)

120591rep

120591env


1=


1 1198792]LCM = 6




) (B4)


120576k1 = k1 = (



120576k2 = k2 = (


sin 120579 sin120594)

(B5)

C Repetitive Time


119864 (119905) =

119873

sum

119896=1

119864119896cos (120596

119896119905) (C1)


119864 (119905 + 120591rep) = 119864 (119905) (C2)

120591rep = [1198791 1198792 119879119873]LCM (C3)

where 119879119896= 2120587120596



120591rep =2120587

(1205961 1205962 120596

119873)GCD

(C4)


1205961+ 1205962+ sdot sdot sdot + 120596

119873

=2120587

1198791

+2120587

1198792


119879119873

=2120587

[1198791 1198792 119879

119873]LCM

(1198991+ 1198992+ sdot sdot sdot + 119899

119873)

(C5)


119899119896=[1198791 1198792 119879

119873]LCM

119879119896

(C6)




Δ120596gcd equiv (1205961 1205962 120596119873)GCD

=2120587

[1198791 1198792 119879

119873]LCM

(C7)


120591rep =2120587

Δ120596gcd(C8)


120591int =2120587

Δ120596 (C9)


Δ120596 (120581) = 120596 (120581) minus 120596 (120581 minus Δ120581)

= 119888(3120581)13

minus 119888(3120581)13

(1 minusΔ120581

120581)

13

(C10)


0and

120581 ≃ 1205810

Δ120581

1205810

= (3

119873k minus 1)Δ

1205960

≪ 1

120581 = 1205810+ 119874(

Δ

1205960

)

(C11)


where 1205810= (13)(120596



Δ120596 ≃119888(31205810)13

3

Δ120581

1205810

(C12)

Acknowledgments


References



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








Evac (r 119905) =2

sum

120582=1

int1198893

119896120576 (k 120582)120578 (k 120582)2


+119886lowast


(1)

120578 (k 120582) equiv radic ℎ120596

812058731205980

119886 (k 120582) equiv 119890119894120579(k120582)

(2)


120576 (k 120582) sdot k = 0 (3)


120576 (k 1) sdot 120576 (k 2) = 0 (4)


k sdot r ≪ 1 (5)


119898 = minus1198981205962

0119909 + 119898Γ + 119902119864vac119909 (119905) (6)


31198981198883




119864vac119909 (119905) =2

sum

120582=1

int1198893

119896120576119909(k 120582)

120578 (k 120582)2


(7)


119909 (119905) =119902

119898

2

sum

120582=1

int1198893

119896120576119909(k 120582)

120578 (k 120582)2

times (119886 (k 120582)119862 (k 120582)

119890minus119894120596119905

+119886lowast

(k 120582)119862lowast (k 120582)

119890119894120596119905

)

(8)


+ 1205962

0) minus 119894Γ120596


Γ1205960≪ 1 (9)


120590119909= radic⟨1199092⟩

120579minus ⟨119909⟩2

120579

= radicℎ

21198981205960

(10)

120590119901= radic⟨1199012⟩

120579minus ⟨119901⟩

2

120579= radic

ℎ1198981205960

2 (11)





120590119909120590119901=ℎ

2 (12)







119865120579(119896) = int

+infin

minusinfin

119890minus119894119896119909

119875120579(119909) 119889119909

=

infin

sum

119899=0

(minus119894119896)119899

119899int

+infin

minusinfin

119909119899

119875120579(119909) 119889119909

=

infin

sum

119899=0

(minus119894119896)119899

119899⟨119909119899

⟩120579

(13)


⟨119890plusmn119894(120579(k120582)+120579(k1015840 1205821015840))

⟩120579

= 0

⟨119890plusmn119894(120579(k120582)minus120579(k1015840 1205821015840))

⟩120579

= 12057512058210158401205821205753

(k1015840 minus k) (14)


⟨1199092119898+1

⟩120579

= 0

⟨1199092119898

⟩120579

=(2119898)

1198982119898(

ℎ

21198981205960

)

119898

(15)


119865120579(119896) =

infin

sum

119898=0

(minus119894119896)2119898

(2119898)⟨1199092119898

⟩

=

infin

sum

119898=0

1

119898(minusℎ1198962

41198981205960

)

119898

= exp(minus ℎ

41198981205960

1198962

)

(16)


119875120579(119909) =

1

2120587int

+infin

minusinfin

119890119894119896119909

119865120579(119896) 119889119896

= radic1198981205960

120587ℎexp(minus

1198981205960

ℎ1199092

)

(17)









0minus Δ2 120596

0+



Γ1205962

0≪ Δ (19)


Evac =2

sum

120582=1

sum

(119896119909 119896119910119896119911)

radicℎ120596



120588k =119881

(2120587)3 (21)




For 119894 = 1 119873120581 119895 = 1 119873

120599 and 119899 = 1 119873

120593

120581119894119895119899=(1205960minus Δ2)

3

31198883+ (119894 minus 1) Δ120581

120599119894119895119899= minus1 + (119895 minus 1) Δ120599

120593119894119895119899= 119877(0)

119894119895+ (119899 minus 1) Δ120593

(22)

where 119877(0)119894119895


Δ120581 =[(1205960+ Δ2)

3

31198883


3

31198883

]

119873120581minus 1

(23)

Δ120599 =2

119873120599minus 1

Δ120593 =2120587

119873120593

(24)



119894119895119899 120599119894119895119899 120593119894119895119899) is then used


k119894119895119899= (

119896119909

119896119910

119896119911

) = (

119896119894119895119899

sin (120579119894119895119899) cos (120601

119894119895119899)

119896119894119895119899

sin (120579119894119895119899) sin (120601

119894119895119899)

119896119894119895119899

cos (120579119894119895119899)

) (25)

where

119896119894119895119899= (3120581

119894119895119899)13

120579119894119895119899= cosminus1 (120599

119894119895119899)

120601119894119895119899= 120593119894119895119899

(26)



Δ3






Evac =2

sum

120582=1

sum

(120581120599120593)

radicℎ120596







119881 = (2120587)3

120588k = (2120587)3119873k119881k (29)

where

119881k =4120587

3(1205960+ Δ2

119888)

3

minus4120587

3(1205960minus Δ2

119888)

3

(30)


Evac =2

sum

120582=1

119873120581

sum

119894=1

119873120599

sum

119895=1

119873120593

sum

119899=1

radicℎ120596





120599and 119873






120596

k119894= (

119896119909

119896119910

119896119911

) = (

119896119894sin 120579119894cos120601119894

119896119894sin 120579119894sin120601119894

119896119894cos 120579119894

) (32)

where

119896119894= (3120581119894)13

120579119894= cosminus1 (120599

119894)

120601119894= 120593119894

(33)

120581119894=(1205960minus Δ2)

3

31198883+ (119894 minus 1) Δ120581

120599119894= 119877(1)

119894

120593119894= 119877(2)

119894

(34)



119894is a random


119894is







119896= 1205960119888 In this limit the


Γ1205960≪

Δ

1205960

≪ 1 (35)


Evac =2

sum

120582=1

119873120596

sum

119894=1

radicℎ120596






120576k1198941= (

1205761119909

1205761119910

1205761119911

)

= (


119894sin120594119894

cos 120579119894sin120601119894cos120594119894+ cos120601

119894sin120594119894

minus sin 120579119894cos120594119894

)

120576k1198942= (

1205762119909

1205762119910

1205762119911

)

= (


119894cos120594119894


119894cos120594119894

sin 120579119894sin120594119894

)

(37)



119894 120599119894 120593119894 120594119894) which are


1

05

0

minus05

minus1

1050

minus05minus1

1

050

minus05minus1

y

z

x


120596= 3 times 10




119898 = minus1198981205962

0119909 + 119898Γ + 119902119864vac119909 (119905) (38)



1205980119881

1

2(119886k120582119890

minus119894120596119905

+ 119886lowast

k120582119890119894120596119905

) 120576k120582 (39)



119909 (119905) =119902

119898sum

k120582radicℎ120596

1205980119881

1

2(119886k120582119862k120582

119890minus119894120596119905

+119886lowast

k120582119862lowastk120582

119890119894120596119905

) 120576k120582119909 (40)


+1205962

0)minus119894Γ120596




119898 = 119865 + 119898Γ (41)


31198981198883

)(141205871205980) is the


119898 ≃ 119865 (42)


119898Γ ≃ Γ (43)


119898 ≃ 119865 + Γ (44)


119898 ≃ minus1198981205962

0119909 minus 119898Γ120596

2

0 + 119902119864vac119909 (119905) + 119902Γvac119909 (119905) (45)


119874(1198981205962

0119909) = 119898120596

2

01199090 (46)

119874 (119902119864vac119909) = 1198901198640 (47)

119874(119898Γ1205962

0) = (Γ120596

0)1198981205962

01199090 (48)

119874(119902Γvac119909) = (Γ1205960) 1199021198640 (49)

where 1199090and 119864



0because


0and 119864

0 For a fixed time






119863rms = radic119873119904 sdot Δ119904 (50)



Δ119904 = (12)((119902119898Γ1205963

0)radic(ℎ120596




(2120587)3

119873120596119881k and 119881k ≃ 4120587120596

2

0(Γ1205962

0)1198883 ((29) and (30)) the


0≃ radic3120587radicℎ2119898120596

0


1198640≃ 119863(119864)


2radicℎ1205960

1205980119881

1199090≃ 119863(119909)


2(

119902

119898Γ12059630

radicℎ1205960

1205980119881)

(51)


119874(119898Γ1205962

0) ≃ 119902radic

119873120596

2sdot radic

ℎ1205960

1205980119881

119874 (119902Γvac119909) ≃ (Γ1205960) 119902radic119873120596

2sdot radic

ℎ1205960

1205980119881

(52)



119898 ≃ minus1198981205962

0119909 minus 119898Γ120596

2

0 + 119902119864vac119909 (119905) (53)



119874(1198981205962

0119909) ≫ 119874(119898Γ120596

2

0) ≃ 119874 (119902119864vac119909) (54)



2




119909 (119905) = 119890minusΓ1205962

01199052

(119860119890119894120596119877119905

+ 119860lowast

119890minus119894120596119877119905

)

+119902

119898sum

k120582radicℎ120596

1205980119881

1

2(119886k120582119861k120582

119890minus119894120596119905

+119886lowast

k120582119861lowastk120582

119890119894120596119905

) 120576k120582119909(55)


120596119877equiv 1205960


2)

2

119861k120582 equiv (minus1205962

+ 1205962

0) minus 119894 (Γ120596

2

0) 120596

119886k120582 equiv 119890119894120579k120582

(56)



120591tran =2

Γ12059620

(57)


120591int =2120587

Δ120596 (58)


Δ120596 ≃119888(31205810)13

3

Δ120581

1205810

(59)

where 1205810equiv (13)(120596

0119888)3







2

10

minus1

minus1

minus215 16 17 18 19


cuum

fiel

d (V

m)

002

0

minus002

Vacu

um fi

eld

(Vm

)

times10minus8


times10minus12

ParticleField

1

0

17 171 172 173Time (s)

Time (s)

Posit

ion

(m)

Posit

ion

(m) 120591(X)

mod

)(E120591mod




0) to avoid


119902 = 119890

119898 = 10minus4

119898119890

1205960= 1016 rads

Δ = 220 times Γ1205962

0

(60)


Γ1205962





10minus4





119909 (119905)

=119902

119898120596119877

int

119905

minusinfin

119864vac119909 (1199051015840

) 119890minusΓ1205962

0(119905minus1199051015840)2 sin (120596

119877(119905 minus 1199051015840

)) 1198891199051015840

(61)







)


0for the particle






6

4

2


Position (m)

Prob

abili

ty d

ensit

y

times107

6

4

2



Prob

abili

ty d

ensit

yQMSED

QMSED

times1025

times10minus26

2

1

0

minus1

minus212 18 24

Posit

ion

(m)


times10minus26

2

1

0

minus1

minus2178 1781

Posit

ion

(m)


times10minus26

times10minus8

1782


120596= 2 times 10



119909 (119905) =119902

119898sum

k120582radicℎ120596

1205980119881

119886k120582119861k120582

119890minus119894120596119905

120576k120582119909 (62)




119909 (119905) = 119860 (119905) 119890minus1198941205960119905

119860 (119905) = sum

k120582(120576k120582119909

119902

119898radicℎ120596

1205980119881

119886k120582119861k120582

)119890minus119894(120596minus120596

0)119905

(63)





2

1

0

minus1

minus205 1 15 2 25 3

Posit

ion

(m)

times10minus8

times10minus12

Prob

abili

ty d

ensit

y 6

4

2

0minus3 0 3

times107 15

10

5

0minus3 0 3

times1078

6

4

2

0minus3 0 3

times107

Time (s)




negligible

119860 (119905 + Δ119905) ≃ 119860 (119905) for Δ119905 ≪ 120591coh (64)




3

15

0

minus15


Real

Imag

inar

y

A(t0)

Δ

A(t0 + Δt)



0) (blue



the coherence time



119875 (119909) = sum

119860

119875119860(119909) = int119875

119860(119909) 119891 (119860) 119889119860 (65)







5

0

minus51 3 5 7 9

times10minus8

times10minus11

Posit

ion

(m)

Time (s)

2

0

minus2461 462 463 464

times10minus8

times10minus11

Posit

ion

(m)

Time (s)

10

5


times107


Prob

abili

ty d

ensit

y




(rare)

= 15 times 107

Small amplitude

3120591coh

Δs


119904= 15 times 10




120596= 2 times 10


6

4

2


times107

times10minus8

Prob

abili

ty d

ensit

y

QMSED

120590x ≃ 761 times 10minus9

6

4

2


times1025

times10minus26

Prob

abili

ty d

ensit

y


QMSED

120590p ≃ 693 times 10minus27 5

0

minus50 2 4Po

sitio

n (m

)

Position (m)

Time (s)

times10minus8

times10minus12

Trial-15

0

minus50 2 4

Posit

ion

(m)

Time (s)

times10minus8

times10minus12

Trial-25

0

minus50 2 4

Posit

ion

(m)

Time (s)

times10minus8

times10minus12

Trial-3


119901= 2 times 10


120596= 2 times 10






120596= 5 times 10


119901= 1198791119879119901 where 119879





4

2

0

minus2

minus40 02 04 06 08 1 12 14

Time (s)

Posit

ion

(m)


times10minus8

times10minus12

4

2

0

minus2

minus40 02 04 06 08 1 12 14

Time (s)

Posit

ion

(m)


times10minus12

times10minus26


10 30 50 70 90

10

20

30

40


Com

puta

tion

time (

hr)

Np = 100 k = 05 kNp = 100 k = 1 kNp = 200 k = 05 k

N120596

N120596

N120596


119901(number


0 200 400 600 800 10000

14

12

34

1

Np = 200 k

Ener

gy (ℏ

1205960)

N120596



119901=


2

0






120596


5 Conclusions


120596= 5 times 10




Vacuum field

SEDRadiationdamping

X

X

2

1

0

minus1

minus2

Time (s)

times10minus8

times10minus12

Posit

ion

(m)

2

1

0

minus1

minus2649647645

Time (s)

times10minus8

times10minus13

Posit

ion

(m)

2

1

0

minus1

minus2

times10minus8

Posit

ion

(m)

CMCM

Probability density

SED (theory)

0 5 10

2

1

0

minus1

minus2

times10minus8

Posit

ion

(m)


SED (simulation)

2 3 4 5 6 7

times107

times107














Appendices




E (r 119905)

=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)


B (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896 (k times 120576 (k 120582))




119860(k 120582)119890119894120579(k120582)

E (r 119905)

=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)


+119860lowast



B (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896120585 (k 120582)


+119860lowast


(A2)


⟨119906⟩120579=1205980

2⟨|E|2⟩

120579

+1

21205830

⟨|B|2⟩120579

(A3)


0is the vacuum



and |B|2 using (A2)

|E|2 (r 119905)

= E (r 119905)Elowast (r 119905)

=1

4sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120576 (k 120582) sdot 120576 (k1015840 1205821015840)) 119891

|B|2 (r 119905)

= B (r 119905)Blowast (r 119905)

=1

41198882sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120585 (k 120582) sdot 120585 (k1015840 1205821015840)) 119891

(A4)

where 119891 equiv 119891(k 120582 k1015840 1205821015840 r 119905)

119891 (k 120582 k1015840 1205821015840 r 119905)



+ 119860 (k 120582) 119860 (k1015840 1205821015840) 119890119894(ksdotrminus120596119905+120579(k120582))


+ 119860lowast



+ 119860lowast



(A5)


⟨119890plusmn119894(120579(k120582)+120579(k1015840 1205821015840))

⟩120579

= 0

⟨119890plusmn119894(120579(k120582)minus120579(k1015840 1205821015840))

⟩120579

= 12057512058210158401205821205753

(k1015840 minus k) (A6)


⟨119891 (k 120582 k1015840 1205821015840 r 119905)⟩120579



12057512058210158401205821205753

(k1015840 minus k)

+ 119860lowast



12057512058210158401205821205753

(k1015840 minus k)

(A7)



(A4) and (A7)

⟨|E|2⟩120579

=1

4sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120576 (k 120582) sdot 120576 (k1015840 1205821015840)) ⟨119891⟩120579

=1

2

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2

⟨|B|2⟩120579

=1

41198882sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120585 (k 120582) sdot 120585 (k1015840 1205821015840)) ⟨119891⟩120579

=1

21198882

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2

(A8)


120579

in unbounded space

⟨119906⟩120579=1205980

2

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2 (A9)


119906vac =1

119881sum

k120582

ℎ120596

2

=1

119881sum

k120582

ℎ120596

2(119871119909

2120587Δ119896119909)(

119871119910

2120587Δ119896119910)(

119871119911

2120587Δ119896119911)

=

2

sum

120582=1

1

(2120587)3sum

kΔ3

119896ℎ120596

2

(A10)



3


119906vac =2

sum

120582=1

1

(2120587)3int1198893

119896ℎ120596

2 (A11)


1003816100381610038161003816119860vac (k 120582)10038161003816100381610038162

=ℎ120596

(2120587)3

1205980

(A12)


119860vac (k 120582) = radicℎ120596

812058731205980

(A13)


=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)radic ℎ120596

812058731205980


119890119894120579(k120582)


119890minus119894120579(k120582)

)

Bvac (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896120585 (k 120582)radic ℎ120596

812058731205980


119890119894120579(k120582)


119890minus119894120579(k120582)

)

(A14)

which is (1)


E (r 119905) = 1

2sum

k120582(119860k120582119890



) 120576k120582

B (r 119905) = 1

2119888sum

k120582(119860k120582119890



) 120585k120582

(A15)



120579= ⟨119890119894120579k120582⟩120579⟨119890plusmn119894120579k1015840 1205821015840 ⟩

120579=

0)

⟨119890plusmn119894(120579k120582+120579k1015840 1205821015840 )⟩

120579

= 0

⟨119890plusmn119894(120579k120582minus120579k1015840 1205821015840 )⟩

120579

= 1205751205821015840120582120575k1015840 k

(A16)


⟨119906⟩120579=1205980

2sum

k120582

1003816100381610038161003816119860k12058210038161003816100381610038162

(A17)



119906vac =1

119881sum

k120582

ℎ120596

2 (A18)


119860vack120582 = radicℎ120596

1205980119881 (A19)


Evac (r 119905) =1

2sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120576k120582

Bvac (r 119905) =1

2119888sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120585k120582

(A20)



k = (119909

119910

119911

) = (

0

0

119896

) (B1)


k1 = (cos120594sin1205940

) k2 = (minus sin120594cos1205940

) (B2)


= (119911)

120601(119910)

120579

= (


sin120601 cos120601 0

0 0 1

)(

cos 120579 0 sin 1205790 1 0

minus sin 120579 0 cos 120579)

= (


)

(B3)


2

1

0

minus1

minus2

0 3 6 9 12Time (s)

Fiel

d (V

m)

120591rep

120591env


1=


1 1198792]LCM = 6




) (B4)


120576k1 = k1 = (



120576k2 = k2 = (


sin 120579 sin120594)

(B5)

C Repetitive Time


119864 (119905) =

119873

sum

119896=1

119864119896cos (120596

119896119905) (C1)


119864 (119905 + 120591rep) = 119864 (119905) (C2)

120591rep = [1198791 1198792 119879119873]LCM (C3)

where 119879119896= 2120587120596



120591rep =2120587

(1205961 1205962 120596

119873)GCD

(C4)


1205961+ 1205962+ sdot sdot sdot + 120596

119873

=2120587

1198791

+2120587

1198792


119879119873

=2120587

[1198791 1198792 119879

119873]LCM

(1198991+ 1198992+ sdot sdot sdot + 119899

119873)

(C5)


119899119896=[1198791 1198792 119879

119873]LCM

119879119896

(C6)




Δ120596gcd equiv (1205961 1205962 120596119873)GCD

=2120587

[1198791 1198792 119879

119873]LCM

(C7)


120591rep =2120587

Δ120596gcd(C8)


120591int =2120587

Δ120596 (C9)


Δ120596 (120581) = 120596 (120581) minus 120596 (120581 minus Δ120581)

= 119888(3120581)13

minus 119888(3120581)13

(1 minusΔ120581

120581)

13

(C10)


0and

120581 ≃ 1205810

Δ120581

1205810

= (3

119873k minus 1)Δ

1205960

≪ 1

120581 = 1205810+ 119874(

Δ

1205960

)

(C11)


where 1205810= (13)(120596



Δ120596 ≃119888(31205810)13

3

Δ120581

1205810

(C12)

Acknowledgments


References



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





120590119909120590119901=ℎ

2 (12)







119865120579(119896) = int

+infin

minusinfin

119890minus119894119896119909

119875120579(119909) 119889119909

=

infin

sum

119899=0

(minus119894119896)119899

119899int

+infin

minusinfin

119909119899

119875120579(119909) 119889119909

=

infin

sum

119899=0

(minus119894119896)119899

119899⟨119909119899

⟩120579

(13)


⟨119890plusmn119894(120579(k120582)+120579(k1015840 1205821015840))

⟩120579

= 0

⟨119890plusmn119894(120579(k120582)minus120579(k1015840 1205821015840))

⟩120579

= 12057512058210158401205821205753

(k1015840 minus k) (14)


⟨1199092119898+1

⟩120579

= 0

⟨1199092119898

⟩120579

=(2119898)

1198982119898(

ℎ

21198981205960

)

119898

(15)


119865120579(119896) =

infin

sum

119898=0

(minus119894119896)2119898

(2119898)⟨1199092119898

⟩

=

infin

sum

119898=0

1

119898(minusℎ1198962

41198981205960

)

119898

= exp(minus ℎ

41198981205960

1198962

)

(16)


119875120579(119909) =

1

2120587int

+infin

minusinfin

119890119894119896119909

119865120579(119896) 119889119896

= radic1198981205960

120587ℎexp(minus

1198981205960

ℎ1199092

)

(17)









0minus Δ2 120596

0+



Γ1205962

0≪ Δ (19)


Evac =2

sum

120582=1

sum

(119896119909 119896119910119896119911)

radicℎ120596



120588k =119881

(2120587)3 (21)




For 119894 = 1 119873120581 119895 = 1 119873

120599 and 119899 = 1 119873

120593

120581119894119895119899=(1205960minus Δ2)

3

31198883+ (119894 minus 1) Δ120581

120599119894119895119899= minus1 + (119895 minus 1) Δ120599

120593119894119895119899= 119877(0)

119894119895+ (119899 minus 1) Δ120593

(22)

where 119877(0)119894119895


Δ120581 =[(1205960+ Δ2)

3

31198883


3

31198883

]

119873120581minus 1

(23)

Δ120599 =2

119873120599minus 1

Δ120593 =2120587

119873120593

(24)



119894119895119899 120599119894119895119899 120593119894119895119899) is then used


k119894119895119899= (

119896119909

119896119910

119896119911

) = (

119896119894119895119899

sin (120579119894119895119899) cos (120601

119894119895119899)

119896119894119895119899

sin (120579119894119895119899) sin (120601

119894119895119899)

119896119894119895119899

cos (120579119894119895119899)

) (25)

where

119896119894119895119899= (3120581

119894119895119899)13

120579119894119895119899= cosminus1 (120599

119894119895119899)

120601119894119895119899= 120593119894119895119899

(26)



Δ3






Evac =2

sum

120582=1

sum

(120581120599120593)

radicℎ120596







119881 = (2120587)3

120588k = (2120587)3119873k119881k (29)

where

119881k =4120587

3(1205960+ Δ2

119888)

3

minus4120587

3(1205960minus Δ2

119888)

3

(30)


Evac =2

sum

120582=1

119873120581

sum

119894=1

119873120599

sum

119895=1

119873120593

sum

119899=1

radicℎ120596





120599and 119873






120596

k119894= (

119896119909

119896119910

119896119911

) = (

119896119894sin 120579119894cos120601119894

119896119894sin 120579119894sin120601119894

119896119894cos 120579119894

) (32)

where

119896119894= (3120581119894)13

120579119894= cosminus1 (120599

119894)

120601119894= 120593119894

(33)

120581119894=(1205960minus Δ2)

3

31198883+ (119894 minus 1) Δ120581

120599119894= 119877(1)

119894

120593119894= 119877(2)

119894

(34)



119894is a random


119894is







119896= 1205960119888 In this limit the


Γ1205960≪

Δ

1205960

≪ 1 (35)


Evac =2

sum

120582=1

119873120596

sum

119894=1

radicℎ120596






120576k1198941= (

1205761119909

1205761119910

1205761119911

)

= (


119894sin120594119894

cos 120579119894sin120601119894cos120594119894+ cos120601

119894sin120594119894

minus sin 120579119894cos120594119894

)

120576k1198942= (

1205762119909

1205762119910

1205762119911

)

= (


119894cos120594119894


119894cos120594119894

sin 120579119894sin120594119894

)

(37)



119894 120599119894 120593119894 120594119894) which are


1

05

0

minus05

minus1

1050

minus05minus1

1

050

minus05minus1

y

z

x


120596= 3 times 10




119898 = minus1198981205962

0119909 + 119898Γ + 119902119864vac119909 (119905) (38)



1205980119881

1

2(119886k120582119890

minus119894120596119905

+ 119886lowast

k120582119890119894120596119905

) 120576k120582 (39)



119909 (119905) =119902

119898sum

k120582radicℎ120596

1205980119881

1

2(119886k120582119862k120582

119890minus119894120596119905

+119886lowast

k120582119862lowastk120582

119890119894120596119905

) 120576k120582119909 (40)


+1205962

0)minus119894Γ120596




119898 = 119865 + 119898Γ (41)


31198981198883

)(141205871205980) is the


119898 ≃ 119865 (42)


119898Γ ≃ Γ (43)


119898 ≃ 119865 + Γ (44)


119898 ≃ minus1198981205962

0119909 minus 119898Γ120596

2

0 + 119902119864vac119909 (119905) + 119902Γvac119909 (119905) (45)


119874(1198981205962

0119909) = 119898120596

2

01199090 (46)

119874 (119902119864vac119909) = 1198901198640 (47)

119874(119898Γ1205962

0) = (Γ120596

0)1198981205962

01199090 (48)

119874(119902Γvac119909) = (Γ1205960) 1199021198640 (49)

where 1199090and 119864



0because


0and 119864

0 For a fixed time






119863rms = radic119873119904 sdot Δ119904 (50)



Δ119904 = (12)((119902119898Γ1205963

0)radic(ℎ120596




(2120587)3

119873120596119881k and 119881k ≃ 4120587120596

2

0(Γ1205962

0)1198883 ((29) and (30)) the


0≃ radic3120587radicℎ2119898120596

0


1198640≃ 119863(119864)


2radicℎ1205960

1205980119881

1199090≃ 119863(119909)


2(

119902

119898Γ12059630

radicℎ1205960

1205980119881)

(51)


119874(119898Γ1205962

0) ≃ 119902radic

119873120596

2sdot radic

ℎ1205960

1205980119881

119874 (119902Γvac119909) ≃ (Γ1205960) 119902radic119873120596

2sdot radic

ℎ1205960

1205980119881

(52)



119898 ≃ minus1198981205962

0119909 minus 119898Γ120596

2

0 + 119902119864vac119909 (119905) (53)



119874(1198981205962

0119909) ≫ 119874(119898Γ120596

2

0) ≃ 119874 (119902119864vac119909) (54)



2




119909 (119905) = 119890minusΓ1205962

01199052

(119860119890119894120596119877119905

+ 119860lowast

119890minus119894120596119877119905

)

+119902

119898sum

k120582radicℎ120596

1205980119881

1

2(119886k120582119861k120582

119890minus119894120596119905

+119886lowast

k120582119861lowastk120582

119890119894120596119905

) 120576k120582119909(55)


120596119877equiv 1205960


2)

2

119861k120582 equiv (minus1205962

+ 1205962

0) minus 119894 (Γ120596

2

0) 120596

119886k120582 equiv 119890119894120579k120582

(56)



120591tran =2

Γ12059620

(57)


120591int =2120587

Δ120596 (58)


Δ120596 ≃119888(31205810)13

3

Δ120581

1205810

(59)

where 1205810equiv (13)(120596

0119888)3







2

10

minus1

minus1

minus215 16 17 18 19


cuum

fiel

d (V

m)

002

0

minus002

Vacu

um fi

eld

(Vm

)

times10minus8


times10minus12

ParticleField

1

0

17 171 172 173Time (s)

Time (s)

Posit

ion

(m)

Posit

ion

(m) 120591(X)

mod

)(E120591mod




0) to avoid


119902 = 119890

119898 = 10minus4

119898119890

1205960= 1016 rads

Δ = 220 times Γ1205962

0

(60)


Γ1205962





10minus4





119909 (119905)

=119902

119898120596119877

int

119905

minusinfin

119864vac119909 (1199051015840

) 119890minusΓ1205962

0(119905minus1199051015840)2 sin (120596

119877(119905 minus 1199051015840

)) 1198891199051015840

(61)







)


0for the particle






6

4

2


Position (m)

Prob

abili

ty d

ensit

y

times107

6

4

2



Prob

abili

ty d

ensit

yQMSED

QMSED

times1025

times10minus26

2

1

0

minus1

minus212 18 24

Posit

ion

(m)


times10minus26

2

1

0

minus1

minus2178 1781

Posit

ion

(m)


times10minus26

times10minus8

1782


120596= 2 times 10



119909 (119905) =119902

119898sum

k120582radicℎ120596

1205980119881

119886k120582119861k120582

119890minus119894120596119905

120576k120582119909 (62)




119909 (119905) = 119860 (119905) 119890minus1198941205960119905

119860 (119905) = sum

k120582(120576k120582119909

119902

119898radicℎ120596

1205980119881

119886k120582119861k120582

)119890minus119894(120596minus120596

0)119905

(63)





2

1

0

minus1

minus205 1 15 2 25 3

Posit

ion

(m)

times10minus8

times10minus12

Prob

abili

ty d

ensit

y 6

4

2

0minus3 0 3

times107 15

10

5

0minus3 0 3

times1078

6

4

2

0minus3 0 3

times107

Time (s)




negligible

119860 (119905 + Δ119905) ≃ 119860 (119905) for Δ119905 ≪ 120591coh (64)




3

15

0

minus15


Real

Imag

inar

y

A(t0)

Δ

A(t0 + Δt)



0) (blue



the coherence time



119875 (119909) = sum

119860

119875119860(119909) = int119875

119860(119909) 119891 (119860) 119889119860 (65)







5

0

minus51 3 5 7 9

times10minus8

times10minus11

Posit

ion

(m)

Time (s)

2

0

minus2461 462 463 464

times10minus8

times10minus11

Posit

ion

(m)

Time (s)

10

5


times107


Prob

abili

ty d

ensit

y




(rare)

= 15 times 107

Small amplitude

3120591coh

Δs


119904= 15 times 10




120596= 2 times 10


6

4

2


times107

times10minus8

Prob

abili

ty d

ensit

y

QMSED

120590x ≃ 761 times 10minus9

6

4

2


times1025

times10minus26

Prob

abili

ty d

ensit

y


QMSED

120590p ≃ 693 times 10minus27 5

0

minus50 2 4Po

sitio

n (m

)

Position (m)

Time (s)

times10minus8

times10minus12

Trial-15

0

minus50 2 4

Posit

ion

(m)

Time (s)

times10minus8

times10minus12

Trial-25

0

minus50 2 4

Posit

ion

(m)

Time (s)

times10minus8

times10minus12

Trial-3


119901= 2 times 10


120596= 2 times 10






120596= 5 times 10


119901= 1198791119879119901 where 119879





4

2

0

minus2

minus40 02 04 06 08 1 12 14

Time (s)

Posit

ion

(m)


times10minus8

times10minus12

4

2

0

minus2

minus40 02 04 06 08 1 12 14

Time (s)

Posit

ion

(m)


times10minus12

times10minus26


10 30 50 70 90

10

20

30

40


Com

puta

tion

time (

hr)

Np = 100 k = 05 kNp = 100 k = 1 kNp = 200 k = 05 k

N120596

N120596

N120596


119901(number


0 200 400 600 800 10000

14

12

34

1

Np = 200 k

Ener

gy (ℏ

1205960)

N120596



119901=


2

0






120596


5 Conclusions


120596= 5 times 10




Vacuum field

SEDRadiationdamping

X

X

2

1

0

minus1

minus2

Time (s)

times10minus8

times10minus12

Posit

ion

(m)

2

1

0

minus1

minus2649647645

Time (s)

times10minus8

times10minus13

Posit

ion

(m)

2

1

0

minus1

minus2

times10minus8

Posit

ion

(m)

CMCM

Probability density

SED (theory)

0 5 10

2

1

0

minus1

minus2

times10minus8

Posit

ion

(m)


SED (simulation)

2 3 4 5 6 7

times107

times107














Appendices




E (r 119905)

=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)


B (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896 (k times 120576 (k 120582))




119860(k 120582)119890119894120579(k120582)

E (r 119905)

=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)


+119860lowast



B (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896120585 (k 120582)


+119860lowast


(A2)


⟨119906⟩120579=1205980

2⟨|E|2⟩

120579

+1

21205830

⟨|B|2⟩120579

(A3)


0is the vacuum



and |B|2 using (A2)

|E|2 (r 119905)

= E (r 119905)Elowast (r 119905)

=1

4sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120576 (k 120582) sdot 120576 (k1015840 1205821015840)) 119891

|B|2 (r 119905)

= B (r 119905)Blowast (r 119905)

=1

41198882sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120585 (k 120582) sdot 120585 (k1015840 1205821015840)) 119891

(A4)

where 119891 equiv 119891(k 120582 k1015840 1205821015840 r 119905)

119891 (k 120582 k1015840 1205821015840 r 119905)



+ 119860 (k 120582) 119860 (k1015840 1205821015840) 119890119894(ksdotrminus120596119905+120579(k120582))


+ 119860lowast



+ 119860lowast



(A5)


⟨119890plusmn119894(120579(k120582)+120579(k1015840 1205821015840))

⟩120579

= 0

⟨119890plusmn119894(120579(k120582)minus120579(k1015840 1205821015840))

⟩120579

= 12057512058210158401205821205753

(k1015840 minus k) (A6)


⟨119891 (k 120582 k1015840 1205821015840 r 119905)⟩120579



12057512058210158401205821205753

(k1015840 minus k)

+ 119860lowast



12057512058210158401205821205753

(k1015840 minus k)

(A7)



(A4) and (A7)

⟨|E|2⟩120579

=1

4sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120576 (k 120582) sdot 120576 (k1015840 1205821015840)) ⟨119891⟩120579

=1

2

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2

⟨|B|2⟩120579

=1

41198882sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120585 (k 120582) sdot 120585 (k1015840 1205821015840)) ⟨119891⟩120579

=1

21198882

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2

(A8)


120579

in unbounded space

⟨119906⟩120579=1205980

2

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2 (A9)


119906vac =1

119881sum

k120582

ℎ120596

2

=1

119881sum

k120582

ℎ120596

2(119871119909

2120587Δ119896119909)(

119871119910

2120587Δ119896119910)(

119871119911

2120587Δ119896119911)

=

2

sum

120582=1

1

(2120587)3sum

kΔ3

119896ℎ120596

2

(A10)



3


119906vac =2

sum

120582=1

1

(2120587)3int1198893

119896ℎ120596

2 (A11)


1003816100381610038161003816119860vac (k 120582)10038161003816100381610038162

=ℎ120596

(2120587)3

1205980

(A12)


119860vac (k 120582) = radicℎ120596

812058731205980

(A13)


=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)radic ℎ120596

812058731205980


119890119894120579(k120582)


119890minus119894120579(k120582)

)

Bvac (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896120585 (k 120582)radic ℎ120596

812058731205980


119890119894120579(k120582)


119890minus119894120579(k120582)

)

(A14)

which is (1)


E (r 119905) = 1

2sum

k120582(119860k120582119890



) 120576k120582

B (r 119905) = 1

2119888sum

k120582(119860k120582119890



) 120585k120582

(A15)



120579= ⟨119890119894120579k120582⟩120579⟨119890plusmn119894120579k1015840 1205821015840 ⟩

120579=

0)

⟨119890plusmn119894(120579k120582+120579k1015840 1205821015840 )⟩

120579

= 0

⟨119890plusmn119894(120579k120582minus120579k1015840 1205821015840 )⟩

120579

= 1205751205821015840120582120575k1015840 k

(A16)


⟨119906⟩120579=1205980

2sum

k120582

1003816100381610038161003816119860k12058210038161003816100381610038162

(A17)



119906vac =1

119881sum

k120582

ℎ120596

2 (A18)


119860vack120582 = radicℎ120596

1205980119881 (A19)


Evac (r 119905) =1

2sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120576k120582

Bvac (r 119905) =1

2119888sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120585k120582

(A20)



k = (119909

119910

119911

) = (

0

0

119896

) (B1)


k1 = (cos120594sin1205940

) k2 = (minus sin120594cos1205940

) (B2)


= (119911)

120601(119910)

120579

= (


sin120601 cos120601 0

0 0 1

)(

cos 120579 0 sin 1205790 1 0

minus sin 120579 0 cos 120579)

= (


)

(B3)


2

1

0

minus1

minus2

0 3 6 9 12Time (s)

Fiel

d (V

m)

120591rep

120591env


1=


1 1198792]LCM = 6




) (B4)


120576k1 = k1 = (



120576k2 = k2 = (


sin 120579 sin120594)

(B5)

C Repetitive Time


119864 (119905) =

119873

sum

119896=1

119864119896cos (120596

119896119905) (C1)


119864 (119905 + 120591rep) = 119864 (119905) (C2)

120591rep = [1198791 1198792 119879119873]LCM (C3)

where 119879119896= 2120587120596



120591rep =2120587

(1205961 1205962 120596

119873)GCD

(C4)


1205961+ 1205962+ sdot sdot sdot + 120596

119873

=2120587

1198791

+2120587

1198792


119879119873

=2120587

[1198791 1198792 119879

119873]LCM

(1198991+ 1198992+ sdot sdot sdot + 119899

119873)

(C5)


119899119896=[1198791 1198792 119879

119873]LCM

119879119896

(C6)




Δ120596gcd equiv (1205961 1205962 120596119873)GCD

=2120587

[1198791 1198792 119879

119873]LCM

(C7)


120591rep =2120587

Δ120596gcd(C8)


120591int =2120587

Δ120596 (C9)


Δ120596 (120581) = 120596 (120581) minus 120596 (120581 minus Δ120581)

= 119888(3120581)13

minus 119888(3120581)13

(1 minusΔ120581

120581)

13

(C10)


0and

120581 ≃ 1205810

Δ120581

1205810

= (3

119873k minus 1)Δ

1205960

≪ 1

120581 = 1205810+ 119874(

Δ

1205960

)

(C11)


where 1205810= (13)(120596



Δ120596 ≃119888(31205810)13

3

Δ120581

1205810

(C12)

Acknowledgments


References



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





For 119894 = 1 119873120581 119895 = 1 119873

120599 and 119899 = 1 119873

120593

120581119894119895119899=(1205960minus Δ2)

3

31198883+ (119894 minus 1) Δ120581

120599119894119895119899= minus1 + (119895 minus 1) Δ120599

120593119894119895119899= 119877(0)

119894119895+ (119899 minus 1) Δ120593

(22)

where 119877(0)119894119895


Δ120581 =[(1205960+ Δ2)

3

31198883


3

31198883

]

119873120581minus 1

(23)

Δ120599 =2

119873120599minus 1

Δ120593 =2120587

119873120593

(24)



119894119895119899 120599119894119895119899 120593119894119895119899) is then used


k119894119895119899= (

119896119909

119896119910

119896119911

) = (

119896119894119895119899

sin (120579119894119895119899) cos (120601

119894119895119899)

119896119894119895119899

sin (120579119894119895119899) sin (120601

119894119895119899)

119896119894119895119899

cos (120579119894119895119899)

) (25)

where

119896119894119895119899= (3120581

119894119895119899)13

120579119894119895119899= cosminus1 (120599

119894119895119899)

120601119894119895119899= 120593119894119895119899

(26)



Δ3






Evac =2

sum

120582=1

sum

(120581120599120593)

radicℎ120596







119881 = (2120587)3

120588k = (2120587)3119873k119881k (29)

where

119881k =4120587

3(1205960+ Δ2

119888)

3

minus4120587

3(1205960minus Δ2

119888)

3

(30)


Evac =2

sum

120582=1

119873120581

sum

119894=1

119873120599

sum

119895=1

119873120593

sum

119899=1

radicℎ120596





120599and 119873






120596

k119894= (

119896119909

119896119910

119896119911

) = (

119896119894sin 120579119894cos120601119894

119896119894sin 120579119894sin120601119894

119896119894cos 120579119894

) (32)

where

119896119894= (3120581119894)13

120579119894= cosminus1 (120599

119894)

120601119894= 120593119894

(33)

120581119894=(1205960minus Δ2)

3

31198883+ (119894 minus 1) Δ120581

120599119894= 119877(1)

119894

120593119894= 119877(2)

119894

(34)



119894is a random


119894is







119896= 1205960119888 In this limit the


Γ1205960≪

Δ

1205960

≪ 1 (35)


Evac =2

sum

120582=1

119873120596

sum

119894=1

radicℎ120596






120576k1198941= (

1205761119909

1205761119910

1205761119911

)

= (


119894sin120594119894

cos 120579119894sin120601119894cos120594119894+ cos120601

119894sin120594119894

minus sin 120579119894cos120594119894

)

120576k1198942= (

1205762119909

1205762119910

1205762119911

)

= (


119894cos120594119894


119894cos120594119894

sin 120579119894sin120594119894

)

(37)



119894 120599119894 120593119894 120594119894) which are


1

05

0

minus05

minus1

1050

minus05minus1

1

050

minus05minus1

y

z

x


120596= 3 times 10




119898 = minus1198981205962

0119909 + 119898Γ + 119902119864vac119909 (119905) (38)



1205980119881

1

2(119886k120582119890

minus119894120596119905

+ 119886lowast

k120582119890119894120596119905

) 120576k120582 (39)



119909 (119905) =119902

119898sum

k120582radicℎ120596

1205980119881

1

2(119886k120582119862k120582

119890minus119894120596119905

+119886lowast

k120582119862lowastk120582

119890119894120596119905

) 120576k120582119909 (40)


+1205962

0)minus119894Γ120596




119898 = 119865 + 119898Γ (41)


31198981198883

)(141205871205980) is the


119898 ≃ 119865 (42)


119898Γ ≃ Γ (43)


119898 ≃ 119865 + Γ (44)


119898 ≃ minus1198981205962

0119909 minus 119898Γ120596

2

0 + 119902119864vac119909 (119905) + 119902Γvac119909 (119905) (45)


119874(1198981205962

0119909) = 119898120596

2

01199090 (46)

119874 (119902119864vac119909) = 1198901198640 (47)

119874(119898Γ1205962

0) = (Γ120596

0)1198981205962

01199090 (48)

119874(119902Γvac119909) = (Γ1205960) 1199021198640 (49)

where 1199090and 119864



0because


0and 119864

0 For a fixed time






119863rms = radic119873119904 sdot Δ119904 (50)



Δ119904 = (12)((119902119898Γ1205963

0)radic(ℎ120596




(2120587)3

119873120596119881k and 119881k ≃ 4120587120596

2

0(Γ1205962

0)1198883 ((29) and (30)) the


0≃ radic3120587radicℎ2119898120596

0


1198640≃ 119863(119864)


2radicℎ1205960

1205980119881

1199090≃ 119863(119909)


2(

119902

119898Γ12059630

radicℎ1205960

1205980119881)

(51)


119874(119898Γ1205962

0) ≃ 119902radic

119873120596

2sdot radic

ℎ1205960

1205980119881

119874 (119902Γvac119909) ≃ (Γ1205960) 119902radic119873120596

2sdot radic

ℎ1205960

1205980119881

(52)



119898 ≃ minus1198981205962

0119909 minus 119898Γ120596

2

0 + 119902119864vac119909 (119905) (53)



119874(1198981205962

0119909) ≫ 119874(119898Γ120596

2

0) ≃ 119874 (119902119864vac119909) (54)



2




119909 (119905) = 119890minusΓ1205962

01199052

(119860119890119894120596119877119905

+ 119860lowast

119890minus119894120596119877119905

)

+119902

119898sum

k120582radicℎ120596

1205980119881

1

2(119886k120582119861k120582

119890minus119894120596119905

+119886lowast

k120582119861lowastk120582

119890119894120596119905

) 120576k120582119909(55)


120596119877equiv 1205960


2)

2

119861k120582 equiv (minus1205962

+ 1205962

0) minus 119894 (Γ120596

2

0) 120596

119886k120582 equiv 119890119894120579k120582

(56)



120591tran =2

Γ12059620

(57)


120591int =2120587

Δ120596 (58)


Δ120596 ≃119888(31205810)13

3

Δ120581

1205810

(59)

where 1205810equiv (13)(120596

0119888)3







2

10

minus1

minus1

minus215 16 17 18 19


cuum

fiel

d (V

m)

002

0

minus002

Vacu

um fi

eld

(Vm

)

times10minus8


times10minus12

ParticleField

1

0

17 171 172 173Time (s)

Time (s)

Posit

ion

(m)

Posit

ion

(m) 120591(X)

mod

)(E120591mod




0) to avoid


119902 = 119890

119898 = 10minus4

119898119890

1205960= 1016 rads

Δ = 220 times Γ1205962

0

(60)


Γ1205962





10minus4





119909 (119905)

=119902

119898120596119877

int

119905

minusinfin

119864vac119909 (1199051015840

) 119890minusΓ1205962

0(119905minus1199051015840)2 sin (120596

119877(119905 minus 1199051015840

)) 1198891199051015840

(61)







)


0for the particle






6

4

2


Position (m)

Prob

abili

ty d

ensit

y

times107

6

4

2



Prob

abili

ty d

ensit

yQMSED

QMSED

times1025

times10minus26

2

1

0

minus1

minus212 18 24

Posit

ion

(m)


times10minus26

2

1

0

minus1

minus2178 1781

Posit

ion

(m)


times10minus26

times10minus8

1782


120596= 2 times 10



119909 (119905) =119902

119898sum

k120582radicℎ120596

1205980119881

119886k120582119861k120582

119890minus119894120596119905

120576k120582119909 (62)




119909 (119905) = 119860 (119905) 119890minus1198941205960119905

119860 (119905) = sum

k120582(120576k120582119909

119902

119898radicℎ120596

1205980119881

119886k120582119861k120582

)119890minus119894(120596minus120596

0)119905

(63)





2

1

0

minus1

minus205 1 15 2 25 3

Posit

ion

(m)

times10minus8

times10minus12

Prob

abili

ty d

ensit

y 6

4

2

0minus3 0 3

times107 15

10

5

0minus3 0 3

times1078

6

4

2

0minus3 0 3

times107

Time (s)




negligible

119860 (119905 + Δ119905) ≃ 119860 (119905) for Δ119905 ≪ 120591coh (64)




3

15

0

minus15


Real

Imag

inar

y

A(t0)

Δ

A(t0 + Δt)



0) (blue



the coherence time



119875 (119909) = sum

119860

119875119860(119909) = int119875

119860(119909) 119891 (119860) 119889119860 (65)







5

0

minus51 3 5 7 9

times10minus8

times10minus11

Posit

ion

(m)

Time (s)

2

0

minus2461 462 463 464

times10minus8

times10minus11

Posit

ion

(m)

Time (s)

10

5


times107


Prob

abili

ty d

ensit

y




(rare)

= 15 times 107

Small amplitude

3120591coh

Δs


119904= 15 times 10




120596= 2 times 10


6

4

2


times107

times10minus8

Prob

abili

ty d

ensit

y

QMSED

120590x ≃ 761 times 10minus9

6

4

2


times1025

times10minus26

Prob

abili

ty d

ensit

y


QMSED

120590p ≃ 693 times 10minus27 5

0

minus50 2 4Po

sitio

n (m

)

Position (m)

Time (s)

times10minus8

times10minus12

Trial-15

0

minus50 2 4

Posit

ion

(m)

Time (s)

times10minus8

times10minus12

Trial-25

0

minus50 2 4

Posit

ion

(m)

Time (s)

times10minus8

times10minus12

Trial-3


119901= 2 times 10


120596= 2 times 10






120596= 5 times 10


119901= 1198791119879119901 where 119879





4

2

0

minus2

minus40 02 04 06 08 1 12 14

Time (s)

Posit

ion

(m)


times10minus8

times10minus12

4

2

0

minus2

minus40 02 04 06 08 1 12 14

Time (s)

Posit

ion

(m)


times10minus12

times10minus26


10 30 50 70 90

10

20

30

40


Com

puta

tion

time (

hr)

Np = 100 k = 05 kNp = 100 k = 1 kNp = 200 k = 05 k

N120596

N120596

N120596


119901(number


0 200 400 600 800 10000

14

12

34

1

Np = 200 k

Ener

gy (ℏ

1205960)

N120596



119901=


2

0






120596


5 Conclusions


120596= 5 times 10




Vacuum field

SEDRadiationdamping

X

X

2

1

0

minus1

minus2

Time (s)

times10minus8

times10minus12

Posit

ion

(m)

2

1

0

minus1

minus2649647645

Time (s)

times10minus8

times10minus13

Posit

ion

(m)

2

1

0

minus1

minus2

times10minus8

Posit

ion

(m)

CMCM

Probability density

SED (theory)

0 5 10

2

1

0

minus1

minus2

times10minus8

Posit

ion

(m)


SED (simulation)

2 3 4 5 6 7

times107

times107














Appendices




E (r 119905)

=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)


B (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896 (k times 120576 (k 120582))




119860(k 120582)119890119894120579(k120582)

E (r 119905)

=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)


+119860lowast



B (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896120585 (k 120582)


+119860lowast


(A2)


⟨119906⟩120579=1205980

2⟨|E|2⟩

120579

+1

21205830

⟨|B|2⟩120579

(A3)


0is the vacuum



and |B|2 using (A2)

|E|2 (r 119905)

= E (r 119905)Elowast (r 119905)

=1

4sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120576 (k 120582) sdot 120576 (k1015840 1205821015840)) 119891

|B|2 (r 119905)

= B (r 119905)Blowast (r 119905)

=1

41198882sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120585 (k 120582) sdot 120585 (k1015840 1205821015840)) 119891

(A4)

where 119891 equiv 119891(k 120582 k1015840 1205821015840 r 119905)

119891 (k 120582 k1015840 1205821015840 r 119905)



+ 119860 (k 120582) 119860 (k1015840 1205821015840) 119890119894(ksdotrminus120596119905+120579(k120582))


+ 119860lowast



+ 119860lowast



(A5)


⟨119890plusmn119894(120579(k120582)+120579(k1015840 1205821015840))

⟩120579

= 0

⟨119890plusmn119894(120579(k120582)minus120579(k1015840 1205821015840))

⟩120579

= 12057512058210158401205821205753

(k1015840 minus k) (A6)


⟨119891 (k 120582 k1015840 1205821015840 r 119905)⟩120579



12057512058210158401205821205753

(k1015840 minus k)

+ 119860lowast



12057512058210158401205821205753

(k1015840 minus k)

(A7)



(A4) and (A7)

⟨|E|2⟩120579

=1

4sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120576 (k 120582) sdot 120576 (k1015840 1205821015840)) ⟨119891⟩120579

=1

2

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2

⟨|B|2⟩120579

=1

41198882sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120585 (k 120582) sdot 120585 (k1015840 1205821015840)) ⟨119891⟩120579

=1

21198882

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2

(A8)


120579

in unbounded space

⟨119906⟩120579=1205980

2

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2 (A9)


119906vac =1

119881sum

k120582

ℎ120596

2

=1

119881sum

k120582

ℎ120596

2(119871119909

2120587Δ119896119909)(

119871119910

2120587Δ119896119910)(

119871119911

2120587Δ119896119911)

=

2

sum

120582=1

1

(2120587)3sum

kΔ3

119896ℎ120596

2

(A10)



3


119906vac =2

sum

120582=1

1

(2120587)3int1198893

119896ℎ120596

2 (A11)


1003816100381610038161003816119860vac (k 120582)10038161003816100381610038162

=ℎ120596

(2120587)3

1205980

(A12)


119860vac (k 120582) = radicℎ120596

812058731205980

(A13)


=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)radic ℎ120596

812058731205980


119890119894120579(k120582)


119890minus119894120579(k120582)

)

Bvac (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896120585 (k 120582)radic ℎ120596

812058731205980


119890119894120579(k120582)


119890minus119894120579(k120582)

)

(A14)

which is (1)


E (r 119905) = 1

2sum

k120582(119860k120582119890



) 120576k120582

B (r 119905) = 1

2119888sum

k120582(119860k120582119890



) 120585k120582

(A15)



120579= ⟨119890119894120579k120582⟩120579⟨119890plusmn119894120579k1015840 1205821015840 ⟩

120579=

0)

⟨119890plusmn119894(120579k120582+120579k1015840 1205821015840 )⟩

120579

= 0

⟨119890plusmn119894(120579k120582minus120579k1015840 1205821015840 )⟩

120579

= 1205751205821015840120582120575k1015840 k

(A16)


⟨119906⟩120579=1205980

2sum

k120582

1003816100381610038161003816119860k12058210038161003816100381610038162

(A17)



119906vac =1

119881sum

k120582

ℎ120596

2 (A18)


119860vack120582 = radicℎ120596

1205980119881 (A19)


Evac (r 119905) =1

2sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120576k120582

Bvac (r 119905) =1

2119888sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120585k120582

(A20)



k = (119909

119910

119911

) = (

0

0

119896

) (B1)


k1 = (cos120594sin1205940

) k2 = (minus sin120594cos1205940

) (B2)


= (119911)

120601(119910)

120579

= (


sin120601 cos120601 0

0 0 1

)(

cos 120579 0 sin 1205790 1 0

minus sin 120579 0 cos 120579)

= (


)

(B3)


2

1

0

minus1

minus2

0 3 6 9 12Time (s)

Fiel

d (V

m)

120591rep

120591env


1=


1 1198792]LCM = 6




) (B4)


120576k1 = k1 = (



120576k2 = k2 = (


sin 120579 sin120594)

(B5)

C Repetitive Time


119864 (119905) =

119873

sum

119896=1

119864119896cos (120596

119896119905) (C1)


119864 (119905 + 120591rep) = 119864 (119905) (C2)

120591rep = [1198791 1198792 119879119873]LCM (C3)

where 119879119896= 2120587120596



120591rep =2120587

(1205961 1205962 120596

119873)GCD

(C4)


1205961+ 1205962+ sdot sdot sdot + 120596

119873

=2120587

1198791

+2120587

1198792


119879119873

=2120587

[1198791 1198792 119879

119873]LCM

(1198991+ 1198992+ sdot sdot sdot + 119899

119873)

(C5)


119899119896=[1198791 1198792 119879

119873]LCM

119879119896

(C6)




Δ120596gcd equiv (1205961 1205962 120596119873)GCD

=2120587

[1198791 1198792 119879

119873]LCM

(C7)


120591rep =2120587

Δ120596gcd(C8)


120591int =2120587

Δ120596 (C9)


Δ120596 (120581) = 120596 (120581) minus 120596 (120581 minus Δ120581)

= 119888(3120581)13

minus 119888(3120581)13

(1 minusΔ120581

120581)

13

(C10)


0and

120581 ≃ 1205810

Δ120581

1205810

= (3

119873k minus 1)Δ

1205960

≪ 1

120581 = 1205810+ 119874(

Δ

1205960

)

(C11)


where 1205810= (13)(120596



Δ120596 ≃119888(31205810)13

3

Δ120581

1205810

(C12)

Acknowledgments


References



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119894is a random


119894is







119896= 1205960119888 In this limit the


Γ1205960≪

Δ

1205960

≪ 1 (35)


Evac =2

sum

120582=1

119873120596

sum

119894=1

radicℎ120596






120576k1198941= (

1205761119909

1205761119910

1205761119911

)

= (


119894sin120594119894

cos 120579119894sin120601119894cos120594119894+ cos120601

119894sin120594119894

minus sin 120579119894cos120594119894

)

120576k1198942= (

1205762119909

1205762119910

1205762119911

)

= (


119894cos120594119894


119894cos120594119894

sin 120579119894sin120594119894

)

(37)



119894 120599119894 120593119894 120594119894) which are


1

05

0

minus05

minus1

1050

minus05minus1

1

050

minus05minus1

y

z

x


120596= 3 times 10




119898 = minus1198981205962

0119909 + 119898Γ + 119902119864vac119909 (119905) (38)



1205980119881

1

2(119886k120582119890

minus119894120596119905

+ 119886lowast

k120582119890119894120596119905

) 120576k120582 (39)



119909 (119905) =119902

119898sum

k120582radicℎ120596

1205980119881

1

2(119886k120582119862k120582

119890minus119894120596119905

+119886lowast

k120582119862lowastk120582

119890119894120596119905

) 120576k120582119909 (40)


+1205962

0)minus119894Γ120596




119898 = 119865 + 119898Γ (41)


31198981198883

)(141205871205980) is the


119898 ≃ 119865 (42)


119898Γ ≃ Γ (43)


119898 ≃ 119865 + Γ (44)


119898 ≃ minus1198981205962

0119909 minus 119898Γ120596

2

0 + 119902119864vac119909 (119905) + 119902Γvac119909 (119905) (45)


119874(1198981205962

0119909) = 119898120596

2

01199090 (46)

119874 (119902119864vac119909) = 1198901198640 (47)

119874(119898Γ1205962

0) = (Γ120596

0)1198981205962

01199090 (48)

119874(119902Γvac119909) = (Γ1205960) 1199021198640 (49)

where 1199090and 119864



0because


0and 119864

0 For a fixed time






119863rms = radic119873119904 sdot Δ119904 (50)



Δ119904 = (12)((119902119898Γ1205963

0)radic(ℎ120596




(2120587)3

119873120596119881k and 119881k ≃ 4120587120596

2

0(Γ1205962

0)1198883 ((29) and (30)) the


0≃ radic3120587radicℎ2119898120596

0


1198640≃ 119863(119864)


2radicℎ1205960

1205980119881

1199090≃ 119863(119909)


2(

119902

119898Γ12059630

radicℎ1205960

1205980119881)

(51)


119874(119898Γ1205962

0) ≃ 119902radic

119873120596

2sdot radic

ℎ1205960

1205980119881

119874 (119902Γvac119909) ≃ (Γ1205960) 119902radic119873120596

2sdot radic

ℎ1205960

1205980119881

(52)



119898 ≃ minus1198981205962

0119909 minus 119898Γ120596

2

0 + 119902119864vac119909 (119905) (53)



119874(1198981205962

0119909) ≫ 119874(119898Γ120596

2

0) ≃ 119874 (119902119864vac119909) (54)



2




119909 (119905) = 119890minusΓ1205962

01199052

(119860119890119894120596119877119905

+ 119860lowast

119890minus119894120596119877119905

)

+119902

119898sum

k120582radicℎ120596

1205980119881

1

2(119886k120582119861k120582

119890minus119894120596119905

+119886lowast

k120582119861lowastk120582

119890119894120596119905

) 120576k120582119909(55)


120596119877equiv 1205960


2)

2

119861k120582 equiv (minus1205962

+ 1205962

0) minus 119894 (Γ120596

2

0) 120596

119886k120582 equiv 119890119894120579k120582

(56)



120591tran =2

Γ12059620

(57)


120591int =2120587

Δ120596 (58)


Δ120596 ≃119888(31205810)13

3

Δ120581

1205810

(59)

where 1205810equiv (13)(120596

0119888)3







2

10

minus1

minus1

minus215 16 17 18 19


cuum

fiel

d (V

m)

002

0

minus002

Vacu

um fi

eld

(Vm

)

times10minus8


times10minus12

ParticleField

1

0

17 171 172 173Time (s)

Time (s)

Posit

ion

(m)

Posit

ion

(m) 120591(X)

mod

)(E120591mod




0) to avoid


119902 = 119890

119898 = 10minus4

119898119890

1205960= 1016 rads

Δ = 220 times Γ1205962

0

(60)


Γ1205962





10minus4





119909 (119905)

=119902

119898120596119877

int

119905

minusinfin

119864vac119909 (1199051015840

) 119890minusΓ1205962

0(119905minus1199051015840)2 sin (120596

119877(119905 minus 1199051015840

)) 1198891199051015840

(61)







)


0for the particle






6

4

2


Position (m)

Prob

abili

ty d

ensit

y

times107

6

4

2



Prob

abili

ty d

ensit

yQMSED

QMSED

times1025

times10minus26

2

1

0

minus1

minus212 18 24

Posit

ion

(m)


times10minus26

2

1

0

minus1

minus2178 1781

Posit

ion

(m)


times10minus26

times10minus8

1782


120596= 2 times 10



119909 (119905) =119902

119898sum

k120582radicℎ120596

1205980119881

119886k120582119861k120582

119890minus119894120596119905

120576k120582119909 (62)




119909 (119905) = 119860 (119905) 119890minus1198941205960119905

119860 (119905) = sum

k120582(120576k120582119909

119902

119898radicℎ120596

1205980119881

119886k120582119861k120582

)119890minus119894(120596minus120596

0)119905

(63)





2

1

0

minus1

minus205 1 15 2 25 3

Posit

ion

(m)

times10minus8

times10minus12

Prob

abili

ty d

ensit

y 6

4

2

0minus3 0 3

times107 15

10

5

0minus3 0 3

times1078

6

4

2

0minus3 0 3

times107

Time (s)




negligible

119860 (119905 + Δ119905) ≃ 119860 (119905) for Δ119905 ≪ 120591coh (64)




3

15

0

minus15


Real

Imag

inar

y

A(t0)

Δ

A(t0 + Δt)



0) (blue



the coherence time



119875 (119909) = sum

119860

119875119860(119909) = int119875

119860(119909) 119891 (119860) 119889119860 (65)







5

0

minus51 3 5 7 9

times10minus8

times10minus11

Posit

ion

(m)

Time (s)

2

0

minus2461 462 463 464

times10minus8

times10minus11

Posit

ion

(m)

Time (s)

10

5


times107


Prob

abili

ty d

ensit

y




(rare)

= 15 times 107

Small amplitude

3120591coh

Δs


119904= 15 times 10




120596= 2 times 10


6

4

2


times107

times10minus8

Prob

abili

ty d

ensit

y

QMSED

120590x ≃ 761 times 10minus9

6

4

2


times1025

times10minus26

Prob

abili

ty d

ensit

y


QMSED

120590p ≃ 693 times 10minus27 5

0

minus50 2 4Po

sitio

n (m

)

Position (m)

Time (s)

times10minus8

times10minus12

Trial-15

0

minus50 2 4

Posit

ion

(m)

Time (s)

times10minus8

times10minus12

Trial-25

0

minus50 2 4

Posit

ion

(m)

Time (s)

times10minus8

times10minus12

Trial-3


119901= 2 times 10


120596= 2 times 10






120596= 5 times 10


119901= 1198791119879119901 where 119879





4

2

0

minus2

minus40 02 04 06 08 1 12 14

Time (s)

Posit

ion

(m)


times10minus8

times10minus12

4

2

0

minus2

minus40 02 04 06 08 1 12 14

Time (s)

Posit

ion

(m)


times10minus12

times10minus26


10 30 50 70 90

10

20

30

40


Com

puta

tion

time (

hr)

Np = 100 k = 05 kNp = 100 k = 1 kNp = 200 k = 05 k

N120596

N120596

N120596


119901(number


0 200 400 600 800 10000

14

12

34

1

Np = 200 k

Ener

gy (ℏ

1205960)

N120596



119901=


2

0






120596


5 Conclusions


120596= 5 times 10




Vacuum field

SEDRadiationdamping

X

X

2

1

0

minus1

minus2

Time (s)

times10minus8

times10minus12

Posit

ion

(m)

2

1

0

minus1

minus2649647645

Time (s)

times10minus8

times10minus13

Posit

ion

(m)

2

1

0

minus1

minus2

times10minus8

Posit

ion

(m)

CMCM

Probability density

SED (theory)

0 5 10

2

1

0

minus1

minus2

times10minus8

Posit

ion

(m)


SED (simulation)

2 3 4 5 6 7

times107

times107














Appendices




E (r 119905)

=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)


B (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896 (k times 120576 (k 120582))




119860(k 120582)119890119894120579(k120582)

E (r 119905)

=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)


+119860lowast



B (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896120585 (k 120582)


+119860lowast


(A2)


⟨119906⟩120579=1205980

2⟨|E|2⟩

120579

+1

21205830

⟨|B|2⟩120579

(A3)


0is the vacuum



and |B|2 using (A2)

|E|2 (r 119905)

= E (r 119905)Elowast (r 119905)

=1

4sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120576 (k 120582) sdot 120576 (k1015840 1205821015840)) 119891

|B|2 (r 119905)

= B (r 119905)Blowast (r 119905)

=1

41198882sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120585 (k 120582) sdot 120585 (k1015840 1205821015840)) 119891

(A4)

where 119891 equiv 119891(k 120582 k1015840 1205821015840 r 119905)

119891 (k 120582 k1015840 1205821015840 r 119905)



+ 119860 (k 120582) 119860 (k1015840 1205821015840) 119890119894(ksdotrminus120596119905+120579(k120582))


+ 119860lowast



+ 119860lowast



(A5)


⟨119890plusmn119894(120579(k120582)+120579(k1015840 1205821015840))

⟩120579

= 0

⟨119890plusmn119894(120579(k120582)minus120579(k1015840 1205821015840))

⟩120579

= 12057512058210158401205821205753

(k1015840 minus k) (A6)


⟨119891 (k 120582 k1015840 1205821015840 r 119905)⟩120579



12057512058210158401205821205753

(k1015840 minus k)

+ 119860lowast



12057512058210158401205821205753

(k1015840 minus k)

(A7)



(A4) and (A7)

⟨|E|2⟩120579

=1

4sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120576 (k 120582) sdot 120576 (k1015840 1205821015840)) ⟨119891⟩120579

=1

2

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2

⟨|B|2⟩120579

=1

41198882sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120585 (k 120582) sdot 120585 (k1015840 1205821015840)) ⟨119891⟩120579

=1

21198882

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2

(A8)


120579

in unbounded space

⟨119906⟩120579=1205980

2

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2 (A9)


119906vac =1

119881sum

k120582

ℎ120596

2

=1

119881sum

k120582

ℎ120596

2(119871119909

2120587Δ119896119909)(

119871119910

2120587Δ119896119910)(

119871119911

2120587Δ119896119911)

=

2

sum

120582=1

1

(2120587)3sum

kΔ3

119896ℎ120596

2

(A10)



3


119906vac =2

sum

120582=1

1

(2120587)3int1198893

119896ℎ120596

2 (A11)


1003816100381610038161003816119860vac (k 120582)10038161003816100381610038162

=ℎ120596

(2120587)3

1205980

(A12)


119860vac (k 120582) = radicℎ120596

812058731205980

(A13)


=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)radic ℎ120596

812058731205980


119890119894120579(k120582)


119890minus119894120579(k120582)

)

Bvac (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896120585 (k 120582)radic ℎ120596

812058731205980


119890119894120579(k120582)


119890minus119894120579(k120582)

)

(A14)

which is (1)


E (r 119905) = 1

2sum

k120582(119860k120582119890



) 120576k120582

B (r 119905) = 1

2119888sum

k120582(119860k120582119890



) 120585k120582

(A15)



120579= ⟨119890119894120579k120582⟩120579⟨119890plusmn119894120579k1015840 1205821015840 ⟩

120579=

0)

⟨119890plusmn119894(120579k120582+120579k1015840 1205821015840 )⟩

120579

= 0

⟨119890plusmn119894(120579k120582minus120579k1015840 1205821015840 )⟩

120579

= 1205751205821015840120582120575k1015840 k

(A16)


⟨119906⟩120579=1205980

2sum

k120582

1003816100381610038161003816119860k12058210038161003816100381610038162

(A17)



119906vac =1

119881sum

k120582

ℎ120596

2 (A18)


119860vack120582 = radicℎ120596

1205980119881 (A19)


Evac (r 119905) =1

2sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120576k120582

Bvac (r 119905) =1

2119888sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120585k120582

(A20)



k = (119909

119910

119911

) = (

0

0

119896

) (B1)


k1 = (cos120594sin1205940

) k2 = (minus sin120594cos1205940

) (B2)


= (119911)

120601(119910)

120579

= (


sin120601 cos120601 0

0 0 1

)(

cos 120579 0 sin 1205790 1 0

minus sin 120579 0 cos 120579)

= (


)

(B3)


2

1

0

minus1

minus2

0 3 6 9 12Time (s)

Fiel

d (V

m)

120591rep

120591env


1=


1 1198792]LCM = 6




) (B4)


120576k1 = k1 = (



120576k2 = k2 = (


sin 120579 sin120594)

(B5)

C Repetitive Time


119864 (119905) =

119873

sum

119896=1

119864119896cos (120596

119896119905) (C1)


119864 (119905 + 120591rep) = 119864 (119905) (C2)

120591rep = [1198791 1198792 119879119873]LCM (C3)

where 119879119896= 2120587120596



120591rep =2120587

(1205961 1205962 120596

119873)GCD

(C4)


1205961+ 1205962+ sdot sdot sdot + 120596

119873

=2120587

1198791

+2120587

1198792


119879119873

=2120587

[1198791 1198792 119879

119873]LCM

(1198991+ 1198992+ sdot sdot sdot + 119899

119873)

(C5)


119899119896=[1198791 1198792 119879

119873]LCM

119879119896

(C6)




Δ120596gcd equiv (1205961 1205962 120596119873)GCD

=2120587

[1198791 1198792 119879

119873]LCM

(C7)


120591rep =2120587

Δ120596gcd(C8)


120591int =2120587

Δ120596 (C9)


Δ120596 (120581) = 120596 (120581) minus 120596 (120581 minus Δ120581)

= 119888(3120581)13

minus 119888(3120581)13

(1 minusΔ120581

120581)

13

(C10)


0and

120581 ≃ 1205810

Δ120581

1205810

= (3

119873k minus 1)Δ

1205960

≪ 1

120581 = 1205810+ 119874(

Δ

1205960

)

(C11)


where 1205810= (13)(120596



Δ120596 ≃119888(31205810)13

3

Δ120581

1205810

(C12)

Acknowledgments


References



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119898 = 119865 + 119898Γ (41)


31198981198883

)(141205871205980) is the


119898 ≃ 119865 (42)


119898Γ ≃ Γ (43)


119898 ≃ 119865 + Γ (44)


119898 ≃ minus1198981205962

0119909 minus 119898Γ120596

2

0 + 119902119864vac119909 (119905) + 119902Γvac119909 (119905) (45)


119874(1198981205962

0119909) = 119898120596

2

01199090 (46)

119874 (119902119864vac119909) = 1198901198640 (47)

119874(119898Γ1205962

0) = (Γ120596

0)1198981205962

01199090 (48)

119874(119902Γvac119909) = (Γ1205960) 1199021198640 (49)

where 1199090and 119864



0because


0and 119864

0 For a fixed time






119863rms = radic119873119904 sdot Δ119904 (50)



Δ119904 = (12)((119902119898Γ1205963

0)radic(ℎ120596




(2120587)3

119873120596119881k and 119881k ≃ 4120587120596

2

0(Γ1205962

0)1198883 ((29) and (30)) the


0≃ radic3120587radicℎ2119898120596

0


1198640≃ 119863(119864)


2radicℎ1205960

1205980119881

1199090≃ 119863(119909)


2(

119902

119898Γ12059630

radicℎ1205960

1205980119881)

(51)


119874(119898Γ1205962

0) ≃ 119902radic

119873120596

2sdot radic

ℎ1205960

1205980119881

119874 (119902Γvac119909) ≃ (Γ1205960) 119902radic119873120596

2sdot radic

ℎ1205960

1205980119881

(52)



119898 ≃ minus1198981205962

0119909 minus 119898Γ120596

2

0 + 119902119864vac119909 (119905) (53)



119874(1198981205962

0119909) ≫ 119874(119898Γ120596

2

0) ≃ 119874 (119902119864vac119909) (54)



2




119909 (119905) = 119890minusΓ1205962

01199052

(119860119890119894120596119877119905

+ 119860lowast

119890minus119894120596119877119905

)

+119902

119898sum

k120582radicℎ120596

1205980119881

1

2(119886k120582119861k120582

119890minus119894120596119905

+119886lowast

k120582119861lowastk120582

119890119894120596119905

) 120576k120582119909(55)


120596119877equiv 1205960


2)

2

119861k120582 equiv (minus1205962

+ 1205962

0) minus 119894 (Γ120596

2

0) 120596

119886k120582 equiv 119890119894120579k120582

(56)



120591tran =2

Γ12059620

(57)


120591int =2120587

Δ120596 (58)


Δ120596 ≃119888(31205810)13

3

Δ120581

1205810

(59)

where 1205810equiv (13)(120596

0119888)3







2

10

minus1

minus1

minus215 16 17 18 19


cuum

fiel

d (V

m)

002

0

minus002

Vacu

um fi

eld

(Vm

)

times10minus8


times10minus12

ParticleField

1

0

17 171 172 173Time (s)

Time (s)

Posit

ion

(m)

Posit

ion

(m) 120591(X)

mod

)(E120591mod




0) to avoid


119902 = 119890

119898 = 10minus4

119898119890

1205960= 1016 rads

Δ = 220 times Γ1205962

0

(60)


Γ1205962





10minus4





119909 (119905)

=119902

119898120596119877

int

119905

minusinfin

119864vac119909 (1199051015840

) 119890minusΓ1205962

0(119905minus1199051015840)2 sin (120596

119877(119905 minus 1199051015840

)) 1198891199051015840

(61)







)


0for the particle






6

4

2


Position (m)

Prob

abili

ty d

ensit

y

times107

6

4

2



Prob

abili

ty d

ensit

yQMSED

QMSED

times1025

times10minus26

2

1

0

minus1

minus212 18 24

Posit

ion

(m)


times10minus26

2

1

0

minus1

minus2178 1781

Posit

ion

(m)


times10minus26

times10minus8

1782


120596= 2 times 10



119909 (119905) =119902

119898sum

k120582radicℎ120596

1205980119881

119886k120582119861k120582

119890minus119894120596119905

120576k120582119909 (62)




119909 (119905) = 119860 (119905) 119890minus1198941205960119905

119860 (119905) = sum

k120582(120576k120582119909

119902

119898radicℎ120596

1205980119881

119886k120582119861k120582

)119890minus119894(120596minus120596

0)119905

(63)





2

1

0

minus1

minus205 1 15 2 25 3

Posit

ion

(m)

times10minus8

times10minus12

Prob

abili

ty d

ensit

y 6

4

2

0minus3 0 3

times107 15

10

5

0minus3 0 3

times1078

6

4

2

0minus3 0 3

times107

Time (s)




negligible

119860 (119905 + Δ119905) ≃ 119860 (119905) for Δ119905 ≪ 120591coh (64)




3

15

0

minus15


Real

Imag

inar

y

A(t0)

Δ

A(t0 + Δt)



0) (blue



the coherence time



119875 (119909) = sum

119860

119875119860(119909) = int119875

119860(119909) 119891 (119860) 119889119860 (65)







5

0

minus51 3 5 7 9

times10minus8

times10minus11

Posit

ion

(m)

Time (s)

2

0

minus2461 462 463 464

times10minus8

times10minus11

Posit

ion

(m)

Time (s)

10

5


times107


Prob

abili

ty d

ensit

y




(rare)

= 15 times 107

Small amplitude

3120591coh

Δs


119904= 15 times 10




120596= 2 times 10


6

4

2


times107

times10minus8

Prob

abili

ty d

ensit

y

QMSED

120590x ≃ 761 times 10minus9

6

4

2


times1025

times10minus26

Prob

abili

ty d

ensit

y


QMSED

120590p ≃ 693 times 10minus27 5

0

minus50 2 4Po

sitio

n (m

)

Position (m)

Time (s)

times10minus8

times10minus12

Trial-15

0

minus50 2 4

Posit

ion

(m)

Time (s)

times10minus8

times10minus12

Trial-25

0

minus50 2 4

Posit

ion

(m)

Time (s)

times10minus8

times10minus12

Trial-3


119901= 2 times 10


120596= 2 times 10






120596= 5 times 10


119901= 1198791119879119901 where 119879





4

2

0

minus2

minus40 02 04 06 08 1 12 14

Time (s)

Posit

ion

(m)


times10minus8

times10minus12

4

2

0

minus2

minus40 02 04 06 08 1 12 14

Time (s)

Posit

ion

(m)


times10minus12

times10minus26


10 30 50 70 90

10

20

30

40


Com

puta

tion

time (

hr)

Np = 100 k = 05 kNp = 100 k = 1 kNp = 200 k = 05 k

N120596

N120596

N120596


119901(number


0 200 400 600 800 10000

14

12

34

1

Np = 200 k

Ener

gy (ℏ

1205960)

N120596



119901=


2

0






120596


5 Conclusions


120596= 5 times 10




Vacuum field

SEDRadiationdamping

X

X

2

1

0

minus1

minus2

Time (s)

times10minus8

times10minus12

Posit

ion

(m)

2

1

0

minus1

minus2649647645

Time (s)

times10minus8

times10minus13

Posit

ion

(m)

2

1

0

minus1

minus2

times10minus8

Posit

ion

(m)

CMCM

Probability density

SED (theory)

0 5 10

2

1

0

minus1

minus2

times10minus8

Posit

ion

(m)


SED (simulation)

2 3 4 5 6 7

times107

times107














Appendices




E (r 119905)

=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)


B (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896 (k times 120576 (k 120582))




119860(k 120582)119890119894120579(k120582)

E (r 119905)

=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)


+119860lowast



B (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896120585 (k 120582)


+119860lowast


(A2)


⟨119906⟩120579=1205980

2⟨|E|2⟩

120579

+1

21205830

⟨|B|2⟩120579

(A3)


0is the vacuum



and |B|2 using (A2)

|E|2 (r 119905)

= E (r 119905)Elowast (r 119905)

=1

4sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120576 (k 120582) sdot 120576 (k1015840 1205821015840)) 119891

|B|2 (r 119905)

= B (r 119905)Blowast (r 119905)

=1

41198882sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120585 (k 120582) sdot 120585 (k1015840 1205821015840)) 119891

(A4)

where 119891 equiv 119891(k 120582 k1015840 1205821015840 r 119905)

119891 (k 120582 k1015840 1205821015840 r 119905)



+ 119860 (k 120582) 119860 (k1015840 1205821015840) 119890119894(ksdotrminus120596119905+120579(k120582))


+ 119860lowast



+ 119860lowast



(A5)


⟨119890plusmn119894(120579(k120582)+120579(k1015840 1205821015840))

⟩120579

= 0

⟨119890plusmn119894(120579(k120582)minus120579(k1015840 1205821015840))

⟩120579

= 12057512058210158401205821205753

(k1015840 minus k) (A6)


⟨119891 (k 120582 k1015840 1205821015840 r 119905)⟩120579



12057512058210158401205821205753

(k1015840 minus k)

+ 119860lowast



12057512058210158401205821205753

(k1015840 minus k)

(A7)



(A4) and (A7)

⟨|E|2⟩120579

=1

4sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120576 (k 120582) sdot 120576 (k1015840 1205821015840)) ⟨119891⟩120579

=1

2

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2

⟨|B|2⟩120579

=1

41198882sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120585 (k 120582) sdot 120585 (k1015840 1205821015840)) ⟨119891⟩120579

=1

21198882

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2

(A8)


120579

in unbounded space

⟨119906⟩120579=1205980

2

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2 (A9)


119906vac =1

119881sum

k120582

ℎ120596

2

=1

119881sum

k120582

ℎ120596

2(119871119909

2120587Δ119896119909)(

119871119910

2120587Δ119896119910)(

119871119911

2120587Δ119896119911)

=

2

sum

120582=1

1

(2120587)3sum

kΔ3

119896ℎ120596

2

(A10)



3


119906vac =2

sum

120582=1

1

(2120587)3int1198893

119896ℎ120596

2 (A11)


1003816100381610038161003816119860vac (k 120582)10038161003816100381610038162

=ℎ120596

(2120587)3

1205980

(A12)


119860vac (k 120582) = radicℎ120596

812058731205980

(A13)


=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)radic ℎ120596

812058731205980


119890119894120579(k120582)


119890minus119894120579(k120582)

)

Bvac (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896120585 (k 120582)radic ℎ120596

812058731205980


119890119894120579(k120582)


119890minus119894120579(k120582)

)

(A14)

which is (1)


E (r 119905) = 1

2sum

k120582(119860k120582119890



) 120576k120582

B (r 119905) = 1

2119888sum

k120582(119860k120582119890



) 120585k120582

(A15)



120579= ⟨119890119894120579k120582⟩120579⟨119890plusmn119894120579k1015840 1205821015840 ⟩

120579=

0)

⟨119890plusmn119894(120579k120582+120579k1015840 1205821015840 )⟩

120579

= 0

⟨119890plusmn119894(120579k120582minus120579k1015840 1205821015840 )⟩

120579

= 1205751205821015840120582120575k1015840 k

(A16)


⟨119906⟩120579=1205980

2sum

k120582

1003816100381610038161003816119860k12058210038161003816100381610038162

(A17)



119906vac =1

119881sum

k120582

ℎ120596

2 (A18)


119860vack120582 = radicℎ120596

1205980119881 (A19)


Evac (r 119905) =1

2sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120576k120582

Bvac (r 119905) =1

2119888sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120585k120582

(A20)



k = (119909

119910

119911

) = (

0

0

119896

) (B1)


k1 = (cos120594sin1205940

) k2 = (minus sin120594cos1205940

) (B2)


= (119911)

120601(119910)

120579

= (


sin120601 cos120601 0

0 0 1

)(

cos 120579 0 sin 1205790 1 0

minus sin 120579 0 cos 120579)

= (


)

(B3)


2

1

0

minus1

minus2

0 3 6 9 12Time (s)

Fiel

d (V

m)

120591rep

120591env


1=


1 1198792]LCM = 6




) (B4)


120576k1 = k1 = (



120576k2 = k2 = (


sin 120579 sin120594)

(B5)

C Repetitive Time


119864 (119905) =

119873

sum

119896=1

119864119896cos (120596

119896119905) (C1)


119864 (119905 + 120591rep) = 119864 (119905) (C2)

120591rep = [1198791 1198792 119879119873]LCM (C3)

where 119879119896= 2120587120596



120591rep =2120587

(1205961 1205962 120596

119873)GCD

(C4)


1205961+ 1205962+ sdot sdot sdot + 120596

119873

=2120587

1198791

+2120587

1198792


119879119873

=2120587

[1198791 1198792 119879

119873]LCM

(1198991+ 1198992+ sdot sdot sdot + 119899

119873)

(C5)


119899119896=[1198791 1198792 119879

119873]LCM

119879119896

(C6)




Δ120596gcd equiv (1205961 1205962 120596119873)GCD

=2120587

[1198791 1198792 119879

119873]LCM

(C7)


120591rep =2120587

Δ120596gcd(C8)


120591int =2120587

Δ120596 (C9)


Δ120596 (120581) = 120596 (120581) minus 120596 (120581 minus Δ120581)

= 119888(3120581)13

minus 119888(3120581)13

(1 minusΔ120581

120581)

13

(C10)


0and

120581 ≃ 1205810

Δ120581

1205810

= (3

119873k minus 1)Δ

1205960

≪ 1

120581 = 1205810+ 119874(

Δ

1205960

)

(C11)


where 1205810= (13)(120596



Δ120596 ≃119888(31205810)13

3

Δ120581

1205810

(C12)

Acknowledgments


References



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





120591tran =2

Γ12059620

(57)


120591int =2120587

Δ120596 (58)


Δ120596 ≃119888(31205810)13

3

Δ120581

1205810

(59)

where 1205810equiv (13)(120596

0119888)3







2

10

minus1

minus1

minus215 16 17 18 19


cuum

fiel

d (V

m)

002

0

minus002

Vacu

um fi

eld

(Vm

)

times10minus8


times10minus12

ParticleField

1

0

17 171 172 173Time (s)

Time (s)

Posit

ion

(m)

Posit

ion

(m) 120591(X)

mod

)(E120591mod




0) to avoid


119902 = 119890

119898 = 10minus4

119898119890

1205960= 1016 rads

Δ = 220 times Γ1205962

0

(60)


Γ1205962





10minus4





119909 (119905)

=119902

119898120596119877

int

119905

minusinfin

119864vac119909 (1199051015840

) 119890minusΓ1205962

0(119905minus1199051015840)2 sin (120596

119877(119905 minus 1199051015840

)) 1198891199051015840

(61)







)


0for the particle






6

4

2


Position (m)

Prob

abili

ty d

ensit

y

times107

6

4

2



Prob

abili

ty d

ensit

yQMSED

QMSED

times1025

times10minus26

2

1

0

minus1

minus212 18 24

Posit

ion

(m)


times10minus26

2

1

0

minus1

minus2178 1781

Posit

ion

(m)


times10minus26

times10minus8

1782


120596= 2 times 10



119909 (119905) =119902

119898sum

k120582radicℎ120596

1205980119881

119886k120582119861k120582

119890minus119894120596119905

120576k120582119909 (62)




119909 (119905) = 119860 (119905) 119890minus1198941205960119905

119860 (119905) = sum

k120582(120576k120582119909

119902

119898radicℎ120596

1205980119881

119886k120582119861k120582

)119890minus119894(120596minus120596

0)119905

(63)





2

1

0

minus1

minus205 1 15 2 25 3

Posit

ion

(m)

times10minus8

times10minus12

Prob

abili

ty d

ensit

y 6

4

2

0minus3 0 3

times107 15

10

5

0minus3 0 3

times1078

6

4

2

0minus3 0 3

times107

Time (s)




negligible

119860 (119905 + Δ119905) ≃ 119860 (119905) for Δ119905 ≪ 120591coh (64)




3

15

0

minus15


Real

Imag

inar

y

A(t0)

Δ

A(t0 + Δt)



0) (blue



the coherence time



119875 (119909) = sum

119860

119875119860(119909) = int119875

119860(119909) 119891 (119860) 119889119860 (65)







5

0

minus51 3 5 7 9

times10minus8

times10minus11

Posit

ion

(m)

Time (s)

2

0

minus2461 462 463 464

times10minus8

times10minus11

Posit

ion

(m)

Time (s)

10

5


times107


Prob

abili

ty d

ensit

y




(rare)

= 15 times 107

Small amplitude

3120591coh

Δs


119904= 15 times 10




120596= 2 times 10


6

4

2


times107

times10minus8

Prob

abili

ty d

ensit

y

QMSED

120590x ≃ 761 times 10minus9

6

4

2


times1025

times10minus26

Prob

abili

ty d

ensit

y


QMSED

120590p ≃ 693 times 10minus27 5

0

minus50 2 4Po

sitio

n (m

)

Position (m)

Time (s)

times10minus8

times10minus12

Trial-15

0

minus50 2 4

Posit

ion

(m)

Time (s)

times10minus8

times10minus12

Trial-25

0

minus50 2 4

Posit

ion

(m)

Time (s)

times10minus8

times10minus12

Trial-3


119901= 2 times 10


120596= 2 times 10






120596= 5 times 10


119901= 1198791119879119901 where 119879





4

2

0

minus2

minus40 02 04 06 08 1 12 14

Time (s)

Posit

ion

(m)


times10minus8

times10minus12

4

2

0

minus2

minus40 02 04 06 08 1 12 14

Time (s)

Posit

ion

(m)


times10minus12

times10minus26


10 30 50 70 90

10

20

30

40


Com

puta

tion

time (

hr)

Np = 100 k = 05 kNp = 100 k = 1 kNp = 200 k = 05 k

N120596

N120596

N120596


119901(number


0 200 400 600 800 10000

14

12

34

1

Np = 200 k

Ener

gy (ℏ

1205960)

N120596



119901=


2

0






120596


5 Conclusions


120596= 5 times 10




Vacuum field

SEDRadiationdamping

X

X

2

1

0

minus1

minus2

Time (s)

times10minus8

times10minus12

Posit

ion

(m)

2

1

0

minus1

minus2649647645

Time (s)

times10minus8

times10minus13

Posit

ion

(m)

2

1

0

minus1

minus2

times10minus8

Posit

ion

(m)

CMCM

Probability density

SED (theory)

0 5 10

2

1

0

minus1

minus2

times10minus8

Posit

ion

(m)


SED (simulation)

2 3 4 5 6 7

times107

times107














Appendices




E (r 119905)

=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)


B (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896 (k times 120576 (k 120582))




119860(k 120582)119890119894120579(k120582)

E (r 119905)

=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)


+119860lowast



B (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896120585 (k 120582)


+119860lowast


(A2)


⟨119906⟩120579=1205980

2⟨|E|2⟩

120579

+1

21205830

⟨|B|2⟩120579

(A3)


0is the vacuum



and |B|2 using (A2)

|E|2 (r 119905)

= E (r 119905)Elowast (r 119905)

=1

4sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120576 (k 120582) sdot 120576 (k1015840 1205821015840)) 119891

|B|2 (r 119905)

= B (r 119905)Blowast (r 119905)

=1

41198882sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120585 (k 120582) sdot 120585 (k1015840 1205821015840)) 119891

(A4)

where 119891 equiv 119891(k 120582 k1015840 1205821015840 r 119905)

119891 (k 120582 k1015840 1205821015840 r 119905)



+ 119860 (k 120582) 119860 (k1015840 1205821015840) 119890119894(ksdotrminus120596119905+120579(k120582))


+ 119860lowast



+ 119860lowast



(A5)


⟨119890plusmn119894(120579(k120582)+120579(k1015840 1205821015840))

⟩120579

= 0

⟨119890plusmn119894(120579(k120582)minus120579(k1015840 1205821015840))

⟩120579

= 12057512058210158401205821205753

(k1015840 minus k) (A6)


⟨119891 (k 120582 k1015840 1205821015840 r 119905)⟩120579



12057512058210158401205821205753

(k1015840 minus k)

+ 119860lowast



12057512058210158401205821205753

(k1015840 minus k)

(A7)



(A4) and (A7)

⟨|E|2⟩120579

=1

4sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120576 (k 120582) sdot 120576 (k1015840 1205821015840)) ⟨119891⟩120579

=1

2

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2

⟨|B|2⟩120579

=1

41198882sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120585 (k 120582) sdot 120585 (k1015840 1205821015840)) ⟨119891⟩120579

=1

21198882

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2

(A8)


120579

in unbounded space

⟨119906⟩120579=1205980

2

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2 (A9)


119906vac =1

119881sum

k120582

ℎ120596

2

=1

119881sum

k120582

ℎ120596

2(119871119909

2120587Δ119896119909)(

119871119910

2120587Δ119896119910)(

119871119911

2120587Δ119896119911)

=

2

sum

120582=1

1

(2120587)3sum

kΔ3

119896ℎ120596

2

(A10)



3


119906vac =2

sum

120582=1

1

(2120587)3int1198893

119896ℎ120596

2 (A11)


1003816100381610038161003816119860vac (k 120582)10038161003816100381610038162

=ℎ120596

(2120587)3

1205980

(A12)


119860vac (k 120582) = radicℎ120596

812058731205980

(A13)


=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)radic ℎ120596

812058731205980


119890119894120579(k120582)


119890minus119894120579(k120582)

)

Bvac (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896120585 (k 120582)radic ℎ120596

812058731205980


119890119894120579(k120582)


119890minus119894120579(k120582)

)

(A14)

which is (1)


E (r 119905) = 1

2sum

k120582(119860k120582119890



) 120576k120582

B (r 119905) = 1

2119888sum

k120582(119860k120582119890



) 120585k120582

(A15)



120579= ⟨119890119894120579k120582⟩120579⟨119890plusmn119894120579k1015840 1205821015840 ⟩

120579=

0)

⟨119890plusmn119894(120579k120582+120579k1015840 1205821015840 )⟩

120579

= 0

⟨119890plusmn119894(120579k120582minus120579k1015840 1205821015840 )⟩

120579

= 1205751205821015840120582120575k1015840 k

(A16)


⟨119906⟩120579=1205980

2sum

k120582

1003816100381610038161003816119860k12058210038161003816100381610038162

(A17)



119906vac =1

119881sum

k120582

ℎ120596

2 (A18)


119860vack120582 = radicℎ120596

1205980119881 (A19)


Evac (r 119905) =1

2sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120576k120582

Bvac (r 119905) =1

2119888sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120585k120582

(A20)



k = (119909

119910

119911

) = (

0

0

119896

) (B1)


k1 = (cos120594sin1205940

) k2 = (minus sin120594cos1205940

) (B2)


= (119911)

120601(119910)

120579

= (


sin120601 cos120601 0

0 0 1

)(

cos 120579 0 sin 1205790 1 0

minus sin 120579 0 cos 120579)

= (


)

(B3)


2

1

0

minus1

minus2

0 3 6 9 12Time (s)

Fiel

d (V

m)

120591rep

120591env


1=


1 1198792]LCM = 6




) (B4)


120576k1 = k1 = (



120576k2 = k2 = (


sin 120579 sin120594)

(B5)

C Repetitive Time


119864 (119905) =

119873

sum

119896=1

119864119896cos (120596

119896119905) (C1)


119864 (119905 + 120591rep) = 119864 (119905) (C2)

120591rep = [1198791 1198792 119879119873]LCM (C3)

where 119879119896= 2120587120596



120591rep =2120587

(1205961 1205962 120596

119873)GCD

(C4)


1205961+ 1205962+ sdot sdot sdot + 120596

119873

=2120587

1198791

+2120587

1198792


119879119873

=2120587

[1198791 1198792 119879

119873]LCM

(1198991+ 1198992+ sdot sdot sdot + 119899

119873)

(C5)


119899119896=[1198791 1198792 119879

119873]LCM

119879119896

(C6)




Δ120596gcd equiv (1205961 1205962 120596119873)GCD

=2120587

[1198791 1198792 119879

119873]LCM

(C7)


120591rep =2120587

Δ120596gcd(C8)


120591int =2120587

Δ120596 (C9)


Δ120596 (120581) = 120596 (120581) minus 120596 (120581 minus Δ120581)

= 119888(3120581)13

minus 119888(3120581)13

(1 minusΔ120581

120581)

13

(C10)


0and

120581 ≃ 1205810

Δ120581

1205810

= (3

119873k minus 1)Δ

1205960

≪ 1

120581 = 1205810+ 119874(

Δ

1205960

)

(C11)


where 1205810= (13)(120596



Δ120596 ≃119888(31205810)13

3

Δ120581

1205810

(C12)

Acknowledgments


References



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119909 (119905)

=119902

119898120596119877

int

119905

minusinfin

119864vac119909 (1199051015840

) 119890minusΓ1205962

0(119905minus1199051015840)2 sin (120596

119877(119905 minus 1199051015840

)) 1198891199051015840

(61)







)


0for the particle






6

4

2


Position (m)

Prob

abili

ty d

ensit

y

times107

6

4

2



Prob

abili

ty d

ensit

yQMSED

QMSED

times1025

times10minus26

2

1

0

minus1

minus212 18 24

Posit

ion

(m)


times10minus26

2

1

0

minus1

minus2178 1781

Posit

ion

(m)


times10minus26

times10minus8

1782


120596= 2 times 10



119909 (119905) =119902

119898sum

k120582radicℎ120596

1205980119881

119886k120582119861k120582

119890minus119894120596119905

120576k120582119909 (62)




119909 (119905) = 119860 (119905) 119890minus1198941205960119905

119860 (119905) = sum

k120582(120576k120582119909

119902

119898radicℎ120596

1205980119881

119886k120582119861k120582

)119890minus119894(120596minus120596

0)119905

(63)





2

1

0

minus1

minus205 1 15 2 25 3

Posit

ion

(m)

times10minus8

times10minus12

Prob

abili

ty d

ensit

y 6

4

2

0minus3 0 3

times107 15

10

5

0minus3 0 3

times1078

6

4

2

0minus3 0 3

times107

Time (s)




negligible

119860 (119905 + Δ119905) ≃ 119860 (119905) for Δ119905 ≪ 120591coh (64)




3

15

0

minus15


Real

Imag

inar

y

A(t0)

Δ

A(t0 + Δt)



0) (blue



the coherence time



119875 (119909) = sum

119860

119875119860(119909) = int119875

119860(119909) 119891 (119860) 119889119860 (65)







5

0

minus51 3 5 7 9

times10minus8

times10minus11

Posit

ion

(m)

Time (s)

2

0

minus2461 462 463 464

times10minus8

times10minus11

Posit

ion

(m)

Time (s)

10

5


times107


Prob

abili

ty d

ensit

y




(rare)

= 15 times 107

Small amplitude

3120591coh

Δs


119904= 15 times 10




120596= 2 times 10


6

4

2


times107

times10minus8

Prob

abili

ty d

ensit

y

QMSED

120590x ≃ 761 times 10minus9

6

4

2


times1025

times10minus26

Prob

abili

ty d

ensit

y


QMSED

120590p ≃ 693 times 10minus27 5

0

minus50 2 4Po

sitio

n (m

)

Position (m)

Time (s)

times10minus8

times10minus12

Trial-15

0

minus50 2 4

Posit

ion

(m)

Time (s)

times10minus8

times10minus12

Trial-25

0

minus50 2 4

Posit

ion

(m)

Time (s)

times10minus8

times10minus12

Trial-3


119901= 2 times 10


120596= 2 times 10






120596= 5 times 10


119901= 1198791119879119901 where 119879





4

2

0

minus2

minus40 02 04 06 08 1 12 14

Time (s)

Posit

ion

(m)


times10minus8

times10minus12

4

2

0

minus2

minus40 02 04 06 08 1 12 14

Time (s)

Posit

ion

(m)


times10minus12

times10minus26


10 30 50 70 90

10

20

30

40


Com

puta

tion

time (

hr)

Np = 100 k = 05 kNp = 100 k = 1 kNp = 200 k = 05 k

N120596

N120596

N120596


119901(number


0 200 400 600 800 10000

14

12

34

1

Np = 200 k

Ener

gy (ℏ

1205960)

N120596



119901=


2

0






120596


5 Conclusions


120596= 5 times 10




Vacuum field

SEDRadiationdamping

X

X

2

1

0

minus1

minus2

Time (s)

times10minus8

times10minus12

Posit

ion

(m)

2

1

0

minus1

minus2649647645

Time (s)

times10minus8

times10minus13

Posit

ion

(m)

2

1

0

minus1

minus2

times10minus8

Posit

ion

(m)

CMCM

Probability density

SED (theory)

0 5 10

2

1

0

minus1

minus2

times10minus8

Posit

ion

(m)


SED (simulation)

2 3 4 5 6 7

times107

times107














Appendices




E (r 119905)

=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)


B (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896 (k times 120576 (k 120582))




119860(k 120582)119890119894120579(k120582)

E (r 119905)

=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)


+119860lowast



B (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896120585 (k 120582)


+119860lowast


(A2)


⟨119906⟩120579=1205980

2⟨|E|2⟩

120579

+1

21205830

⟨|B|2⟩120579

(A3)


0is the vacuum



and |B|2 using (A2)

|E|2 (r 119905)

= E (r 119905)Elowast (r 119905)

=1

4sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120576 (k 120582) sdot 120576 (k1015840 1205821015840)) 119891

|B|2 (r 119905)

= B (r 119905)Blowast (r 119905)

=1

41198882sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120585 (k 120582) sdot 120585 (k1015840 1205821015840)) 119891

(A4)

where 119891 equiv 119891(k 120582 k1015840 1205821015840 r 119905)

119891 (k 120582 k1015840 1205821015840 r 119905)



+ 119860 (k 120582) 119860 (k1015840 1205821015840) 119890119894(ksdotrminus120596119905+120579(k120582))


+ 119860lowast



+ 119860lowast



(A5)


⟨119890plusmn119894(120579(k120582)+120579(k1015840 1205821015840))

⟩120579

= 0

⟨119890plusmn119894(120579(k120582)minus120579(k1015840 1205821015840))

⟩120579

= 12057512058210158401205821205753

(k1015840 minus k) (A6)


⟨119891 (k 120582 k1015840 1205821015840 r 119905)⟩120579



12057512058210158401205821205753

(k1015840 minus k)

+ 119860lowast



12057512058210158401205821205753

(k1015840 minus k)

(A7)



(A4) and (A7)

⟨|E|2⟩120579

=1

4sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120576 (k 120582) sdot 120576 (k1015840 1205821015840)) ⟨119891⟩120579

=1

2

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2

⟨|B|2⟩120579

=1

41198882sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120585 (k 120582) sdot 120585 (k1015840 1205821015840)) ⟨119891⟩120579

=1

21198882

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2

(A8)


120579

in unbounded space

⟨119906⟩120579=1205980

2

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2 (A9)


119906vac =1

119881sum

k120582

ℎ120596

2

=1

119881sum

k120582

ℎ120596

2(119871119909

2120587Δ119896119909)(

119871119910

2120587Δ119896119910)(

119871119911

2120587Δ119896119911)

=

2

sum

120582=1

1

(2120587)3sum

kΔ3

119896ℎ120596

2

(A10)



3


119906vac =2

sum

120582=1

1

(2120587)3int1198893

119896ℎ120596

2 (A11)


1003816100381610038161003816119860vac (k 120582)10038161003816100381610038162

=ℎ120596

(2120587)3

1205980

(A12)


119860vac (k 120582) = radicℎ120596

812058731205980

(A13)


=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)radic ℎ120596

812058731205980


119890119894120579(k120582)


119890minus119894120579(k120582)

)

Bvac (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896120585 (k 120582)radic ℎ120596

812058731205980


119890119894120579(k120582)


119890minus119894120579(k120582)

)

(A14)

which is (1)


E (r 119905) = 1

2sum

k120582(119860k120582119890



) 120576k120582

B (r 119905) = 1

2119888sum

k120582(119860k120582119890



) 120585k120582

(A15)



120579= ⟨119890119894120579k120582⟩120579⟨119890plusmn119894120579k1015840 1205821015840 ⟩

120579=

0)

⟨119890plusmn119894(120579k120582+120579k1015840 1205821015840 )⟩

120579

= 0

⟨119890plusmn119894(120579k120582minus120579k1015840 1205821015840 )⟩

120579

= 1205751205821015840120582120575k1015840 k

(A16)


⟨119906⟩120579=1205980

2sum

k120582

1003816100381610038161003816119860k12058210038161003816100381610038162

(A17)



119906vac =1

119881sum

k120582

ℎ120596

2 (A18)


119860vack120582 = radicℎ120596

1205980119881 (A19)


Evac (r 119905) =1

2sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120576k120582

Bvac (r 119905) =1

2119888sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120585k120582

(A20)



k = (119909

119910

119911

) = (

0

0

119896

) (B1)


k1 = (cos120594sin1205940

) k2 = (minus sin120594cos1205940

) (B2)


= (119911)

120601(119910)

120579

= (


sin120601 cos120601 0

0 0 1

)(

cos 120579 0 sin 1205790 1 0

minus sin 120579 0 cos 120579)

= (


)

(B3)


2

1

0

minus1

minus2

0 3 6 9 12Time (s)

Fiel

d (V

m)

120591rep

120591env


1=


1 1198792]LCM = 6




) (B4)


120576k1 = k1 = (



120576k2 = k2 = (


sin 120579 sin120594)

(B5)

C Repetitive Time


119864 (119905) =

119873

sum

119896=1

119864119896cos (120596

119896119905) (C1)


119864 (119905 + 120591rep) = 119864 (119905) (C2)

120591rep = [1198791 1198792 119879119873]LCM (C3)

where 119879119896= 2120587120596



120591rep =2120587

(1205961 1205962 120596

119873)GCD

(C4)


1205961+ 1205962+ sdot sdot sdot + 120596

119873

=2120587

1198791

+2120587

1198792


119879119873

=2120587

[1198791 1198792 119879

119873]LCM

(1198991+ 1198992+ sdot sdot sdot + 119899

119873)

(C5)


119899119896=[1198791 1198792 119879

119873]LCM

119879119896

(C6)




Δ120596gcd equiv (1205961 1205962 120596119873)GCD

=2120587

[1198791 1198792 119879

119873]LCM

(C7)


120591rep =2120587

Δ120596gcd(C8)


120591int =2120587

Δ120596 (C9)


Δ120596 (120581) = 120596 (120581) minus 120596 (120581 minus Δ120581)

= 119888(3120581)13

minus 119888(3120581)13

(1 minusΔ120581

120581)

13

(C10)


0and

120581 ≃ 1205810

Δ120581

1205810

= (3

119873k minus 1)Δ

1205960

≪ 1

120581 = 1205810+ 119874(

Δ

1205960

)

(C11)


where 1205810= (13)(120596



Δ120596 ≃119888(31205810)13

3

Δ120581

1205810

(C12)

Acknowledgments


References



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




2

1

0

minus1

minus205 1 15 2 25 3

Posit

ion

(m)

times10minus8

times10minus12

Prob

abili

ty d

ensit

y 6

4

2

0minus3 0 3

times107 15

10

5

0minus3 0 3

times1078

6

4

2

0minus3 0 3

times107

Time (s)




negligible

119860 (119905 + Δ119905) ≃ 119860 (119905) for Δ119905 ≪ 120591coh (64)




3

15

0

minus15


Real

Imag

inar

y

A(t0)

Δ

A(t0 + Δt)



0) (blue



the coherence time



119875 (119909) = sum

119860

119875119860(119909) = int119875

119860(119909) 119891 (119860) 119889119860 (65)







5

0

minus51 3 5 7 9

times10minus8

times10minus11

Posit

ion

(m)

Time (s)

2

0

minus2461 462 463 464

times10minus8

times10minus11

Posit

ion

(m)

Time (s)

10

5


times107


Prob

abili

ty d

ensit

y




(rare)

= 15 times 107

Small amplitude

3120591coh

Δs


119904= 15 times 10




120596= 2 times 10


6

4

2


times107

times10minus8

Prob

abili

ty d

ensit

y

QMSED

120590x ≃ 761 times 10minus9

6

4

2


times1025

times10minus26

Prob

abili

ty d

ensit

y


QMSED

120590p ≃ 693 times 10minus27 5

0

minus50 2 4Po

sitio

n (m

)

Position (m)

Time (s)

times10minus8

times10minus12

Trial-15

0

minus50 2 4

Posit

ion

(m)

Time (s)

times10minus8

times10minus12

Trial-25

0

minus50 2 4

Posit

ion

(m)

Time (s)

times10minus8

times10minus12

Trial-3


119901= 2 times 10


120596= 2 times 10






120596= 5 times 10


119901= 1198791119879119901 where 119879





4

2

0

minus2

minus40 02 04 06 08 1 12 14

Time (s)

Posit

ion

(m)


times10minus8

times10minus12

4

2

0

minus2

minus40 02 04 06 08 1 12 14

Time (s)

Posit

ion

(m)


times10minus12

times10minus26


10 30 50 70 90

10

20

30

40


Com

puta

tion

time (

hr)

Np = 100 k = 05 kNp = 100 k = 1 kNp = 200 k = 05 k

N120596

N120596

N120596


119901(number


0 200 400 600 800 10000

14

12

34

1

Np = 200 k

Ener

gy (ℏ

1205960)

N120596



119901=


2

0






120596


5 Conclusions


120596= 5 times 10




Vacuum field

SEDRadiationdamping

X

X

2

1

0

minus1

minus2

Time (s)

times10minus8

times10minus12

Posit

ion

(m)

2

1

0

minus1

minus2649647645

Time (s)

times10minus8

times10minus13

Posit

ion

(m)

2

1

0

minus1

minus2

times10minus8

Posit

ion

(m)

CMCM

Probability density

SED (theory)

0 5 10

2

1

0

minus1

minus2

times10minus8

Posit

ion

(m)


SED (simulation)

2 3 4 5 6 7

times107

times107














Appendices




E (r 119905)

=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)


B (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896 (k times 120576 (k 120582))




119860(k 120582)119890119894120579(k120582)

E (r 119905)

=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)


+119860lowast



B (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896120585 (k 120582)


+119860lowast


(A2)


⟨119906⟩120579=1205980

2⟨|E|2⟩

120579

+1

21205830

⟨|B|2⟩120579

(A3)


0is the vacuum



and |B|2 using (A2)

|E|2 (r 119905)

= E (r 119905)Elowast (r 119905)

=1

4sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120576 (k 120582) sdot 120576 (k1015840 1205821015840)) 119891

|B|2 (r 119905)

= B (r 119905)Blowast (r 119905)

=1

41198882sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120585 (k 120582) sdot 120585 (k1015840 1205821015840)) 119891

(A4)

where 119891 equiv 119891(k 120582 k1015840 1205821015840 r 119905)

119891 (k 120582 k1015840 1205821015840 r 119905)



+ 119860 (k 120582) 119860 (k1015840 1205821015840) 119890119894(ksdotrminus120596119905+120579(k120582))


+ 119860lowast



+ 119860lowast



(A5)


⟨119890plusmn119894(120579(k120582)+120579(k1015840 1205821015840))

⟩120579

= 0

⟨119890plusmn119894(120579(k120582)minus120579(k1015840 1205821015840))

⟩120579

= 12057512058210158401205821205753

(k1015840 minus k) (A6)


⟨119891 (k 120582 k1015840 1205821015840 r 119905)⟩120579



12057512058210158401205821205753

(k1015840 minus k)

+ 119860lowast



12057512058210158401205821205753

(k1015840 minus k)

(A7)



(A4) and (A7)

⟨|E|2⟩120579

=1

4sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120576 (k 120582) sdot 120576 (k1015840 1205821015840)) ⟨119891⟩120579

=1

2

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2

⟨|B|2⟩120579

=1

41198882sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120585 (k 120582) sdot 120585 (k1015840 1205821015840)) ⟨119891⟩120579

=1

21198882

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2

(A8)


120579

in unbounded space

⟨119906⟩120579=1205980

2

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2 (A9)


119906vac =1

119881sum

k120582

ℎ120596

2

=1

119881sum

k120582

ℎ120596

2(119871119909

2120587Δ119896119909)(

119871119910

2120587Δ119896119910)(

119871119911

2120587Δ119896119911)

=

2

sum

120582=1

1

(2120587)3sum

kΔ3

119896ℎ120596

2

(A10)



3


119906vac =2

sum

120582=1

1

(2120587)3int1198893

119896ℎ120596

2 (A11)


1003816100381610038161003816119860vac (k 120582)10038161003816100381610038162

=ℎ120596

(2120587)3

1205980

(A12)


119860vac (k 120582) = radicℎ120596

812058731205980

(A13)


=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)radic ℎ120596

812058731205980


119890119894120579(k120582)


119890minus119894120579(k120582)

)

Bvac (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896120585 (k 120582)radic ℎ120596

812058731205980


119890119894120579(k120582)


119890minus119894120579(k120582)

)

(A14)

which is (1)


E (r 119905) = 1

2sum

k120582(119860k120582119890



) 120576k120582

B (r 119905) = 1

2119888sum

k120582(119860k120582119890



) 120585k120582

(A15)



120579= ⟨119890119894120579k120582⟩120579⟨119890plusmn119894120579k1015840 1205821015840 ⟩

120579=

0)

⟨119890plusmn119894(120579k120582+120579k1015840 1205821015840 )⟩

120579

= 0

⟨119890plusmn119894(120579k120582minus120579k1015840 1205821015840 )⟩

120579

= 1205751205821015840120582120575k1015840 k

(A16)


⟨119906⟩120579=1205980

2sum

k120582

1003816100381610038161003816119860k12058210038161003816100381610038162

(A17)



119906vac =1

119881sum

k120582

ℎ120596

2 (A18)


119860vack120582 = radicℎ120596

1205980119881 (A19)


Evac (r 119905) =1

2sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120576k120582

Bvac (r 119905) =1

2119888sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120585k120582

(A20)



k = (119909

119910

119911

) = (

0

0

119896

) (B1)


k1 = (cos120594sin1205940

) k2 = (minus sin120594cos1205940

) (B2)


= (119911)

120601(119910)

120579

= (


sin120601 cos120601 0

0 0 1

)(

cos 120579 0 sin 1205790 1 0

minus sin 120579 0 cos 120579)

= (


)

(B3)


2

1

0

minus1

minus2

0 3 6 9 12Time (s)

Fiel

d (V

m)

120591rep

120591env


1=


1 1198792]LCM = 6




) (B4)


120576k1 = k1 = (



120576k2 = k2 = (


sin 120579 sin120594)

(B5)

C Repetitive Time


119864 (119905) =

119873

sum

119896=1

119864119896cos (120596

119896119905) (C1)


119864 (119905 + 120591rep) = 119864 (119905) (C2)

120591rep = [1198791 1198792 119879119873]LCM (C3)

where 119879119896= 2120587120596



120591rep =2120587

(1205961 1205962 120596

119873)GCD

(C4)


1205961+ 1205962+ sdot sdot sdot + 120596

119873

=2120587

1198791

+2120587

1198792


119879119873

=2120587

[1198791 1198792 119879

119873]LCM

(1198991+ 1198992+ sdot sdot sdot + 119899

119873)

(C5)


119899119896=[1198791 1198792 119879

119873]LCM

119879119896

(C6)




Δ120596gcd equiv (1205961 1205962 120596119873)GCD

=2120587

[1198791 1198792 119879

119873]LCM

(C7)


120591rep =2120587

Δ120596gcd(C8)


120591int =2120587

Δ120596 (C9)


Δ120596 (120581) = 120596 (120581) minus 120596 (120581 minus Δ120581)

= 119888(3120581)13

minus 119888(3120581)13

(1 minusΔ120581

120581)

13

(C10)


0and

120581 ≃ 1205810

Δ120581

1205810

= (3

119873k minus 1)Δ

1205960

≪ 1

120581 = 1205810+ 119874(

Δ

1205960

)

(C11)


where 1205810= (13)(120596



Δ120596 ≃119888(31205810)13

3

Δ120581

1205810

(C12)

Acknowledgments


References



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




5

0

minus51 3 5 7 9

times10minus8

times10minus11

Posit

ion

(m)

Time (s)

2

0

minus2461 462 463 464

times10minus8

times10minus11

Posit

ion

(m)

Time (s)

10

5


times107


Prob

abili

ty d

ensit

y




(rare)

= 15 times 107

Small amplitude

3120591coh

Δs


119904= 15 times 10




120596= 2 times 10


6

4

2


times107

times10minus8

Prob

abili

ty d

ensit

y

QMSED

120590x ≃ 761 times 10minus9

6

4

2


times1025

times10minus26

Prob

abili

ty d

ensit

y


QMSED

120590p ≃ 693 times 10minus27 5

0

minus50 2 4Po

sitio

n (m

)

Position (m)

Time (s)

times10minus8

times10minus12

Trial-15

0

minus50 2 4

Posit

ion

(m)

Time (s)

times10minus8

times10minus12

Trial-25

0

minus50 2 4

Posit

ion

(m)

Time (s)

times10minus8

times10minus12

Trial-3


119901= 2 times 10


120596= 2 times 10






120596= 5 times 10


119901= 1198791119879119901 where 119879





4

2

0

minus2

minus40 02 04 06 08 1 12 14

Time (s)

Posit

ion

(m)


times10minus8

times10minus12

4

2

0

minus2

minus40 02 04 06 08 1 12 14

Time (s)

Posit

ion

(m)


times10minus12

times10minus26


10 30 50 70 90

10

20

30

40


Com

puta

tion

time (

hr)

Np = 100 k = 05 kNp = 100 k = 1 kNp = 200 k = 05 k

N120596

N120596

N120596


119901(number


0 200 400 600 800 10000

14

12

34

1

Np = 200 k

Ener

gy (ℏ

1205960)

N120596



119901=


2

0






120596


5 Conclusions


120596= 5 times 10




Vacuum field

SEDRadiationdamping

X

X

2

1

0

minus1

minus2

Time (s)

times10minus8

times10minus12

Posit

ion

(m)

2

1

0

minus1

minus2649647645

Time (s)

times10minus8

times10minus13

Posit

ion

(m)

2

1

0

minus1

minus2

times10minus8

Posit

ion

(m)

CMCM

Probability density

SED (theory)

0 5 10

2

1

0

minus1

minus2

times10minus8

Posit

ion

(m)


SED (simulation)

2 3 4 5 6 7

times107

times107














Appendices




E (r 119905)

=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)


B (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896 (k times 120576 (k 120582))




119860(k 120582)119890119894120579(k120582)

E (r 119905)

=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)


+119860lowast



B (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896120585 (k 120582)


+119860lowast


(A2)


⟨119906⟩120579=1205980

2⟨|E|2⟩

120579

+1

21205830

⟨|B|2⟩120579

(A3)


0is the vacuum



and |B|2 using (A2)

|E|2 (r 119905)

= E (r 119905)Elowast (r 119905)

=1

4sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120576 (k 120582) sdot 120576 (k1015840 1205821015840)) 119891

|B|2 (r 119905)

= B (r 119905)Blowast (r 119905)

=1

41198882sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120585 (k 120582) sdot 120585 (k1015840 1205821015840)) 119891

(A4)

where 119891 equiv 119891(k 120582 k1015840 1205821015840 r 119905)

119891 (k 120582 k1015840 1205821015840 r 119905)



+ 119860 (k 120582) 119860 (k1015840 1205821015840) 119890119894(ksdotrminus120596119905+120579(k120582))


+ 119860lowast



+ 119860lowast



(A5)


⟨119890plusmn119894(120579(k120582)+120579(k1015840 1205821015840))

⟩120579

= 0

⟨119890plusmn119894(120579(k120582)minus120579(k1015840 1205821015840))

⟩120579

= 12057512058210158401205821205753

(k1015840 minus k) (A6)


⟨119891 (k 120582 k1015840 1205821015840 r 119905)⟩120579



12057512058210158401205821205753

(k1015840 minus k)

+ 119860lowast



12057512058210158401205821205753

(k1015840 minus k)

(A7)



(A4) and (A7)

⟨|E|2⟩120579

=1

4sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120576 (k 120582) sdot 120576 (k1015840 1205821015840)) ⟨119891⟩120579

=1

2

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2

⟨|B|2⟩120579

=1

41198882sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120585 (k 120582) sdot 120585 (k1015840 1205821015840)) ⟨119891⟩120579

=1

21198882

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2

(A8)


120579

in unbounded space

⟨119906⟩120579=1205980

2

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2 (A9)


119906vac =1

119881sum

k120582

ℎ120596

2

=1

119881sum

k120582

ℎ120596

2(119871119909

2120587Δ119896119909)(

119871119910

2120587Δ119896119910)(

119871119911

2120587Δ119896119911)

=

2

sum

120582=1

1

(2120587)3sum

kΔ3

119896ℎ120596

2

(A10)



3


119906vac =2

sum

120582=1

1

(2120587)3int1198893

119896ℎ120596

2 (A11)


1003816100381610038161003816119860vac (k 120582)10038161003816100381610038162

=ℎ120596

(2120587)3

1205980

(A12)


119860vac (k 120582) = radicℎ120596

812058731205980

(A13)


=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)radic ℎ120596

812058731205980


119890119894120579(k120582)


119890minus119894120579(k120582)

)

Bvac (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896120585 (k 120582)radic ℎ120596

812058731205980


119890119894120579(k120582)


119890minus119894120579(k120582)

)

(A14)

which is (1)


E (r 119905) = 1

2sum

k120582(119860k120582119890



) 120576k120582

B (r 119905) = 1

2119888sum

k120582(119860k120582119890



) 120585k120582

(A15)



120579= ⟨119890119894120579k120582⟩120579⟨119890plusmn119894120579k1015840 1205821015840 ⟩

120579=

0)

⟨119890plusmn119894(120579k120582+120579k1015840 1205821015840 )⟩

120579

= 0

⟨119890plusmn119894(120579k120582minus120579k1015840 1205821015840 )⟩

120579

= 1205751205821015840120582120575k1015840 k

(A16)


⟨119906⟩120579=1205980

2sum

k120582

1003816100381610038161003816119860k12058210038161003816100381610038162

(A17)



119906vac =1

119881sum

k120582

ℎ120596

2 (A18)


119860vack120582 = radicℎ120596

1205980119881 (A19)


Evac (r 119905) =1

2sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120576k120582

Bvac (r 119905) =1

2119888sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120585k120582

(A20)



k = (119909

119910

119911

) = (

0

0

119896

) (B1)


k1 = (cos120594sin1205940

) k2 = (minus sin120594cos1205940

) (B2)


= (119911)

120601(119910)

120579

= (


sin120601 cos120601 0

0 0 1

)(

cos 120579 0 sin 1205790 1 0

minus sin 120579 0 cos 120579)

= (


)

(B3)


2

1

0

minus1

minus2

0 3 6 9 12Time (s)

Fiel

d (V

m)

120591rep

120591env


1=


1 1198792]LCM = 6




) (B4)


120576k1 = k1 = (



120576k2 = k2 = (


sin 120579 sin120594)

(B5)

C Repetitive Time


119864 (119905) =

119873

sum

119896=1

119864119896cos (120596

119896119905) (C1)


119864 (119905 + 120591rep) = 119864 (119905) (C2)

120591rep = [1198791 1198792 119879119873]LCM (C3)

where 119879119896= 2120587120596



120591rep =2120587

(1205961 1205962 120596

119873)GCD

(C4)


1205961+ 1205962+ sdot sdot sdot + 120596

119873

=2120587

1198791

+2120587

1198792


119879119873

=2120587

[1198791 1198792 119879

119873]LCM

(1198991+ 1198992+ sdot sdot sdot + 119899

119873)

(C5)


119899119896=[1198791 1198792 119879

119873]LCM

119879119896

(C6)




Δ120596gcd equiv (1205961 1205962 120596119873)GCD

=2120587

[1198791 1198792 119879

119873]LCM

(C7)


120591rep =2120587

Δ120596gcd(C8)


120591int =2120587

Δ120596 (C9)


Δ120596 (120581) = 120596 (120581) minus 120596 (120581 minus Δ120581)

= 119888(3120581)13

minus 119888(3120581)13

(1 minusΔ120581

120581)

13

(C10)


0and

120581 ≃ 1205810

Δ120581

1205810

= (3

119873k minus 1)Δ

1205960

≪ 1

120581 = 1205810+ 119874(

Δ

1205960

)

(C11)


where 1205810= (13)(120596



Δ120596 ≃119888(31205810)13

3

Δ120581

1205810

(C12)

Acknowledgments


References



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




4

2

0

minus2

minus40 02 04 06 08 1 12 14

Time (s)

Posit

ion

(m)


times10minus8

times10minus12

4

2

0

minus2

minus40 02 04 06 08 1 12 14

Time (s)

Posit

ion

(m)


times10minus12

times10minus26


10 30 50 70 90

10

20

30

40


Com

puta

tion

time (

hr)

Np = 100 k = 05 kNp = 100 k = 1 kNp = 200 k = 05 k

N120596

N120596

N120596


119901(number


0 200 400 600 800 10000

14

12

34

1

Np = 200 k

Ener

gy (ℏ

1205960)

N120596



119901=


2

0






120596


5 Conclusions


120596= 5 times 10




Vacuum field

SEDRadiationdamping

X

X

2

1

0

minus1

minus2

Time (s)

times10minus8

times10minus12

Posit

ion

(m)

2

1

0

minus1

minus2649647645

Time (s)

times10minus8

times10minus13

Posit

ion

(m)

2

1

0

minus1

minus2

times10minus8

Posit

ion

(m)

CMCM

Probability density

SED (theory)

0 5 10

2

1

0

minus1

minus2

times10minus8

Posit

ion

(m)


SED (simulation)

2 3 4 5 6 7

times107

times107














Appendices




E (r 119905)

=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)


B (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896 (k times 120576 (k 120582))




119860(k 120582)119890119894120579(k120582)

E (r 119905)

=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)


+119860lowast



B (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896120585 (k 120582)


+119860lowast


(A2)


⟨119906⟩120579=1205980

2⟨|E|2⟩

120579

+1

21205830

⟨|B|2⟩120579

(A3)


0is the vacuum



and |B|2 using (A2)

|E|2 (r 119905)

= E (r 119905)Elowast (r 119905)

=1

4sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120576 (k 120582) sdot 120576 (k1015840 1205821015840)) 119891

|B|2 (r 119905)

= B (r 119905)Blowast (r 119905)

=1

41198882sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120585 (k 120582) sdot 120585 (k1015840 1205821015840)) 119891

(A4)

where 119891 equiv 119891(k 120582 k1015840 1205821015840 r 119905)

119891 (k 120582 k1015840 1205821015840 r 119905)



+ 119860 (k 120582) 119860 (k1015840 1205821015840) 119890119894(ksdotrminus120596119905+120579(k120582))


+ 119860lowast



+ 119860lowast



(A5)


⟨119890plusmn119894(120579(k120582)+120579(k1015840 1205821015840))

⟩120579

= 0

⟨119890plusmn119894(120579(k120582)minus120579(k1015840 1205821015840))

⟩120579

= 12057512058210158401205821205753

(k1015840 minus k) (A6)


⟨119891 (k 120582 k1015840 1205821015840 r 119905)⟩120579



12057512058210158401205821205753

(k1015840 minus k)

+ 119860lowast



12057512058210158401205821205753

(k1015840 minus k)

(A7)



(A4) and (A7)

⟨|E|2⟩120579

=1

4sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120576 (k 120582) sdot 120576 (k1015840 1205821015840)) ⟨119891⟩120579

=1

2

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2

⟨|B|2⟩120579

=1

41198882sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120585 (k 120582) sdot 120585 (k1015840 1205821015840)) ⟨119891⟩120579

=1

21198882

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2

(A8)


120579

in unbounded space

⟨119906⟩120579=1205980

2

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2 (A9)


119906vac =1

119881sum

k120582

ℎ120596

2

=1

119881sum

k120582

ℎ120596

2(119871119909

2120587Δ119896119909)(

119871119910

2120587Δ119896119910)(

119871119911

2120587Δ119896119911)

=

2

sum

120582=1

1

(2120587)3sum

kΔ3

119896ℎ120596

2

(A10)



3


119906vac =2

sum

120582=1

1

(2120587)3int1198893

119896ℎ120596

2 (A11)


1003816100381610038161003816119860vac (k 120582)10038161003816100381610038162

=ℎ120596

(2120587)3

1205980

(A12)


119860vac (k 120582) = radicℎ120596

812058731205980

(A13)


=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)radic ℎ120596

812058731205980


119890119894120579(k120582)


119890minus119894120579(k120582)

)

Bvac (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896120585 (k 120582)radic ℎ120596

812058731205980


119890119894120579(k120582)


119890minus119894120579(k120582)

)

(A14)

which is (1)


E (r 119905) = 1

2sum

k120582(119860k120582119890



) 120576k120582

B (r 119905) = 1

2119888sum

k120582(119860k120582119890



) 120585k120582

(A15)



120579= ⟨119890119894120579k120582⟩120579⟨119890plusmn119894120579k1015840 1205821015840 ⟩

120579=

0)

⟨119890plusmn119894(120579k120582+120579k1015840 1205821015840 )⟩

120579

= 0

⟨119890plusmn119894(120579k120582minus120579k1015840 1205821015840 )⟩

120579

= 1205751205821015840120582120575k1015840 k

(A16)


⟨119906⟩120579=1205980

2sum

k120582

1003816100381610038161003816119860k12058210038161003816100381610038162

(A17)



119906vac =1

119881sum

k120582

ℎ120596

2 (A18)


119860vack120582 = radicℎ120596

1205980119881 (A19)


Evac (r 119905) =1

2sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120576k120582

Bvac (r 119905) =1

2119888sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120585k120582

(A20)



k = (119909

119910

119911

) = (

0

0

119896

) (B1)


k1 = (cos120594sin1205940

) k2 = (minus sin120594cos1205940

) (B2)


= (119911)

120601(119910)

120579

= (


sin120601 cos120601 0

0 0 1

)(

cos 120579 0 sin 1205790 1 0

minus sin 120579 0 cos 120579)

= (


)

(B3)


2

1

0

minus1

minus2

0 3 6 9 12Time (s)

Fiel

d (V

m)

120591rep

120591env


1=


1 1198792]LCM = 6




) (B4)


120576k1 = k1 = (



120576k2 = k2 = (


sin 120579 sin120594)

(B5)

C Repetitive Time


119864 (119905) =

119873

sum

119896=1

119864119896cos (120596

119896119905) (C1)


119864 (119905 + 120591rep) = 119864 (119905) (C2)

120591rep = [1198791 1198792 119879119873]LCM (C3)

where 119879119896= 2120587120596



120591rep =2120587

(1205961 1205962 120596

119873)GCD

(C4)


1205961+ 1205962+ sdot sdot sdot + 120596

119873

=2120587

1198791

+2120587

1198792


119879119873

=2120587

[1198791 1198792 119879

119873]LCM

(1198991+ 1198992+ sdot sdot sdot + 119899

119873)

(C5)


119899119896=[1198791 1198792 119879

119873]LCM

119879119896

(C6)




Δ120596gcd equiv (1205961 1205962 120596119873)GCD

=2120587

[1198791 1198792 119879

119873]LCM

(C7)


120591rep =2120587

Δ120596gcd(C8)


120591int =2120587

Δ120596 (C9)


Δ120596 (120581) = 120596 (120581) minus 120596 (120581 minus Δ120581)

= 119888(3120581)13

minus 119888(3120581)13

(1 minusΔ120581

120581)

13

(C10)


0and

120581 ≃ 1205810

Δ120581

1205810

= (3

119873k minus 1)Δ

1205960

≪ 1

120581 = 1205810+ 119874(

Δ

1205960

)

(C11)


where 1205810= (13)(120596



Δ120596 ≃119888(31205810)13

3

Δ120581

1205810

(C12)

Acknowledgments


References



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Vacuum field

SEDRadiationdamping

X

X

2

1

0

minus1

minus2

Time (s)

times10minus8

times10minus12

Posit

ion

(m)

2

1

0

minus1

minus2649647645

Time (s)

times10minus8

times10minus13

Posit

ion

(m)

2

1

0

minus1

minus2

times10minus8

Posit

ion

(m)

CMCM

Probability density

SED (theory)

0 5 10

2

1

0

minus1

minus2

times10minus8

Posit

ion

(m)


SED (simulation)

2 3 4 5 6 7

times107

times107














Appendices




E (r 119905)

=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)


B (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896 (k times 120576 (k 120582))




119860(k 120582)119890119894120579(k120582)

E (r 119905)

=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)


+119860lowast



B (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896120585 (k 120582)


+119860lowast


(A2)


⟨119906⟩120579=1205980

2⟨|E|2⟩

120579

+1

21205830

⟨|B|2⟩120579

(A3)


0is the vacuum



and |B|2 using (A2)

|E|2 (r 119905)

= E (r 119905)Elowast (r 119905)

=1

4sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120576 (k 120582) sdot 120576 (k1015840 1205821015840)) 119891

|B|2 (r 119905)

= B (r 119905)Blowast (r 119905)

=1

41198882sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120585 (k 120582) sdot 120585 (k1015840 1205821015840)) 119891

(A4)

where 119891 equiv 119891(k 120582 k1015840 1205821015840 r 119905)

119891 (k 120582 k1015840 1205821015840 r 119905)



+ 119860 (k 120582) 119860 (k1015840 1205821015840) 119890119894(ksdotrminus120596119905+120579(k120582))


+ 119860lowast



+ 119860lowast



(A5)


⟨119890plusmn119894(120579(k120582)+120579(k1015840 1205821015840))

⟩120579

= 0

⟨119890plusmn119894(120579(k120582)minus120579(k1015840 1205821015840))

⟩120579

= 12057512058210158401205821205753

(k1015840 minus k) (A6)


⟨119891 (k 120582 k1015840 1205821015840 r 119905)⟩120579



12057512058210158401205821205753

(k1015840 minus k)

+ 119860lowast



12057512058210158401205821205753

(k1015840 minus k)

(A7)



(A4) and (A7)

⟨|E|2⟩120579

=1

4sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120576 (k 120582) sdot 120576 (k1015840 1205821015840)) ⟨119891⟩120579

=1

2

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2

⟨|B|2⟩120579

=1

41198882sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120585 (k 120582) sdot 120585 (k1015840 1205821015840)) ⟨119891⟩120579

=1

21198882

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2

(A8)


120579

in unbounded space

⟨119906⟩120579=1205980

2

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2 (A9)


119906vac =1

119881sum

k120582

ℎ120596

2

=1

119881sum

k120582

ℎ120596

2(119871119909

2120587Δ119896119909)(

119871119910

2120587Δ119896119910)(

119871119911

2120587Δ119896119911)

=

2

sum

120582=1

1

(2120587)3sum

kΔ3

119896ℎ120596

2

(A10)



3


119906vac =2

sum

120582=1

1

(2120587)3int1198893

119896ℎ120596

2 (A11)


1003816100381610038161003816119860vac (k 120582)10038161003816100381610038162

=ℎ120596

(2120587)3

1205980

(A12)


119860vac (k 120582) = radicℎ120596

812058731205980

(A13)


=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)radic ℎ120596

812058731205980


119890119894120579(k120582)


119890minus119894120579(k120582)

)

Bvac (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896120585 (k 120582)radic ℎ120596

812058731205980


119890119894120579(k120582)


119890minus119894120579(k120582)

)

(A14)

which is (1)


E (r 119905) = 1

2sum

k120582(119860k120582119890



) 120576k120582

B (r 119905) = 1

2119888sum

k120582(119860k120582119890



) 120585k120582

(A15)



120579= ⟨119890119894120579k120582⟩120579⟨119890plusmn119894120579k1015840 1205821015840 ⟩

120579=

0)

⟨119890plusmn119894(120579k120582+120579k1015840 1205821015840 )⟩

120579

= 0

⟨119890plusmn119894(120579k120582minus120579k1015840 1205821015840 )⟩

120579

= 1205751205821015840120582120575k1015840 k

(A16)


⟨119906⟩120579=1205980

2sum

k120582

1003816100381610038161003816119860k12058210038161003816100381610038162

(A17)



119906vac =1

119881sum

k120582

ℎ120596

2 (A18)


119860vack120582 = radicℎ120596

1205980119881 (A19)


Evac (r 119905) =1

2sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120576k120582

Bvac (r 119905) =1

2119888sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120585k120582

(A20)



k = (119909

119910

119911

) = (

0

0

119896

) (B1)


k1 = (cos120594sin1205940

) k2 = (minus sin120594cos1205940

) (B2)


= (119911)

120601(119910)

120579

= (


sin120601 cos120601 0

0 0 1

)(

cos 120579 0 sin 1205790 1 0

minus sin 120579 0 cos 120579)

= (


)

(B3)


2

1

0

minus1

minus2

0 3 6 9 12Time (s)

Fiel

d (V

m)

120591rep

120591env


1=


1 1198792]LCM = 6




) (B4)


120576k1 = k1 = (



120576k2 = k2 = (


sin 120579 sin120594)

(B5)

C Repetitive Time


119864 (119905) =

119873

sum

119896=1

119864119896cos (120596

119896119905) (C1)


119864 (119905 + 120591rep) = 119864 (119905) (C2)

120591rep = [1198791 1198792 119879119873]LCM (C3)

where 119879119896= 2120587120596



120591rep =2120587

(1205961 1205962 120596

119873)GCD

(C4)


1205961+ 1205962+ sdot sdot sdot + 120596

119873

=2120587

1198791

+2120587

1198792


119879119873

=2120587

[1198791 1198792 119879

119873]LCM

(1198991+ 1198992+ sdot sdot sdot + 119899

119873)

(C5)


119899119896=[1198791 1198792 119879

119873]LCM

119879119896

(C6)




Δ120596gcd equiv (1205961 1205962 120596119873)GCD

=2120587

[1198791 1198792 119879

119873]LCM

(C7)


120591rep =2120587

Δ120596gcd(C8)


120591int =2120587

Δ120596 (C9)


Δ120596 (120581) = 120596 (120581) minus 120596 (120581 minus Δ120581)

= 119888(3120581)13

minus 119888(3120581)13

(1 minusΔ120581

120581)

13

(C10)


0and

120581 ≃ 1205810

Δ120581

1205810

= (3

119873k minus 1)Δ

1205960

≪ 1

120581 = 1205810+ 119874(

Δ

1205960

)

(C11)


where 1205810= (13)(120596



Δ120596 ≃119888(31205810)13

3

Δ120581

1205810

(C12)

Acknowledgments


References



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








Appendices




E (r 119905)

=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)


B (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896 (k times 120576 (k 120582))




119860(k 120582)119890119894120579(k120582)

E (r 119905)

=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)


+119860lowast



B (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896120585 (k 120582)


+119860lowast


(A2)


⟨119906⟩120579=1205980

2⟨|E|2⟩

120579

+1

21205830

⟨|B|2⟩120579

(A3)


0is the vacuum



and |B|2 using (A2)

|E|2 (r 119905)

= E (r 119905)Elowast (r 119905)

=1

4sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120576 (k 120582) sdot 120576 (k1015840 1205821015840)) 119891

|B|2 (r 119905)

= B (r 119905)Blowast (r 119905)

=1

41198882sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120585 (k 120582) sdot 120585 (k1015840 1205821015840)) 119891

(A4)

where 119891 equiv 119891(k 120582 k1015840 1205821015840 r 119905)

119891 (k 120582 k1015840 1205821015840 r 119905)



+ 119860 (k 120582) 119860 (k1015840 1205821015840) 119890119894(ksdotrminus120596119905+120579(k120582))


+ 119860lowast



+ 119860lowast



(A5)


⟨119890plusmn119894(120579(k120582)+120579(k1015840 1205821015840))

⟩120579

= 0

⟨119890plusmn119894(120579(k120582)minus120579(k1015840 1205821015840))

⟩120579

= 12057512058210158401205821205753

(k1015840 minus k) (A6)


⟨119891 (k 120582 k1015840 1205821015840 r 119905)⟩120579



12057512058210158401205821205753

(k1015840 minus k)

+ 119860lowast



12057512058210158401205821205753

(k1015840 minus k)

(A7)



(A4) and (A7)

⟨|E|2⟩120579

=1

4sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120576 (k 120582) sdot 120576 (k1015840 1205821015840)) ⟨119891⟩120579

=1

2

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2

⟨|B|2⟩120579

=1

41198882sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120585 (k 120582) sdot 120585 (k1015840 1205821015840)) ⟨119891⟩120579

=1

21198882

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2

(A8)


120579

in unbounded space

⟨119906⟩120579=1205980

2

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2 (A9)


119906vac =1

119881sum

k120582

ℎ120596

2

=1

119881sum

k120582

ℎ120596

2(119871119909

2120587Δ119896119909)(

119871119910

2120587Δ119896119910)(

119871119911

2120587Δ119896119911)

=

2

sum

120582=1

1

(2120587)3sum

kΔ3

119896ℎ120596

2

(A10)



3


119906vac =2

sum

120582=1

1

(2120587)3int1198893

119896ℎ120596

2 (A11)


1003816100381610038161003816119860vac (k 120582)10038161003816100381610038162

=ℎ120596

(2120587)3

1205980

(A12)


119860vac (k 120582) = radicℎ120596

812058731205980

(A13)


=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)radic ℎ120596

812058731205980


119890119894120579(k120582)


119890minus119894120579(k120582)

)

Bvac (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896120585 (k 120582)radic ℎ120596

812058731205980


119890119894120579(k120582)


119890minus119894120579(k120582)

)

(A14)

which is (1)


E (r 119905) = 1

2sum

k120582(119860k120582119890



) 120576k120582

B (r 119905) = 1

2119888sum

k120582(119860k120582119890



) 120585k120582

(A15)



120579= ⟨119890119894120579k120582⟩120579⟨119890plusmn119894120579k1015840 1205821015840 ⟩

120579=

0)

⟨119890plusmn119894(120579k120582+120579k1015840 1205821015840 )⟩

120579

= 0

⟨119890plusmn119894(120579k120582minus120579k1015840 1205821015840 )⟩

120579

= 1205751205821015840120582120575k1015840 k

(A16)


⟨119906⟩120579=1205980

2sum

k120582

1003816100381610038161003816119860k12058210038161003816100381610038162

(A17)



119906vac =1

119881sum

k120582

ℎ120596

2 (A18)


119860vack120582 = radicℎ120596

1205980119881 (A19)


Evac (r 119905) =1

2sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120576k120582

Bvac (r 119905) =1

2119888sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120585k120582

(A20)



k = (119909

119910

119911

) = (

0

0

119896

) (B1)


k1 = (cos120594sin1205940

) k2 = (minus sin120594cos1205940

) (B2)


= (119911)

120601(119910)

120579

= (


sin120601 cos120601 0

0 0 1

)(

cos 120579 0 sin 1205790 1 0

minus sin 120579 0 cos 120579)

= (


)

(B3)


2

1

0

minus1

minus2

0 3 6 9 12Time (s)

Fiel

d (V

m)

120591rep

120591env


1=


1 1198792]LCM = 6




) (B4)


120576k1 = k1 = (



120576k2 = k2 = (


sin 120579 sin120594)

(B5)

C Repetitive Time


119864 (119905) =

119873

sum

119896=1

119864119896cos (120596

119896119905) (C1)


119864 (119905 + 120591rep) = 119864 (119905) (C2)

120591rep = [1198791 1198792 119879119873]LCM (C3)

where 119879119896= 2120587120596



120591rep =2120587

(1205961 1205962 120596

119873)GCD

(C4)


1205961+ 1205962+ sdot sdot sdot + 120596

119873

=2120587

1198791

+2120587

1198792


119879119873

=2120587

[1198791 1198792 119879

119873]LCM

(1198991+ 1198992+ sdot sdot sdot + 119899

119873)

(C5)


119899119896=[1198791 1198792 119879

119873]LCM

119879119896

(C6)




Δ120596gcd equiv (1205961 1205962 120596119873)GCD

=2120587

[1198791 1198792 119879

119873]LCM

(C7)


120591rep =2120587

Δ120596gcd(C8)


120591int =2120587

Δ120596 (C9)


Δ120596 (120581) = 120596 (120581) minus 120596 (120581 minus Δ120581)

= 119888(3120581)13

minus 119888(3120581)13

(1 minusΔ120581

120581)

13

(C10)


0and

120581 ≃ 1205810

Δ120581

1205810

= (3

119873k minus 1)Δ

1205960

≪ 1

120581 = 1205810+ 119874(

Δ

1205960

)

(C11)


where 1205810= (13)(120596



Δ120596 ≃119888(31205810)13

3

Δ120581

1205810

(C12)

Acknowledgments


References



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




B (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896120585 (k 120582)


+119860lowast


(A2)


⟨119906⟩120579=1205980

2⟨|E|2⟩

120579

+1

21205830

⟨|B|2⟩120579

(A3)


0is the vacuum



and |B|2 using (A2)

|E|2 (r 119905)

= E (r 119905)Elowast (r 119905)

=1

4sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120576 (k 120582) sdot 120576 (k1015840 1205821015840)) 119891

|B|2 (r 119905)

= B (r 119905)Blowast (r 119905)

=1

41198882sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120585 (k 120582) sdot 120585 (k1015840 1205821015840)) 119891

(A4)

where 119891 equiv 119891(k 120582 k1015840 1205821015840 r 119905)

119891 (k 120582 k1015840 1205821015840 r 119905)



+ 119860 (k 120582) 119860 (k1015840 1205821015840) 119890119894(ksdotrminus120596119905+120579(k120582))


+ 119860lowast



+ 119860lowast



(A5)


⟨119890plusmn119894(120579(k120582)+120579(k1015840 1205821015840))

⟩120579

= 0

⟨119890plusmn119894(120579(k120582)minus120579(k1015840 1205821015840))

⟩120579

= 12057512058210158401205821205753

(k1015840 minus k) (A6)


⟨119891 (k 120582 k1015840 1205821015840 r 119905)⟩120579



12057512058210158401205821205753

(k1015840 minus k)

+ 119860lowast



12057512058210158401205821205753

(k1015840 minus k)

(A7)



(A4) and (A7)

⟨|E|2⟩120579

=1

4sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120576 (k 120582) sdot 120576 (k1015840 1205821015840)) ⟨119891⟩120579

=1

2

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2

⟨|B|2⟩120579

=1

41198882sum

1205821205821015840

int1198893

119896int1198893

1198961015840

(120585 (k 120582) sdot 120585 (k1015840 1205821015840)) ⟨119891⟩120579

=1

21198882

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2

(A8)


120579

in unbounded space

⟨119906⟩120579=1205980

2

2

sum

120582=1

int1198893

119896 |119860 (k 120582)|2 (A9)


119906vac =1

119881sum

k120582

ℎ120596

2

=1

119881sum

k120582

ℎ120596

2(119871119909

2120587Δ119896119909)(

119871119910

2120587Δ119896119910)(

119871119911

2120587Δ119896119911)

=

2

sum

120582=1

1

(2120587)3sum

kΔ3

119896ℎ120596

2

(A10)



3


119906vac =2

sum

120582=1

1

(2120587)3int1198893

119896ℎ120596

2 (A11)


1003816100381610038161003816119860vac (k 120582)10038161003816100381610038162

=ℎ120596

(2120587)3

1205980

(A12)


119860vac (k 120582) = radicℎ120596

812058731205980

(A13)


=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)radic ℎ120596

812058731205980


119890119894120579(k120582)


119890minus119894120579(k120582)

)

Bvac (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896120585 (k 120582)radic ℎ120596

812058731205980


119890119894120579(k120582)


119890minus119894120579(k120582)

)

(A14)

which is (1)


E (r 119905) = 1

2sum

k120582(119860k120582119890



) 120576k120582

B (r 119905) = 1

2119888sum

k120582(119860k120582119890



) 120585k120582

(A15)



120579= ⟨119890119894120579k120582⟩120579⟨119890plusmn119894120579k1015840 1205821015840 ⟩

120579=

0)

⟨119890plusmn119894(120579k120582+120579k1015840 1205821015840 )⟩

120579

= 0

⟨119890plusmn119894(120579k120582minus120579k1015840 1205821015840 )⟩

120579

= 1205751205821015840120582120575k1015840 k

(A16)


⟨119906⟩120579=1205980

2sum

k120582

1003816100381610038161003816119860k12058210038161003816100381610038162

(A17)



119906vac =1

119881sum

k120582

ℎ120596

2 (A18)


119860vack120582 = radicℎ120596

1205980119881 (A19)


Evac (r 119905) =1

2sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120576k120582

Bvac (r 119905) =1

2119888sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120585k120582

(A20)



k = (119909

119910

119911

) = (

0

0

119896

) (B1)


k1 = (cos120594sin1205940

) k2 = (minus sin120594cos1205940

) (B2)


= (119911)

120601(119910)

120579

= (


sin120601 cos120601 0

0 0 1

)(

cos 120579 0 sin 1205790 1 0

minus sin 120579 0 cos 120579)

= (


)

(B3)


2

1

0

minus1

minus2

0 3 6 9 12Time (s)

Fiel

d (V

m)

120591rep

120591env


1=


1 1198792]LCM = 6




) (B4)


120576k1 = k1 = (



120576k2 = k2 = (


sin 120579 sin120594)

(B5)

C Repetitive Time


119864 (119905) =

119873

sum

119896=1

119864119896cos (120596

119896119905) (C1)


119864 (119905 + 120591rep) = 119864 (119905) (C2)

120591rep = [1198791 1198792 119879119873]LCM (C3)

where 119879119896= 2120587120596



120591rep =2120587

(1205961 1205962 120596

119873)GCD

(C4)


1205961+ 1205962+ sdot sdot sdot + 120596

119873

=2120587

1198791

+2120587

1198792


119879119873

=2120587

[1198791 1198792 119879

119873]LCM

(1198991+ 1198992+ sdot sdot sdot + 119899

119873)

(C5)


119899119896=[1198791 1198792 119879

119873]LCM

119879119896

(C6)




Δ120596gcd equiv (1205961 1205962 120596119873)GCD

=2120587

[1198791 1198792 119879

119873]LCM

(C7)


120591rep =2120587

Δ120596gcd(C8)


120591int =2120587

Δ120596 (C9)


Δ120596 (120581) = 120596 (120581) minus 120596 (120581 minus Δ120581)

= 119888(3120581)13

minus 119888(3120581)13

(1 minusΔ120581

120581)

13

(C10)


0and

120581 ≃ 1205810

Δ120581

1205810

= (3

119873k minus 1)Δ

1205960

≪ 1

120581 = 1205810+ 119874(

Δ

1205960

)

(C11)


where 1205810= (13)(120596



Δ120596 ≃119888(31205810)13

3

Δ120581

1205810

(C12)

Acknowledgments


References



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





3


119906vac =2

sum

120582=1

1

(2120587)3int1198893

119896ℎ120596

2 (A11)


1003816100381610038161003816119860vac (k 120582)10038161003816100381610038162

=ℎ120596

(2120587)3

1205980

(A12)


119860vac (k 120582) = radicℎ120596

812058731205980

(A13)


=1

2

2

sum

120582=1

int1198893

119896120576 (k 120582)radic ℎ120596

812058731205980


119890119894120579(k120582)


119890minus119894120579(k120582)

)

Bvac (r 119905)

=1

2119888

2

sum

120582=1

int1198893

119896120585 (k 120582)radic ℎ120596

812058731205980


119890119894120579(k120582)


119890minus119894120579(k120582)

)

(A14)

which is (1)


E (r 119905) = 1

2sum

k120582(119860k120582119890



) 120576k120582

B (r 119905) = 1

2119888sum

k120582(119860k120582119890



) 120585k120582

(A15)



120579= ⟨119890119894120579k120582⟩120579⟨119890plusmn119894120579k1015840 1205821015840 ⟩

120579=

0)

⟨119890plusmn119894(120579k120582+120579k1015840 1205821015840 )⟩

120579

= 0

⟨119890plusmn119894(120579k120582minus120579k1015840 1205821015840 )⟩

120579

= 1205751205821015840120582120575k1015840 k

(A16)


⟨119906⟩120579=1205980

2sum

k120582

1003816100381610038161003816119860k12058210038161003816100381610038162

(A17)



119906vac =1

119881sum

k120582

ℎ120596

2 (A18)


119860vack120582 = radicℎ120596

1205980119881 (A19)


Evac (r 119905) =1

2sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120576k120582

Bvac (r 119905) =1

2119888sum

k120582(radic

ℎ120596

1205980119881119890119894(ksdotrminus120596119905)

119890119894120579k120582 + 119888119888) 120585k120582

(A20)



k = (119909

119910

119911

) = (

0

0

119896

) (B1)


k1 = (cos120594sin1205940

) k2 = (minus sin120594cos1205940

) (B2)


= (119911)

120601(119910)

120579

= (


sin120601 cos120601 0

0 0 1

)(

cos 120579 0 sin 1205790 1 0

minus sin 120579 0 cos 120579)

= (


)

(B3)


2

1

0

minus1

minus2

0 3 6 9 12Time (s)

Fiel

d (V

m)

120591rep

120591env


1=


1 1198792]LCM = 6




) (B4)


120576k1 = k1 = (



120576k2 = k2 = (


sin 120579 sin120594)

(B5)

C Repetitive Time


119864 (119905) =

119873

sum

119896=1

119864119896cos (120596

119896119905) (C1)


119864 (119905 + 120591rep) = 119864 (119905) (C2)

120591rep = [1198791 1198792 119879119873]LCM (C3)

where 119879119896= 2120587120596



120591rep =2120587

(1205961 1205962 120596

119873)GCD

(C4)


1205961+ 1205962+ sdot sdot sdot + 120596

119873

=2120587

1198791

+2120587

1198792


119879119873

=2120587

[1198791 1198792 119879

119873]LCM

(1198991+ 1198992+ sdot sdot sdot + 119899

119873)

(C5)


119899119896=[1198791 1198792 119879

119873]LCM

119879119896

(C6)




Δ120596gcd equiv (1205961 1205962 120596119873)GCD

=2120587

[1198791 1198792 119879

119873]LCM

(C7)


120591rep =2120587

Δ120596gcd(C8)


120591int =2120587

Δ120596 (C9)


Δ120596 (120581) = 120596 (120581) minus 120596 (120581 minus Δ120581)

= 119888(3120581)13

minus 119888(3120581)13

(1 minusΔ120581

120581)

13

(C10)


0and

120581 ≃ 1205810

Δ120581

1205810

= (3

119873k minus 1)Δ

1205960

≪ 1

120581 = 1205810+ 119874(

Δ

1205960

)

(C11)


where 1205810= (13)(120596



Δ120596 ≃119888(31205810)13

3

Δ120581

1205810

(C12)

Acknowledgments


References



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




2

1

0

minus1

minus2

0 3 6 9 12Time (s)

Fiel

d (V

m)

120591rep

120591env


1=


1 1198792]LCM = 6




) (B4)


120576k1 = k1 = (



120576k2 = k2 = (


sin 120579 sin120594)

(B5)

C Repetitive Time


119864 (119905) =

119873

sum

119896=1

119864119896cos (120596

119896119905) (C1)


119864 (119905 + 120591rep) = 119864 (119905) (C2)

120591rep = [1198791 1198792 119879119873]LCM (C3)

where 119879119896= 2120587120596



120591rep =2120587

(1205961 1205962 120596

119873)GCD

(C4)


1205961+ 1205962+ sdot sdot sdot + 120596

119873

=2120587

1198791

+2120587

1198792


119879119873

=2120587

[1198791 1198792 119879

119873]LCM

(1198991+ 1198992+ sdot sdot sdot + 119899

119873)

(C5)


119899119896=[1198791 1198792 119879

119873]LCM

119879119896

(C6)




Δ120596gcd equiv (1205961 1205962 120596119873)GCD

=2120587

[1198791 1198792 119879

119873]LCM

(C7)


120591rep =2120587

Δ120596gcd(C8)


120591int =2120587

Δ120596 (C9)


Δ120596 (120581) = 120596 (120581) minus 120596 (120581 minus Δ120581)

= 119888(3120581)13

minus 119888(3120581)13

(1 minusΔ120581

120581)

13

(C10)


0and

120581 ≃ 1205810

Δ120581

1205810

= (3

119873k minus 1)Δ

1205960

≪ 1

120581 = 1205810+ 119874(

Δ

1205960

)

(C11)


where 1205810= (13)(120596



Δ120596 ≃119888(31205810)13

3

Δ120581

1205810

(C12)

Acknowledgments


References



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




where 1205810= (13)(120596



Δ120596 ≃119888(31205810)13

3

Δ120581

1205810

(C12)

Acknowledgments


References



























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



research article dynamics underlying the gaussian ...tions of an sed harmonic oscillator are a...

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