review article quantum analysis on time behavior of a … · 2019. 7. 30. · review article...

Review ArticleQuantum Analysis on Time Behavior ofa Lengthening Pendulum

Jeong Ryeol Choi and Ji Nny Song

Department of Radiologic Technology Daegu Health College Buk-gu Daegu 41453 Republic of Korea

Correspondence should be addressed to Jeong Ryeol Choi choiardorhanmailnet

Received 22 August 2015 Accepted 26 October 2015

Academic Editor Ashok Chatterjee

Copyright copy 2016 J R Choi and J N Song This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Quantum properties of a lengthening pendulum are studied under the assumption that the length of the string increases at a steadyrate Advanced analysis for various physical problems in several types of quantum states such as propagators Wigner distributionfunctions energy eigenvalues probability densities and dispersions of physical quantities is carried out using quantum wavefunctions of the system In particular the time behavior of Gaussian-type wave packets is investigated in detail The probabilitydensity for a Gaussian wave packet displaced in the positive 120579 at 119905 = 0 oscillates back and forth from the center (120579 = 0)This phenomenon is very similar to the classical motion of the pendulum As a consequence we can confirm that there is acorrespondence between its quantum and classical behaviorsWhenwe analyze a dynamical system in view of quantummechanicsthe quantum and classical correspondence is very important in order for the associated quantum theory to be valid and viable

1 Introduction

If the Hamiltonian of a system is dependent on time it isclassified as a time-dependent Hamiltonian system (TDHS)Typical examples of TDHSs include a damped harmonicoscillator [1ndash3] a harmonic oscillator driven by a time-vary-ing force [4 5] and a harmonic oscillator with time-depend-ent parameters such as time-dependent mass [6ndash9] andorfrequency [10ndash12] A powerful method for solving quan-tum solutions of a TDHS is the invariant operator method[13ndash17]

Among various types of TDHSs we study quantumdynamical properties of a lengthening pendulum [18ndash23] inthis paper This kind of pendulum is an interesting topicas a fundamental nonstationary mechanical system and itsstudy may enable us to acquire elementary methods formanipulating more complicated general dynamical systemsThe derivation of quantum mechanical solutions of suchtime-varying swinging objects and the analysis of them withhigh precision demands exquisite theory of quantum physicsand chaos based on advanced mathematical techniques

Much attention has been paid to the classical and quan-tum problems of the lengthening pendulum up to now Thetime variation of angular and linear amplitudes has beenanalyzed by Brearley for a simple lengthening pendulumunder the assumption that the amplitude of the oscillationis small [18] The characteristics of quantum wave func-tions in Fock state for the lengthening pendulum havebeen investigated by Um et al [21] The research group ofMcMillan et al [23] studied a parametric amplification ofits oscillatory motion and the physical mechanism associatedwith the energy transfer to the pendulum via a time-varyingparameter (the length of pendulum) under the assumptionthat the lengthening pendulum is driven by an arbitraryforce Nevertheless as far as we know the exact analyticalsolutions have not been obtained yet due to the diffi-culty in mathematical treatment of the differential equationdescribing the motion of the pendulum which involves asinusoidal function The derivation of quantum solutionsof the system with higher accuracy than what we can doby hand requires the aid of computer programs such asMATLAB [23]

Hindawi Publishing CorporationPhysics Research InternationalVolume 2016 Article ID 1696105 9 pageshttpdxdoiorg10115520161696105

2 Physics Research International

The pendulum or the lengthening thereof not only is agoodmathematicalmodel for practicing a nontrivial problemin mechanics but also can possibly be applied to manyactual systems A gravitational pendulum which has beenused as a seismic sensor of translation and rotation is oneof its well-known applications [24] Usually seismologistsuse a kind of sensing device (pendulum) in order to detectearthquakewaves which ismore complicated than the simplependulum The theory of the lengthening pendulum canalso be applied to analyzing a drifting spiral motion of acharged particle in mirror-confined plasma [25] In this casea position-dependent magnetic field which is a dense andweak magnetic field is responsible for the change of therotational radius along its spiral motion

We will study in this work several quantum states ofthe lengthening pendulum beyond the Fock state We willcompare quantummotions of the pendulumwith its classicalbehaviors Quantized energy and its dispersion will be inves-tigated via a rigorous development of the quantum theoryassociated with the pendulum

This paper is organized as follows The fundamentalmechanics of the lengthening pendulum is surveyed inSection 2 and various quantum mechanical properties ofthe system are analyzed in the subsequent sections Sec-tion 3 is devoted to the study of the time behavior of thequantized energy Path integral formulation of quantummechanics for the system is treated in Section 4 Wignerdistribution function (WDF) of the system is investigatedin Section 5 Quantum properties of the superposition oftwo neighboring states are discussed in Section 6 The cor-respondence between classical and quantum mechanics forthe pendulum is demonstrated in Section 7 through Gaus-sian wave packet description of the system The con-cluding remarks are given in Section 8 which is the last sec-tion

2 Fundamentals ofthe Lengthening Pendulum

Now we are carrying out a survey of the fundamentals ofthe lengthening pendulum Let us consider a pendulum of arelatively massive object hung by a vertical string from a fixedceiling and assume that it swings from its fixed equilibriumposition Suppose that the length of the pendulum increaseswith a constant rate 119896 from its initial length 119897

0

119897 (119905) = 1198970+ 119896119905 (1)

In case 119896 is sufficiently small the pendulum undergoes anadiabatic change and such case will only be considered in thisreview

We do not regard the quantum effects of the 119897-componentfor the sake of simplicity Then the Hamiltonian consideredup to the second order in 120579 is given by [21]

( 120579) =

2

120579

21198981198972 (119905)+1

2119898119892119897 (119905)

2

+H (119905) (2)

where H(119905) = 11989811989622 minus 119898119892119897(119905) A classical solution of the

system can be represented as [22]

120579 (119905) = 119888119872 (119905) 119890119894[120574(119905)+1198880] (3)

Here 119888 and 1198880are arbitrary constants and

119872(119905) =1

radic119897 (119905)[1198692

1(120585 (119905)) + 119873

2

1(120585 (119905))]

12

120574 (119905) = tanminus11198731 (120585 (119905))1198691 (120585 (119905))

(4)

where 119869 and 119873 are Bessel functions of the first kind and thesecond kind respectively and 120585(119905) = 2[119892119897(119905)]

12119896 A more

general solution for this system is given in [21]The construction of the annihilation and creation oper-

ators is useful for developing a quantum theory of a TDHSAccording to the invariant operator theory [15] the formulaeof them are somewhat different from those of the simpleharmonic oscillator For instance the annihilation operatoris given by [21]

= radic1

2ℏ (119905)1198981198972(119905)

[1198981198972(119905) ( (119905) minus 119894

(119905)

119872 (119905))

+ 119894120579]

(5)

Of course the Hermitian adjoint dagger of the above equationplays the role of the creation operatorThese operators satisfythe boson commutation relation such that [ dagger] = 1

If we use the property of this commutation relation wecan easily derive the wave functions in Fock state whichare represented in terms of some time functions 120601

119899(120579 119905) and

time-dependent phases 120598119899(119905) [21 22]

120595119899(120579 119905) = 120601

119899(120579 119905) 119890

119894120598119899(119905) (6)

where the quantum number is given by 119899 = 1 2 3 (All subsequent 119899 will follow this convention unless weparticularly specify it)The explicit forms of 120601

119899(120579 119905) and 120598

119899(119905)

are given by

120601119899(120579 119905)

=1

radic2119899119899(Λ (119905)

120587)

14

119867119899(radicΛ (119905)120579) 119890

minus(12)Λ(119905)120581lowast

1(119905)1205792

120598119899(119905) = minus (119899 +

1

2) 120574 (119905)

(7)

Physics Research International 3

where119867119899is 119899th-order Hermite polynomial and

Λ (119905) = 119875 (119905)radic119892119898 [119897 (119905)]

32

2ℏ [11986921(120585 (119905)) + 119873

2

1(120585 (119905))]

1205811(119905) = 1 + 119894Π (119905)

(8)

with

119875 (119905) = 1198691(120585 (119905)) [119873

0(120585 (119905)) minus 119873

2(120585 (119905))]

minus 1198731(120585 (119905)) [119869

0(120585 (119905)) minus 119869

2(120585 (119905))]

Π (119905) = [119875 (119905)]minus1119876 (119905) minus

119896 [1198692

1(120585 (119905)) + 119873

2

1(120585 (119905))]

radic119892119897 (119905)

119876 (119905) = 1198691(120585 (119905)) [119869

0(120585 (119905)) minus 119869

2(120585 (119905))]

+ 1198731(120585 (119905)) [119873

0(120585 (119905)) minus 119873

2(120585 (119905))]

(9)

The wave functions given in (6) are the basic informationnecessary for studying quantum features of the system Wewill investigate quantum characteristics of the pendulum formore complicated cases in the next section

3 Quantized Energy

Bohr tried to merge quantum and classical mechanics byintroducing a correspondence principle between them Eventhough the results of quantum and classical descriptions fora system more or less overlap under particular limits theirunderlying principles are quite different from each otherThere are significant differences between consequences ofthe two theories when the quantum number is sufficientlylow whereas those with a higher quantum number look verysimilar It is well known that we can compute the probabilityfor finding the pendulumat a particular angle from the squareof the given wave function In cases of higher energy statesthe most probable angle of the pendulum will shift awayfrom the center (120579 = 0) and the intervals between any twoadjacent peaks in the graph of probability densities in Fockstate become narrow Due to this trend associated with theprobability density the behavior of the quantum pendulumin a high energy limit may look like that of the counterpartclassical one

Another difference between them is that while the clas-sical probability density function is confined within the twoclassical turning points the quantum probabilities extendbeyond the classically allowed angle Formore detailed inves-tigations about this consequence we consider the quantumand the classical energies of the system that are given by

119864119899= (119899 +

1

2)ℏ

2[ℏΛ (119905)

1198981198972 (119905)

10038161003816100381610038161205811 (119905)10038161003816100381610038162+119898119892119897 (119905)

ℏΛ (119905)]

+H (119905)

(10)

119864cl =1

21198981198972(119905) 2

+1

2119898119892119897 (119905) 120579

2+H (119905) (11)

Let us evaluate the probability that the pendulum staysoutside the classically allowed region If we denote the ampli-tude of angle as Θ the maximum classical potential energyis given by 119898119892119897(119905)Θ22 +H(119905) Then regarding the classicalturning point of the pendulum associated with the groundstate wave packet we equate the ground state energy 119864

0

with the maximum classical potential energy

1198640=1

2119898119892119897 (119905) Θ

2+H (119905) (12)

Thus by inserting (10) with 119899 = 0 into the left-hand sideof this equation the classical amplitude can be evaluated tobe

Θ = [ℏ

2119898119892119897 (119905)(ℏΛ (119905)

1198981198972 (119905)

10038161003816100381610038161205811 (119905)10038161003816100381610038162+119898119892119897 (119905)

ℏΛ (119905))]

12

(13)

The probability density outside the classically allowed regionis given by

119875outside = 2int

120587

Θ

10038161003816100381610038161205950 (120579 119905)10038161003816100381610038162119889120579 ≃ 2int

infin

Θ

10038161003816100381610038161205950 (120579 119905)10038161003816100381610038162119889120579 (14)

Here factor 2 is multiplied considering the equal contribu-tion of the probability from outside of the opposite turningangle [minus120587 minusΘ] In addition we have supposed that theoscillating amplitude is sufficiently small in order to replacethe integral interval [Θ 120587] in (14) with [Θinfin]This extensionof the scope of integration will also be equally applied infurther similar calculations By substituting (6) with 119899 = 0

into (14) we have

119875outside = radic4

120587[1 minus erf (radicΛ (119905)Θ)] (15)

This is the probability that the quantum pendulum remainswithin the classically forbidden regions

In fact the description of quantized energy for a TDHSsuch as (10) is a delicate problem While some authors [1 311 21 22] including ours have investigated quantized energylevels for specific TDHSs there is another opinion [26 27]that such energy levels do not exist for the case of TDHSsHence explicit demonstrations of the existence of quantizedenergy levels may be an interesting research topic for furtherstudy in the future in this field

4 Propagator

The path integral formulation of quantum mechanics wasfound by Feynman [28] as an alternative quantumdescriptionof dynamical systems Historically this achievement waspartly inspired by Diracrsquos idea of a quantum mechanicaldescription [29] The techniques of path integrals continuedto be advanced until now leading to providing many usefultools for solving problems in quantum mechanics One of


the efficient uses of the path integral method is to evaluatea propagator This achieved great success in both quantummechanics and quantum field theory because a propagatorinvolves all the information of the quantum systemThroughthe analysis of the propagator we can confirm how topropagate quantum wave packets from an initial angle andtime (1205791015840 1199051015840) to a final angle and time (120579 119905)

In terms of wave functions the propagator is given by

119870(120579 119905 1205791015840 1199051015840) =

infin

sum

119899=0

120595119899(120579 119905) 120595

lowast

119899(1205791015840 1199051015840) (16)

Let us calculate the complete analytical form of the propaga-tor of the system using this definition By inserting (6) with(7) we have

119870 = (Λ (119905) Λ (119905

1015840)

1205872)

14

119890minus(12)[Λ(119905)120581

lowast

1(119905)1205792+Λ(1199051015840)1205811(1199051015840)12057910158402]119890minus(1198942)[120574(119905)minus120574(119905

1015840)]

infin

sum

119899=0

119890minus119894119899[120574(119905)minus120574(119905

1015840)]

2119899119899119867119899(radicΛ (119905)120579)119867

119899(radicΛ (1199051015840)120579

1015840) (17)

From the use of Mehlerrsquos formula [30]

infin

sum

119899=0

(1199112)119899

119899119867119899(119909)119867119899(119910)

=1

(1 minus 1199112)12

exp[2119909119910119911 minus (119909

2+ 1199102) 1199112

1 minus 1199112]

(18)

it becomes

119870 =

4radicΛ (119905) Λ (1199051015840)

radic2120587119894 sin [120574 (119905) minus 120574 (1199051015840)]

sdot exp(minus119894

radicΛ (119905) Λ (1199051015840)1205791205791015840

sin [120574 (119905) minus 120574 (1199051015840)])

sdot 119890(1198942)Λ(119905)[Π(119905)+cot[120574(119905)minus120574(1199051015840)]]1205792minusΛ(1199051015840)[Π(1199051015840)minuscot[120574(119905)minus120574(1199051015840)]]12057910158402

(19)

We can use this result in estimating the evolution of quantumstates under the given Hamiltonian of the pendulum and instudying the dynamics of the pendulum with varying lengthMore precisely the propagator enables us to know how theinitial state 120595(1205791015840 1199051015840) evolves to an arbitrary later state 120595(120579 119905)via the following equation

120595 (120579 119905) = int

120587

minus120587

1198891205791015840119870(120579 119905 120579

1015840 1199051015840) 120595 (120579

1015840 1199051015840) (20)

The propagator plays an important role in modern physics

5 Wigner Distribution Function

The investigation of the probability distribution is usefulfor understanding the novel features of quantum mechanicsThe WDF [31] may be the most useful one among vari-ous distribution functions that are needed when we studyquantum statistical mechanics Many quantum mechanical

characteristics of a dynamical system which are absent inclassical mechanics can be derived fromWDF

TheWDFs in phase space are given by

119882119899(120579 119901120579 119905)

=1

120587ℏint

120587

minus120587

120595lowast

119899(120579 + 120572 119905) 120595

119899(120579 minus 120572 119905) 119890

2119894119901120579120572ℏ119889120572

(21)

We replace the range of the integration [minus120587 120587]with [minusinfininfin]

for the same reason as that of the previous integration in (14)Using the integral formula of the form [32]

int

infin

minusinfin

119889119911119890minus1199112

119867119899 (119911 + 119886 + 119887)119867119899 (119911 + 119886 minus 119887)

= 211989912058712119899119871119899[2 (1198872minus 1198862)]

(22)

we can evaluate (21) leading to

119882119899(120579 119901120579 119905) =

(minus1)119899

120587ℏ119890minus119884(120579119901120579 119905)119871

119899[2119884 (120579 119901

120579 119905)] (23)

where

119884 (120579 119901120579 119905) = Λ (119905) [120579

2+ (Π (119905) 120579 minus

119901120579

ℏΛ (119905))

2

] (24)

This can be used to study the probability distribution for120579 and 119901

120579of the pendulum within a quantum mechanical

view point Figure 1 shows the WDF for several values of119905 Exact marginal distributions can be achievable with theuse of a WDF whereas it cannot be done by means ofthe Husimi distribution function [33] It is allowed that theWDF takes negative values as well as positive ones in phasespace The negativity of the WDF is in fact one of theprimary nonclassical effects of quantum mechanics [34 35]The WDF given in (23) enables us to study the quantumstatistical properties of the system in configuration spaceSome imaginary distribution functions that can be used ina wide branch of physics were also proposed in the literatureby other authors [36ndash40] The width of119882

119899in the angle space

becomes small while that in the angular momentum spacebecomes large as time goes by


0

5

0

50

01

0

5

Wn

p 120579

minus5

minus5

minus01

120579

(a)

0

5

0

50

01

0

5

Wn

p 120579

minus5

minus5

minus01

120579

(b)

0

5

0

5

01

0

0

5

Wn

p 120579

minus5

minus5

minus01

120579

(c)

Figure 1 The WDF given in (23) as a function of angle 120579 and angular momentum 119901120579where 119905 = 0 for (a) 119905 = 25 for (b) and 119905 = 5 for (c)

We have used ℏ = 1 119896 = 02 119898 = 1 1198970= 1 119892 = 1 and 119899 = 5 The width of 119882

119899associated with 120579 becomes small with time while that

associated with 119901120579becomes large

6 Superposition of Two Neighboring States

The general solution of the Schrodinger equation is a linearcombination of separated solutions given in (6)

120595 (120579 119905) =

infin

sum

119899=0

119888119899120595119899 (120579 119905) (25)

where sum119899|119888119899|2= 1 To study the energy spectrum in the

state it is necessary to evaluate the eigenvalue equation of theHamiltonian which is

120595 (120579 119905) = 119864120595 (120579 119905) (26)

It is well known that quantum mechanics predicts onlythe probability for happening of particular results among alarge number of mutually independent possible outcomesin physical systems An arbitrary state function representingany dynamical state can be expanded in terms of Fock state

wave functions As a first example let us consider the casethat (25) is a superposition of two neighboring states

120595 (120579 119905) =1

radic21198901198941205750 [120595119899(120579 119905) + 119890

119894120575120595119899+1

(120579 119905)] (27)

where phases 1205750and 120575 are arbitrary real constantsThe expec-

tation values of canonical variables in this state can be easilyevaluated to be [41]

⟨1205951003816100381610038161003816100381610038161003816100381610038161003816120595⟩ = radic119899 + 1(

ℏ

2 (119905)1198981198972 (119905))

12

sdot cos [120574 (119905) minus 120575]

⟨1205951003816100381610038161003816120579

1003816100381610038161003816 120595⟩ =radic119899 + 1(

ℏ1198981198972(119905)

2 (119905))

12

sdot ( (119905)

119872 (119905)cos [120574 (119905) minus 120575] minus (119905) sin [120574 (119905) minus 120575])

(28)


From these equations we can confirm that the expectationvalues of and

120579oscillate with time

Through the similar procedure the expectation value ofthe energy operator can also be obtained as

119864 = (119899 + 1)ℏ

2(ℏΛ (119905)

1198981198972 (119905)

10038161003816100381610038161205811 (119905)10038161003816100381610038162+119898119892119897 (119905)

ℏΛ (119905)) +H (119905) (29)

This is the same as the average energy of the 119899th state and(119899 + 1)th state energies with the same proportion Thisenergy corresponds to that of an instantaneous eigenstate andvaries with time like a classical one However this reducesto a constant value when 119896 = 0 which is associated withthe results of standard quantum mechanics for the simplependulum represented in many text books in this field

7 Gaussian Wave Packets

As a next example we consider the case that the initial statefunction is given by

120595 (120579 0) =4radic1205782

120587119890minus(12)120578

21205792

(30)

where 120578 is some constant which is real In view of quantumtheory the probabilistic predictions are always obtained bysquaring the modulus of a probability amplitude The prob-ability amplitudes that the oscillator stays in 119899th eigenstatesare

119888119899= int

120587

minus120587

120595lowast

119899(120579 0) 120595 (120579 0) 119889120579 (31)

Then using (6) (30) and (31) after replacing the interval ofthe integration [minus120587 120587] with [minusinfininfin] we obtain that

119888119899=

radic119899

21198992 (1198992)[

2radicΛ (0)120578

Λ (0) 1205811 (0) + 1205782(Λ (0) 120581

lowast

1(0) minus 120578

2

Λ (0) 1205811 (0) + 1205782)

119899

]

12

sdot 119890119894(119899+12)120574(0)

(32)

for 119899 = 0 2 4 On the other hand 119888119899= 0 for 119899 = 1 3 5

Hence odd order amplitudes do not contribute to the statefunction

Onemay also possibly consider other types of wave pack-ets described as a superposition of Fock state wave functionsNow we are going to choose a state which seems appropriatefor studying whether there is a quantum and classical corre-spondence concerning the behavior of the lengthening pen-dulum At the early era of quantum mechanics Schrodingertried to explain the relation between quantum and classicalmechanics by introducing coherent wave packets [42] Inorder to find the possibility for the existence of a quantumsolution relevant to a particular wave packet whose centeroscillates with the same period as the classical motion let usconsider a state function that is expressed at 119905 = 0 as

120595 (120579 0) =4radicΛ (0)

120587119890minus(12)Λ(0)(120579minus120573)

2

(33)

where 120573 is a real constant with the dimension of angle Thisis the same as (30) except that 1205782 is replaced by Λ(0) and thecenter of the initial wave packet is displaced by an amount120573 In this case the probability amplitudes 119888

119899can also be

obtained using the same method as that of the first case (thenondisplaced Gaussian case) performed with (31) such that

119888119899=

1

radic2119899119899

120573119899

[1205812(0) 2]

119899+12

sdot 119890(12)[11205812(0)minusΛ(0)]120573

2+119894(119899+12)120574(0)

(34)

where 1205812(119905) = 2 + 119894Π(119905) Here we have supposed that the

increase of the length of string is sufficiently slow With thehelp of (6) and (34) the time evolution of the state function(25) becomes

120595 (120579 119905) =4radicΛ (119905)

120587radic

2

1205812(0)

exp [2120573

1205812(0)

sdot 119890119894[120574(0)minus120574(119905)]

(radicΛ (119905)120579 minus1

2

120573

1205812 (0)

119890119894[120574(0)minus120574(119905)]

)]

sdot 119890(12)[11205812(0)minusΛ(0)]120573

2

119890minus(12)Λ(119905)1205811(119905)120579

2

119890(1198942)[120574(0)minus120574(119905)]

(35)

The corresponding probability density is obtained by squar-ing the above equation Thus a direct evaluation gives

1003816100381610038161003816120595 (120579 119905)10038161003816100381610038162= radic

Λ (119905)

120587

210038161003816100381610038161205812 (0)

1003816100381610038161003816

sdot exp[minusΛ (119905) (120579 minus120573 [119883 (119905) + 119883

lowast(119905)]

radicΛ (119905))

2

]

sdot 119890[4|1205812(0)|

2minusΛ(0)]120573

2

(36)

where 119883(119905) = 119890119894[120574(0)minus120574(119905)]

1205812(0) The wave packet equation

(36) is a time-dependent Gaussian shape We depicted thiswave packet in Figure 2 The wave packet not only convergesto the center (120579 = 0) but also oscillates back and fortharound the center as time goes by The amplitude of suchoscillation is determined depending on the magnitude of 120573This quantum behavior is very similar to the classical motionof the pendulum The width of the wave packet is 1radic2Λ(119905)The probabilities for the contribution of 119899th quantum state120595119899(120579 119905) to the state function are 119875

119899= |119888119899|2 Thus using (34)

we have

119875119899=

1

2119899119899

1205732119899

[10038161003816100381610038161205812 (0)

1003816100381610038161003816 2]2119899+1

119890[2|1205812(0)|

2minusΛ(0)]120573

2

(37)

These probabilities do not vary with time The mean value ofquantum numbers which contribute to the state function is119899 = sum

119899119899119875119899 Using (37) we see that this can be evaluated to

be

119899 =1205732

2 [10038161003816100381610038161205812 (0)

1003816100381610038161003816 2]3119890[4|1205812(0)|

2minusΛ(0)]120573

2

(38)


0102 0

20

40

60

t

001020304

0minus01

minus02

120579

|120595|2

Figure 2 The probability density |120595|2 given in (36) as a function ofangle 120579 and time 119905 for ℏ = 1 119896 = 1 119898 = 1 119897

0= 6 119892 = 1 and

120573 = 05 We see that the wave packet oscillates around the centerlike a classical one

Thus for the case that |120573| is large the contribution of highenergy states is also large

The dispersion of 119899 is given by Δ119899 = [sum1198991198992119875119899minus

(sum119899119899119875119899)2]12 This can be easily calculated as

Δ119899 =

1

4

10038161003816100381610038161205812 (0)1003816100381610038161003816

2

minus exp[( 4

10038161003816100381610038161205812 (0)10038161003816100381610038162minus Λ (0))120573

2]

1205734

[10038161003816100381610038161205812 (0)

100381610038161003816100381624]3

+1205732

2 [10038161003816100381610038161205812 (0)

1003816100381610038161003816 2]3

12

119890(12)[4|1205812(0)|

2minusΛ(0)]120573

2

(39)

Figure 3(a) reveals that Δ119899 increases as 120573 growsWe can also evaluate the dispersion of energy from Δ119864 =

[sum1198991198642

119899119875119899minus (sum119899119864119899119875119899)2]12 Through a mathematical proce-

dure similar to that of the calculation of Δ119899 we have

Δ119864 =ℏ120573

10038161003816100381610038161205812 (0)10038161003816100381610038162[

41205732

10038161003816100381610038161205812 (0)10038161003816100381610038162(ℏΛ (119905)

1198981198972 (119905)

10038161003816100381610038161205811 (119905)10038161003816100381610038162

+119898119892119897 (119905)

ℏΛ (119905))(

10038161003816100381610038161205812 (0)1003816100381610038161003816

2minus 119890(4|1205812(0)|

2minusΛ(0))120573

2

)

+8H (119905)

ℏ(

10038161003816100381610038161205812 (0)1003816100381610038161003816

2minus 1) + 2 (

10038161003816100381610038161205812 (0)1003816100381610038161003816 minus 1)]

12

sdot 119890[2|1205812(0)|

2minusΛ(0)2]120573

2

(40)

From Figure 3(b) we can confirm that Δ119864 increases as 120573grows However Δ119864 decreases with time

00501

015

02 0

10

20

30

40

50

t

0002004006008

0

Δn

120573

(a)

00005001

0015002

0025

ΔE

00501

015

02 0

10

20

30

40

50

t0

120573

(b)

Figure 3The dispersion Δ119899 (39) and Δ119864 (40) as a function of 120573 andtime 119905 for ℏ = 1 119896 = 05 119898 = 1 119897

0= 10 and 119892 = 1 Note that Δ119899

increases as 120573 grows

8 Conclusion

On the basis of the Schrodinger solutions that can beobtained by taking advantage of the ladder operators anddagger various quantum states of the lengthening pendulum

have been investigated We obtained the energy eigenvaluespropagatorWDFs uncertainties and probability densities inseveral types of quantum statesThe quantum behavior of thependulum was analyzed rigorously and compared to that ofthe classical one

The propagator given in (19) entirely determines thequantum behavior of the lengthening pendulum in a grav-itation If we know the wave function at an initial angleand time it is possible to estimate probability amplitude foranother angle at a subsequent time using the propagatorWigner distribution function of the system was derived asshown in (23)This can be a negative value in some regions ofphase space as well as a positive one which signifies a novel


nonclassical feature of the quantum system We can confirmfromFigure 1 that thewidth of119882

119899associatedwith 120579 decreases

as time goes by while that associated with 119901120579increases

This appearance is due to the fact that the uncertainty in120579 decreases with time while that in 119901

120579increases For the

variation of the uncertainty for 120579 and 119901120579 you can refer to [41]

As the length of string increases mechanical energy can-not be conserved because some amount of work is extractedfrom the system [22] This is the reason why the amplitudedecreases with time not only for the classical pendulum butfor the quantum one as well

From (28) we see that the expectation values of canonicalvariables in the superposition of the two neighboring statesoscillate with time The quantum energy in this case varieswith time as shown in (29) For the state that is describedby (30) the odd order probability amplitudes disappear since(30) is an even function about 120579

To investigate the existence of the quantum and classicalcorrespondence for the behavior of the lengthening pendu-lum we have supposed that the form of the state functionat 119905 = 0 is given by (33) The initial angle in this case isdisplaced toward a positive 120579 by an amount 120573 The resultingprobability density is represented in (36) which is a Gaussianshape From Figure 2 we can confirm that this probabilitydensity oscillates back and forth from the center dependingon themagnitude of120573 while its width becomes narrower overtime as the quantum energy decays So to speak its quantumbehavior is very similar to that of the classical oscillationof the pendulum Therefore the correspondence betweenquantum and classical behaviors holds

Indeed the quantum and classical correspondence fora certain quantum theory is a crucial requirement for thevalidity of the associated quantum theory The invariantoperator theory we used here is sound and enables us toobtain reasonable quantum solutions for a time-dependentdynamical system Further we know that it sometimes givesthe exact quantum solutions for a system of which theclassical solutions are completely known

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the Basic Science ResearchProgram of the year 2015 through the National ResearchFoundation of Korea (NRF) funded by the Ministry ofEducation (Grant no NRF-2013R1A1A2062907) Some partsof this work were written and developed on the basis of Chap2 of doctoral thesis of Choi [41]

References

[1] K-H Yeon S-S Kim Y-M Moon S-K Hong C-I Um andT F George ldquoThe quantum under- critical- and over-dampeddriven harmonic oscillatorsrdquo Journal of Physics AMathematicaland General vol 34 no 37 pp 7719ndash7732 2001

[2] D M Greenberger ldquoA new approach to the problem of dissipa-tion in quantum mechanicsrdquo Journal of Mathematical Physicsvol 20 no 5 pp 771ndash780 1979

[3] C-I Um K H Yeon andW H Kahng ldquoThe quantum dampeddriven harmonic oscillatorrdquo Journal of Physics A Mathematicaland General vol 20 no 3 pp 611ndash626 1987

[4] S Klarsfeld and J A Oteo ldquoDriven harmonic oscillators in theadiabatic Magnus approximationrdquo Physical Review A vol 47no 3 pp 1620ndash1624 1993

[5] Y I Salamin ldquoOn the quantum harmonic oscillator drivenby a strong linearly polarized fieldrdquo Journal of Physics AMathematical and General vol 28 no 4 pp 1129ndash1138 1995

[6] CMADantas I A Pedrosa andB Baseia ldquoHarmonic oscilla-torwith time-dependentmass and frequency and a perturbativepotentialrdquo Physical Review A vol 45 no 3 pp 1320ndash1324 1992

[7] I A Pedrosa ldquoExact wave functions of a harmonic oscillatorwith time-dependent mass and frequencyrdquo Physical Review Avol 55 no 4 pp 3219ndash3221 1997

[8] J-Y Ji J K Kim and S P Kim ldquoHeisenberg-picture approachto the exact quantum motion of a time-dependent harmonicoscillatorrdquo Physical Review A vol 51 no 5 pp 4268ndash4271 1995

[9] P G L Leach ldquoHarmonic oscillator with variablemassrdquo Journalof Physics A Mathematical and General vol 16 no 14 pp 3261ndash3269 1983

[10] H J Korsch ldquoDynamical invariants and time-dependent har-monic systemsrdquo Physics Letters A vol 74 no 5 pp 294ndash2961979

[11] J-R Choi and J-Y Oh ldquoOperator method for the numbercoherent and squeezed states of time-dependent Hamiltoniansystemrdquo International Mathematical Journal vol 3 no 9 pp939ndash952 2003

[12] R S Kaushal and H J Korsch ldquoDynamical Noether invariantsfor time-dependent nonlinear systemsrdquo Journal of Mathemati-cal Physics vol 22 no 9 pp 1904ndash1908 1981

[13] H R Lewis Jr ldquoClassical and quantum systems with time-dependent harmonic-oscillator-type Hamiltoniansrdquo PhysicalReview Letters vol 18 no 13 pp 510ndash512 1967

[14] H R Lewis Jr ldquoClass of exact invariants for classical andquantum time-dependent harmonic oscillatorsrdquo Journal ofMathematical Physics vol 9 no 11 pp 1976ndash1986 1968

[15] H R Lewis Jr and W B Riesenfeld ldquoAn exact quantum theoryof the time-dependent harmonic oscillator and of a chargedparticle in a time-dependent electromagnetic fieldrdquo Journal ofMathematical Physics vol 10 no 8 pp 1458ndash1473 1969

[16] I A Malkin V I Manrsquoko and D A Trifonov ldquoCoherent statesand transition probabilities in a time-dependent electromag-netic fieldrdquo Physical Review D vol 2 no 8 pp 1371ndash1385 1970

[17] V V Dodonov and V I Manrsquoko ldquoInvariants and evolutionof nonstationary quantum systemsrdquo in Proceedings of LebedevPhysical Institute 183 M A Markov Ed Nova Science Publish-ers Commack NY USA 1989

[18] M N Brearley ldquoThe simple pendulum with uniformly chang-ing string lengthrdquo Proceedings of the Edinburgh MathematicalSociety vol 15 no 1 pp 61ndash66 1966

[19] M L BoasMathematical Methods in the Physical Sciences JohnWiley amp Sons New York NY USA 2nd edition 1983

[20] L C Andrews Special Function of Mathematics for EngineersSPIE Press Oxford UK 2nd edition 1998

[21] C I Um J R Choi K H Yeon S Zhang and T F GeorgeldquoExact quantum theory of a lengthening pendulumrdquo Journal ofthe Korean Physical Society vol 41 no 5 pp 649ndash654 2002


[22] J R Choi J N Song and S J Hong ldquoSpectrum of quantizedenergy for a lengthening pendulumrdquo AIP Conference Proceed-ings vol 1281 no 1 pp 594ndash598 2010

[23] M McMillan D Blasing and H M Whitney ldquoRadial forcingand Edgar Allan Poersquos lengthening pendulumrdquo American Jour-nal of Physics vol 81 no 9 pp 682ndash687 2013

[24] R D Peters ldquoTutorial on gravitational pendulum theoryapplied to seismic sensing of translation and rotationrdquo Bulletinof the Seismological Society of America vol 99 no 2B pp 1050ndash1063 2009

[25] F F Chen Introduction to Plasma Physics and Controled Fusionvol 1 Springer New York NY USA 2006

[26] R W Hasse ldquoOn the quantum mechanical treatment of dissi-pative systemsrdquo Journal of Mathematical Physics vol 16 no 10pp 2005ndash2011 1975

[27] M A Manrsquoko ldquoAnalogs of time-dependent quantum phenom-ena in optical fibersrdquo Journal of Physics Conference Series vol99 no 1 Article ID 012012 2008

[28] R P Feynman ldquoSpace-time approach to non relativistic quan-tum mechanicsrdquo Reviews of Modern Physics vol 20 no 2 pp367ndash387 1948

[29] P A M Dirac ldquoThe Lagrangian in quantum mechanicsrdquoPhysikalische Zeitschrift der Sowjetunion vol 3 no 1 pp 64ndash72 1933

[30] A Erdely Higher Transcendental Functions vol II McGraw-Hill New York NY USA 1953

[31] E Wigner ldquoOn the quantum correction for thermodynamicequilibriumrdquo Physical Review vol 40 no 5 pp 749ndash759 1932

[32] I S Gradshteyn and M Ryzhik Table of Integrals Series andProducts Academic Press New York NY USA 1980

[33] K Husimi ldquoSome formal properties of the density matrixrdquoProceedings of the Physico-Mathematical Society of Japan vol 22no 5 pp 264ndash314 1940

[34] EM E Zayed A S DaoudM A Al-Laithy and E N NaseemldquoTheWigner distribution function for squeezed vacuum super-posed staterdquo Chaos Solitons amp Fractals vol 24 no 4 pp 967ndash975 2005

[35] J R Choi J N Song and S J Hong ldquoWigner distributionfunction of superposed quantum states for a time-dependentoscillator-like Hamiltonian systemrdquo Journal of Theoretical andApplied Physics vol 6 no 1 article 26 2012

[36] S Khademi and S Nasiri ldquoGauge invariant quantization ofdissipative systems of charged particles in extended phasespacerdquo Iranian Journal of Science and Technology TransactionA Science vol 26 no 1 2002

[37] M Mahmoudi Y I Salamin and C H Keitel ldquoFree-electronquantum signatures in intense laser fieldsrdquo Physical Review Avol 72 no 3 Article ID 033402 2005

[38] R F OrsquoConnell and L Wang ldquoPhase-space representations ofthe Bloch equationrdquo Physical Review A vol 31 no 3 pp 1707ndash1711 1985

[39] R J Glauber ldquoCoherent and incoherent states of the radiationfieldrdquo Physical Review vol 131 no 6 pp 2766ndash2788 1963

[40] J G Kirkwood ldquoQuantum statistics of almost classical assem-bliesrdquo Physical Review vol 44 no 1 pp 31ndash37 1933

[41] J R Choi Quantum mechanical treatment of time-dependenthamiltonians [PhD thesis] Korea University Seoul SouthKorea 2001

[42] L I Schiff Quantum Mechanics McGraw-Hill New York NYUSA 1968

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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FluidsJournal of

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OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


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Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


The pendulum or the lengthening thereof not only is agoodmathematicalmodel for practicing a nontrivial problemin mechanics but also can possibly be applied to manyactual systems A gravitational pendulum which has beenused as a seismic sensor of translation and rotation is oneof its well-known applications [24] Usually seismologistsuse a kind of sensing device (pendulum) in order to detectearthquakewaves which ismore complicated than the simplependulum The theory of the lengthening pendulum canalso be applied to analyzing a drifting spiral motion of acharged particle in mirror-confined plasma [25] In this casea position-dependent magnetic field which is a dense andweak magnetic field is responsible for the change of therotational radius along its spiral motion

We will study in this work several quantum states ofthe lengthening pendulum beyond the Fock state We willcompare quantummotions of the pendulumwith its classicalbehaviors Quantized energy and its dispersion will be inves-tigated via a rigorous development of the quantum theoryassociated with the pendulum

This paper is organized as follows The fundamentalmechanics of the lengthening pendulum is surveyed inSection 2 and various quantum mechanical properties ofthe system are analyzed in the subsequent sections Sec-tion 3 is devoted to the study of the time behavior of thequantized energy Path integral formulation of quantummechanics for the system is treated in Section 4 Wignerdistribution function (WDF) of the system is investigatedin Section 5 Quantum properties of the superposition oftwo neighboring states are discussed in Section 6 The cor-respondence between classical and quantum mechanics forthe pendulum is demonstrated in Section 7 through Gaus-sian wave packet description of the system The con-cluding remarks are given in Section 8 which is the last sec-tion

2 Fundamentals ofthe Lengthening Pendulum

Now we are carrying out a survey of the fundamentals ofthe lengthening pendulum Let us consider a pendulum of arelatively massive object hung by a vertical string from a fixedceiling and assume that it swings from its fixed equilibriumposition Suppose that the length of the pendulum increaseswith a constant rate 119896 from its initial length 119897

0

119897 (119905) = 1198970+ 119896119905 (1)

In case 119896 is sufficiently small the pendulum undergoes anadiabatic change and such case will only be considered in thisreview

We do not regard the quantum effects of the 119897-componentfor the sake of simplicity Then the Hamiltonian consideredup to the second order in 120579 is given by [21]

( 120579) =

2

120579

21198981198972 (119905)+1

2119898119892119897 (119905)

2

+H (119905) (2)

where H(119905) = 11989811989622 minus 119898119892119897(119905) A classical solution of the

system can be represented as [22]

120579 (119905) = 119888119872 (119905) 119890119894[120574(119905)+1198880] (3)

Here 119888 and 1198880are arbitrary constants and

119872(119905) =1

radic119897 (119905)[1198692

1(120585 (119905)) + 119873

2

1(120585 (119905))]

12

120574 (119905) = tanminus11198731 (120585 (119905))1198691 (120585 (119905))

(4)

where 119869 and 119873 are Bessel functions of the first kind and thesecond kind respectively and 120585(119905) = 2[119892119897(119905)]

12119896 A more

general solution for this system is given in [21]The construction of the annihilation and creation oper-

ators is useful for developing a quantum theory of a TDHSAccording to the invariant operator theory [15] the formulaeof them are somewhat different from those of the simpleharmonic oscillator For instance the annihilation operatoris given by [21]

= radic1

2ℏ (119905)1198981198972(119905)

[1198981198972(119905) ( (119905) minus 119894

(119905)

119872 (119905))

+ 119894120579]

(5)

Of course the Hermitian adjoint dagger of the above equationplays the role of the creation operatorThese operators satisfythe boson commutation relation such that [ dagger] = 1

If we use the property of this commutation relation wecan easily derive the wave functions in Fock state whichare represented in terms of some time functions 120601

119899(120579 119905) and

time-dependent phases 120598119899(119905) [21 22]

120595119899(120579 119905) = 120601

119899(120579 119905) 119890

119894120598119899(119905) (6)

where the quantum number is given by 119899 = 1 2 3 (All subsequent 119899 will follow this convention unless weparticularly specify it)The explicit forms of 120601

119899(120579 119905) and 120598

119899(119905)

are given by

120601119899(120579 119905)

=1

radic2119899119899(Λ (119905)

120587)

14

119867119899(radicΛ (119905)120579) 119890

minus(12)Λ(119905)120581lowast

1(119905)1205792

120598119899(119905) = minus (119899 +

1

2) 120574 (119905)

(7)



Λ (119905) = 119875 (119905)radic119892119898 [119897 (119905)]

32

2ℏ [11986921(120585 (119905)) + 119873

2

1(120585 (119905))]

1205811(119905) = 1 + 119894Π (119905)

(8)

with

119875 (119905) = 1198691(120585 (119905)) [119873

0(120585 (119905)) minus 119873

2(120585 (119905))]

minus 1198731(120585 (119905)) [119869

0(120585 (119905)) minus 119869

2(120585 (119905))]

Π (119905) = [119875 (119905)]minus1119876 (119905) minus

119896 [1198692

1(120585 (119905)) + 119873

2

1(120585 (119905))]

radic119892119897 (119905)

119876 (119905) = 1198691(120585 (119905)) [119869

0(120585 (119905)) minus 119869

2(120585 (119905))]

+ 1198731(120585 (119905)) [119873

0(120585 (119905)) minus 119873

2(120585 (119905))]

(9)


3 Quantized Energy



119864119899= (119899 +

1

2)ℏ

2[ℏΛ (119905)

1198981198972 (119905)

10038161003816100381610038161205811 (119905)10038161003816100381610038162+119898119892119897 (119905)

ℏΛ (119905)]

+H (119905)

(10)

119864cl =1

21198981198972(119905) 2

+1

2119898119892119897 (119905) 120579

2+H (119905) (11)


0


1198640=1

2119898119892119897 (119905) Θ

2+H (119905) (12)


Θ = [ℏ

2119898119892119897 (119905)(ℏΛ (119905)

1198981198972 (119905)

10038161003816100381610038161205811 (119905)10038161003816100381610038162+119898119892119897 (119905)

ℏΛ (119905))]

12

(13)



120587

Θ

10038161003816100381610038161205950 (120579 119905)10038161003816100381610038162119889120579 ≃ 2int

infin

Θ

10038161003816100381610038161205950 (120579 119905)10038161003816100381610038162119889120579 (14)


into (14) we have





4 Propagator





119870(120579 119905 1205791015840 1199051015840) =

infin

sum

119899=0

120595119899(120579 119905) 120595

lowast

119899(1205791015840 1199051015840) (16)


119870 = (Λ (119905) Λ (119905

1015840)

1205872)

14

119890minus(12)[Λ(119905)120581

lowast

1(119905)1205792+Λ(1199051015840)1205811(1199051015840)12057910158402]119890minus(1198942)[120574(119905)minus120574(119905

1015840)]

infin

sum

119899=0

119890minus119894119899[120574(119905)minus120574(119905

1015840)]

2119899119899119867119899(radicΛ (119905)120579)119867

119899(radicΛ (1199051015840)120579

1015840) (17)


infin

sum

119899=0

(1199112)119899

119899119867119899(119909)119867119899(119910)

=1

(1 minus 1199112)12

exp[2119909119910119911 minus (119909

2+ 1199102) 1199112

1 minus 1199112]

(18)

it becomes

119870 =

4radicΛ (119905) Λ (1199051015840)

radic2120587119894 sin [120574 (119905) minus 120574 (1199051015840)]


radicΛ (119905) Λ (1199051015840)1205791205791015840

sin [120574 (119905) minus 120574 (1199051015840)])


(19)


120595 (120579 119905) = int

120587

minus120587

1198891205791015840119870(120579 119905 120579

1015840 1199051015840) 120595 (120579

1015840 1199051015840) (20)






119882119899(120579 119901120579 119905)

=1

120587ℏint

120587

minus120587

120595lowast

119899(120579 + 120572 119905) 120595

119899(120579 minus 120572 119905) 119890

2119894119901120579120572ℏ119889120572

(21)



int

infin

minusinfin

119889119911119890minus1199112

119867119899 (119911 + 119886 + 119887)119867119899 (119911 + 119886 minus 119887)

= 211989912058712119899119871119899[2 (1198872minus 1198862)]

(22)


119882119899(120579 119901120579 119905) =

(minus1)119899

120587ℏ119890minus119884(120579119901120579 119905)119871

119899[2119884 (120579 119901

120579 119905)] (23)

where

119884 (120579 119901120579 119905) = Λ (119905) [120579

2+ (Π (119905) 120579 minus

119901120579

ℏΛ (119905))

2

] (24)







0

5

0

50

01

0

5

Wn

p 120579

minus5

minus5

minus01

120579

(a)

0

5

0

50

01

0

5

Wn

p 120579

minus5

minus5

minus01

120579

(b)

0

5

0

5

01

0

0

5

Wn

p 120579

minus5

minus5

minus01

120579

(c)







120595 (120579 119905) =

infin

sum

119899=0

119888119899120595119899 (120579 119905) (25)



120595 (120579 119905) = 119864120595 (120579 119905) (26)



120595 (120579 119905) =1

radic21198901198941205750 [120595119899(120579 119905) + 119890

119894120575120595119899+1

(120579 119905)] (27)



⟨1205951003816100381610038161003816100381610038161003816100381610038161003816120595⟩ = radic119899 + 1(

ℏ

2 (119905)1198981198972 (119905))

12

sdot cos [120574 (119905) minus 120575]

⟨1205951003816100381610038161003816120579

1003816100381610038161003816 120595⟩ =radic119899 + 1(

ℏ1198981198972(119905)

2 (119905))

12

sdot ( (119905)


(28)





119864 = (119899 + 1)ℏ

2(ℏΛ (119905)

1198981198972 (119905)

10038161003816100381610038161205811 (119905)10038161003816100381610038162+119898119892119897 (119905)

ℏΛ (119905)) +H (119905) (29)




120595 (120579 0) =4radic1205782

120587119890minus(12)120578

21205792

(30)


119888119899= int

120587

minus120587

120595lowast

119899(120579 0) 120595 (120579 0) 119889120579 (31)


119888119899=

radic119899

21198992 (1198992)[

2radicΛ (0)120578

Λ (0) 1205811 (0) + 1205782(Λ (0) 120581

lowast

1(0) minus 120578

2

Λ (0) 1205811 (0) + 1205782)

119899

]

12

sdot 119890119894(119899+12)120574(0)

(32)




120595 (120579 0) =4radicΛ (0)

120587119890minus(12)Λ(0)(120579minus120573)

2

(33)


119899can also be


119888119899=

1

radic2119899119899

120573119899

[1205812(0) 2]

119899+12

sdot 119890(12)[11205812(0)minusΛ(0)]120573

2+119894(119899+12)120574(0)

(34)



120595 (120579 119905) =4radicΛ (119905)

120587radic

2

1205812(0)

exp [2120573

1205812(0)

sdot 119890119894[120574(0)minus120574(119905)]

(radicΛ (119905)120579 minus1

2

120573

1205812 (0)

119890119894[120574(0)minus120574(119905)]

)]

sdot 119890(12)[11205812(0)minusΛ(0)]120573

2

119890minus(12)Λ(119905)1205811(119905)120579

2

119890(1198942)[120574(0)minus120574(119905)]

(35)


1003816100381610038161003816120595 (120579 119905)10038161003816100381610038162= radic

Λ (119905)

120587

210038161003816100381610038161205812 (0)

1003816100381610038161003816


lowast(119905)]

radicΛ (119905))

2

]

sdot 119890[4|1205812(0)|

2minusΛ(0)]120573

2

(36)

where 119883(119905) = 119890119894[120574(0)minus120574(119905)]



119899= |119888119899|2 Thus using (34)

we have

119875119899=

1

2119899119899

1205732119899

[10038161003816100381610038161205812 (0)

1003816100381610038161003816 2]2119899+1

119890[2|1205812(0)|

2minusΛ(0)]120573

2

(37)



be

119899 =1205732

2 [10038161003816100381610038161205812 (0)

1003816100381610038161003816 2]3119890[4|1205812(0)|

2minusΛ(0)]120573

2

(38)


0102 0

20

40

60

t

001020304

0minus01

minus02

120579

|120595|2


0= 6 119892 = 1 and





Δ119899 =

1

4

10038161003816100381610038161205812 (0)1003816100381610038161003816

2

minus exp[( 4

10038161003816100381610038161205812 (0)10038161003816100381610038162minus Λ (0))120573

2]

1205734

[10038161003816100381610038161205812 (0)

100381610038161003816100381624]3

+1205732

2 [10038161003816100381610038161205812 (0)

1003816100381610038161003816 2]3

12

119890(12)[4|1205812(0)|

2minusΛ(0)]120573

2

(39)


[sum1198991198642



Δ119864 =ℏ120573

10038161003816100381610038161205812 (0)10038161003816100381610038162[

41205732

10038161003816100381610038161205812 (0)10038161003816100381610038162(ℏΛ (119905)

1198981198972 (119905)

10038161003816100381610038161205811 (119905)10038161003816100381610038162

+119898119892119897 (119905)

ℏΛ (119905))(

10038161003816100381610038161205812 (0)1003816100381610038161003816

2minus 119890(4|1205812(0)|

2minusΛ(0))120573

2

)

+8H (119905)

ℏ(

10038161003816100381610038161205812 (0)1003816100381610038161003816

2minus 1) + 2 (

10038161003816100381610038161205812 (0)1003816100381610038161003816 minus 1)]

12

sdot 119890[2|1205812(0)|

2minusΛ(0)2]120573

2

(40)


00501

015

02 0

10

20

30

40

50

t

0002004006008

0

Δn

120573

(a)

00005001

0015002

0025

ΔE

00501

015

02 0

10

20

30

40

50

t0

120573

(b)


0= 10 and 119892 = 1 Note that Δ119899


8 Conclusion

















Acknowledgments


References

















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





Λ (119905) = 119875 (119905)radic119892119898 [119897 (119905)]

32

2ℏ [11986921(120585 (119905)) + 119873

2

1(120585 (119905))]

1205811(119905) = 1 + 119894Π (119905)

(8)

with

119875 (119905) = 1198691(120585 (119905)) [119873

0(120585 (119905)) minus 119873

2(120585 (119905))]

minus 1198731(120585 (119905)) [119869

0(120585 (119905)) minus 119869

2(120585 (119905))]

Π (119905) = [119875 (119905)]minus1119876 (119905) minus

119896 [1198692

1(120585 (119905)) + 119873

2

1(120585 (119905))]

radic119892119897 (119905)

119876 (119905) = 1198691(120585 (119905)) [119869

0(120585 (119905)) minus 119869

2(120585 (119905))]

+ 1198731(120585 (119905)) [119873

0(120585 (119905)) minus 119873

2(120585 (119905))]

(9)


3 Quantized Energy



119864119899= (119899 +

1

2)ℏ

2[ℏΛ (119905)

1198981198972 (119905)

10038161003816100381610038161205811 (119905)10038161003816100381610038162+119898119892119897 (119905)

ℏΛ (119905)]

+H (119905)

(10)

119864cl =1

21198981198972(119905) 2

+1

2119898119892119897 (119905) 120579

2+H (119905) (11)


0


1198640=1

2119898119892119897 (119905) Θ

2+H (119905) (12)


Θ = [ℏ

2119898119892119897 (119905)(ℏΛ (119905)

1198981198972 (119905)

10038161003816100381610038161205811 (119905)10038161003816100381610038162+119898119892119897 (119905)

ℏΛ (119905))]

12

(13)



120587

Θ

10038161003816100381610038161205950 (120579 119905)10038161003816100381610038162119889120579 ≃ 2int

infin

Θ

10038161003816100381610038161205950 (120579 119905)10038161003816100381610038162119889120579 (14)


into (14) we have





4 Propagator





119870(120579 119905 1205791015840 1199051015840) =

infin

sum

119899=0

120595119899(120579 119905) 120595

lowast

119899(1205791015840 1199051015840) (16)


119870 = (Λ (119905) Λ (119905

1015840)

1205872)

14

119890minus(12)[Λ(119905)120581

lowast

1(119905)1205792+Λ(1199051015840)1205811(1199051015840)12057910158402]119890minus(1198942)[120574(119905)minus120574(119905

1015840)]

infin

sum

119899=0

119890minus119894119899[120574(119905)minus120574(119905

1015840)]

2119899119899119867119899(radicΛ (119905)120579)119867

119899(radicΛ (1199051015840)120579

1015840) (17)


infin

sum

119899=0

(1199112)119899

119899119867119899(119909)119867119899(119910)

=1

(1 minus 1199112)12

exp[2119909119910119911 minus (119909

2+ 1199102) 1199112

1 minus 1199112]

(18)

it becomes

119870 =

4radicΛ (119905) Λ (1199051015840)

radic2120587119894 sin [120574 (119905) minus 120574 (1199051015840)]


radicΛ (119905) Λ (1199051015840)1205791205791015840

sin [120574 (119905) minus 120574 (1199051015840)])


(19)


120595 (120579 119905) = int

120587

minus120587

1198891205791015840119870(120579 119905 120579

1015840 1199051015840) 120595 (120579

1015840 1199051015840) (20)






119882119899(120579 119901120579 119905)

=1

120587ℏint

120587

minus120587

120595lowast

119899(120579 + 120572 119905) 120595

119899(120579 minus 120572 119905) 119890

2119894119901120579120572ℏ119889120572

(21)



int

infin

minusinfin

119889119911119890minus1199112

119867119899 (119911 + 119886 + 119887)119867119899 (119911 + 119886 minus 119887)

= 211989912058712119899119871119899[2 (1198872minus 1198862)]

(22)


119882119899(120579 119901120579 119905) =

(minus1)119899

120587ℏ119890minus119884(120579119901120579 119905)119871

119899[2119884 (120579 119901

120579 119905)] (23)

where

119884 (120579 119901120579 119905) = Λ (119905) [120579

2+ (Π (119905) 120579 minus

119901120579

ℏΛ (119905))

2

] (24)







0

5

0

50

01

0

5

Wn

p 120579

minus5

minus5

minus01

120579

(a)

0

5

0

50

01

0

5

Wn

p 120579

minus5

minus5

minus01

120579

(b)

0

5

0

5

01

0

0

5

Wn

p 120579

minus5

minus5

minus01

120579

(c)







120595 (120579 119905) =

infin

sum

119899=0

119888119899120595119899 (120579 119905) (25)



120595 (120579 119905) = 119864120595 (120579 119905) (26)



120595 (120579 119905) =1

radic21198901198941205750 [120595119899(120579 119905) + 119890

119894120575120595119899+1

(120579 119905)] (27)



⟨1205951003816100381610038161003816100381610038161003816100381610038161003816120595⟩ = radic119899 + 1(

ℏ

2 (119905)1198981198972 (119905))

12

sdot cos [120574 (119905) minus 120575]

⟨1205951003816100381610038161003816120579

1003816100381610038161003816 120595⟩ =radic119899 + 1(

ℏ1198981198972(119905)

2 (119905))

12

sdot ( (119905)


(28)





119864 = (119899 + 1)ℏ

2(ℏΛ (119905)

1198981198972 (119905)

10038161003816100381610038161205811 (119905)10038161003816100381610038162+119898119892119897 (119905)

ℏΛ (119905)) +H (119905) (29)




120595 (120579 0) =4radic1205782

120587119890minus(12)120578

21205792

(30)


119888119899= int

120587

minus120587

120595lowast

119899(120579 0) 120595 (120579 0) 119889120579 (31)


119888119899=

radic119899

21198992 (1198992)[

2radicΛ (0)120578

Λ (0) 1205811 (0) + 1205782(Λ (0) 120581

lowast

1(0) minus 120578

2

Λ (0) 1205811 (0) + 1205782)

119899

]

12

sdot 119890119894(119899+12)120574(0)

(32)




120595 (120579 0) =4radicΛ (0)

120587119890minus(12)Λ(0)(120579minus120573)

2

(33)


119899can also be


119888119899=

1

radic2119899119899

120573119899

[1205812(0) 2]

119899+12

sdot 119890(12)[11205812(0)minusΛ(0)]120573

2+119894(119899+12)120574(0)

(34)



120595 (120579 119905) =4radicΛ (119905)

120587radic

2

1205812(0)

exp [2120573

1205812(0)

sdot 119890119894[120574(0)minus120574(119905)]

(radicΛ (119905)120579 minus1

2

120573

1205812 (0)

119890119894[120574(0)minus120574(119905)]

)]

sdot 119890(12)[11205812(0)minusΛ(0)]120573

2

119890minus(12)Λ(119905)1205811(119905)120579

2

119890(1198942)[120574(0)minus120574(119905)]

(35)


1003816100381610038161003816120595 (120579 119905)10038161003816100381610038162= radic

Λ (119905)

120587

210038161003816100381610038161205812 (0)

1003816100381610038161003816


lowast(119905)]

radicΛ (119905))

2

]

sdot 119890[4|1205812(0)|

2minusΛ(0)]120573

2

(36)

where 119883(119905) = 119890119894[120574(0)minus120574(119905)]



119899= |119888119899|2 Thus using (34)

we have

119875119899=

1

2119899119899

1205732119899

[10038161003816100381610038161205812 (0)

1003816100381610038161003816 2]2119899+1

119890[2|1205812(0)|

2minusΛ(0)]120573

2

(37)



be

119899 =1205732

2 [10038161003816100381610038161205812 (0)

1003816100381610038161003816 2]3119890[4|1205812(0)|

2minusΛ(0)]120573

2

(38)


0102 0

20

40

60

t

001020304

0minus01

minus02

120579

|120595|2


0= 6 119892 = 1 and





Δ119899 =

1

4

10038161003816100381610038161205812 (0)1003816100381610038161003816

2

minus exp[( 4

10038161003816100381610038161205812 (0)10038161003816100381610038162minus Λ (0))120573

2]

1205734

[10038161003816100381610038161205812 (0)

100381610038161003816100381624]3

+1205732

2 [10038161003816100381610038161205812 (0)

1003816100381610038161003816 2]3

12

119890(12)[4|1205812(0)|

2minusΛ(0)]120573

2

(39)


[sum1198991198642



Δ119864 =ℏ120573

10038161003816100381610038161205812 (0)10038161003816100381610038162[

41205732

10038161003816100381610038161205812 (0)10038161003816100381610038162(ℏΛ (119905)

1198981198972 (119905)

10038161003816100381610038161205811 (119905)10038161003816100381610038162

+119898119892119897 (119905)

ℏΛ (119905))(

10038161003816100381610038161205812 (0)1003816100381610038161003816

2minus 119890(4|1205812(0)|

2minusΛ(0))120573

2

)

+8H (119905)

ℏ(

10038161003816100381610038161205812 (0)1003816100381610038161003816

2minus 1) + 2 (

10038161003816100381610038161205812 (0)1003816100381610038161003816 minus 1)]

12

sdot 119890[2|1205812(0)|

2minusΛ(0)2]120573

2

(40)


00501

015

02 0

10

20

30

40

50

t

0002004006008

0

Δn

120573

(a)

00005001

0015002

0025

ΔE

00501

015

02 0

10

20

30

40

50

t0

120573

(b)


0= 10 and 119892 = 1 Note that Δ119899


8 Conclusion

















Acknowledgments


References

















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119870(120579 119905 1205791015840 1199051015840) =

infin

sum

119899=0

120595119899(120579 119905) 120595

lowast

119899(1205791015840 1199051015840) (16)


119870 = (Λ (119905) Λ (119905

1015840)

1205872)

14

119890minus(12)[Λ(119905)120581

lowast

1(119905)1205792+Λ(1199051015840)1205811(1199051015840)12057910158402]119890minus(1198942)[120574(119905)minus120574(119905

1015840)]

infin

sum

119899=0

119890minus119894119899[120574(119905)minus120574(119905

1015840)]

2119899119899119867119899(radicΛ (119905)120579)119867

119899(radicΛ (1199051015840)120579

1015840) (17)


infin

sum

119899=0

(1199112)119899

119899119867119899(119909)119867119899(119910)

=1

(1 minus 1199112)12

exp[2119909119910119911 minus (119909

2+ 1199102) 1199112

1 minus 1199112]

(18)

it becomes

119870 =

4radicΛ (119905) Λ (1199051015840)

radic2120587119894 sin [120574 (119905) minus 120574 (1199051015840)]


radicΛ (119905) Λ (1199051015840)1205791205791015840

sin [120574 (119905) minus 120574 (1199051015840)])


(19)


120595 (120579 119905) = int

120587

minus120587

1198891205791015840119870(120579 119905 120579

1015840 1199051015840) 120595 (120579

1015840 1199051015840) (20)






119882119899(120579 119901120579 119905)

=1

120587ℏint

120587

minus120587

120595lowast

119899(120579 + 120572 119905) 120595

119899(120579 minus 120572 119905) 119890

2119894119901120579120572ℏ119889120572

(21)



int

infin

minusinfin

119889119911119890minus1199112

119867119899 (119911 + 119886 + 119887)119867119899 (119911 + 119886 minus 119887)

= 211989912058712119899119871119899[2 (1198872minus 1198862)]

(22)


119882119899(120579 119901120579 119905) =

(minus1)119899

120587ℏ119890minus119884(120579119901120579 119905)119871

119899[2119884 (120579 119901

120579 119905)] (23)

where

119884 (120579 119901120579 119905) = Λ (119905) [120579

2+ (Π (119905) 120579 minus

119901120579

ℏΛ (119905))

2

] (24)







0

5

0

50

01

0

5

Wn

p 120579

minus5

minus5

minus01

120579

(a)

0

5

0

50

01

0

5

Wn

p 120579

minus5

minus5

minus01

120579

(b)

0

5

0

5

01

0

0

5

Wn

p 120579

minus5

minus5

minus01

120579

(c)







120595 (120579 119905) =

infin

sum

119899=0

119888119899120595119899 (120579 119905) (25)



120595 (120579 119905) = 119864120595 (120579 119905) (26)



120595 (120579 119905) =1

radic21198901198941205750 [120595119899(120579 119905) + 119890

119894120575120595119899+1

(120579 119905)] (27)



⟨1205951003816100381610038161003816100381610038161003816100381610038161003816120595⟩ = radic119899 + 1(

ℏ

2 (119905)1198981198972 (119905))

12

sdot cos [120574 (119905) minus 120575]

⟨1205951003816100381610038161003816120579

1003816100381610038161003816 120595⟩ =radic119899 + 1(

ℏ1198981198972(119905)

2 (119905))

12

sdot ( (119905)


(28)





119864 = (119899 + 1)ℏ

2(ℏΛ (119905)

1198981198972 (119905)

10038161003816100381610038161205811 (119905)10038161003816100381610038162+119898119892119897 (119905)

ℏΛ (119905)) +H (119905) (29)




120595 (120579 0) =4radic1205782

120587119890minus(12)120578

21205792

(30)


119888119899= int

120587

minus120587

120595lowast

119899(120579 0) 120595 (120579 0) 119889120579 (31)


119888119899=

radic119899

21198992 (1198992)[

2radicΛ (0)120578

Λ (0) 1205811 (0) + 1205782(Λ (0) 120581

lowast

1(0) minus 120578

2

Λ (0) 1205811 (0) + 1205782)

119899

]

12

sdot 119890119894(119899+12)120574(0)

(32)




120595 (120579 0) =4radicΛ (0)

120587119890minus(12)Λ(0)(120579minus120573)

2

(33)


119899can also be


119888119899=

1

radic2119899119899

120573119899

[1205812(0) 2]

119899+12

sdot 119890(12)[11205812(0)minusΛ(0)]120573

2+119894(119899+12)120574(0)

(34)



120595 (120579 119905) =4radicΛ (119905)

120587radic

2

1205812(0)

exp [2120573

1205812(0)

sdot 119890119894[120574(0)minus120574(119905)]

(radicΛ (119905)120579 minus1

2

120573

1205812 (0)

119890119894[120574(0)minus120574(119905)]

)]

sdot 119890(12)[11205812(0)minusΛ(0)]120573

2

119890minus(12)Λ(119905)1205811(119905)120579

2

119890(1198942)[120574(0)minus120574(119905)]

(35)


1003816100381610038161003816120595 (120579 119905)10038161003816100381610038162= radic

Λ (119905)

120587

210038161003816100381610038161205812 (0)

1003816100381610038161003816


lowast(119905)]

radicΛ (119905))

2

]

sdot 119890[4|1205812(0)|

2minusΛ(0)]120573

2

(36)

where 119883(119905) = 119890119894[120574(0)minus120574(119905)]



119899= |119888119899|2 Thus using (34)

we have

119875119899=

1

2119899119899

1205732119899

[10038161003816100381610038161205812 (0)

1003816100381610038161003816 2]2119899+1

119890[2|1205812(0)|

2minusΛ(0)]120573

2

(37)



be

119899 =1205732

2 [10038161003816100381610038161205812 (0)

1003816100381610038161003816 2]3119890[4|1205812(0)|

2minusΛ(0)]120573

2

(38)


0102 0

20

40

60

t

001020304

0minus01

minus02

120579

|120595|2


0= 6 119892 = 1 and





Δ119899 =

1

4

10038161003816100381610038161205812 (0)1003816100381610038161003816

2

minus exp[( 4

10038161003816100381610038161205812 (0)10038161003816100381610038162minus Λ (0))120573

2]

1205734

[10038161003816100381610038161205812 (0)

100381610038161003816100381624]3

+1205732

2 [10038161003816100381610038161205812 (0)

1003816100381610038161003816 2]3

12

119890(12)[4|1205812(0)|

2minusΛ(0)]120573

2

(39)


[sum1198991198642



Δ119864 =ℏ120573

10038161003816100381610038161205812 (0)10038161003816100381610038162[

41205732

10038161003816100381610038161205812 (0)10038161003816100381610038162(ℏΛ (119905)

1198981198972 (119905)

10038161003816100381610038161205811 (119905)10038161003816100381610038162

+119898119892119897 (119905)

ℏΛ (119905))(

10038161003816100381610038161205812 (0)1003816100381610038161003816

2minus 119890(4|1205812(0)|

2minusΛ(0))120573

2

)

+8H (119905)

ℏ(

10038161003816100381610038161205812 (0)1003816100381610038161003816

2minus 1) + 2 (

10038161003816100381610038161205812 (0)1003816100381610038161003816 minus 1)]

12

sdot 119890[2|1205812(0)|

2minusΛ(0)2]120573

2

(40)


00501

015

02 0

10

20

30

40

50

t

0002004006008

0

Δn

120573

(a)

00005001

0015002

0025

ΔE

00501

015

02 0

10

20

30

40

50

t0

120573

(b)


0= 10 and 119892 = 1 Note that Δ119899


8 Conclusion

















Acknowledgments


References

















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0

5

0

50

01

0

5

Wn

p 120579

minus5

minus5

minus01

120579

(a)

0

5

0

50

01

0

5

Wn

p 120579

minus5

minus5

minus01

120579

(b)

0

5

0

5

01

0

0

5

Wn

p 120579

minus5

minus5

minus01

120579

(c)







120595 (120579 119905) =

infin

sum

119899=0

119888119899120595119899 (120579 119905) (25)



120595 (120579 119905) = 119864120595 (120579 119905) (26)



120595 (120579 119905) =1

radic21198901198941205750 [120595119899(120579 119905) + 119890

119894120575120595119899+1

(120579 119905)] (27)



⟨1205951003816100381610038161003816100381610038161003816100381610038161003816120595⟩ = radic119899 + 1(

ℏ

2 (119905)1198981198972 (119905))

12

sdot cos [120574 (119905) minus 120575]

⟨1205951003816100381610038161003816120579

1003816100381610038161003816 120595⟩ =radic119899 + 1(

ℏ1198981198972(119905)

2 (119905))

12

sdot ( (119905)


(28)





119864 = (119899 + 1)ℏ

2(ℏΛ (119905)

1198981198972 (119905)

10038161003816100381610038161205811 (119905)10038161003816100381610038162+119898119892119897 (119905)

ℏΛ (119905)) +H (119905) (29)




120595 (120579 0) =4radic1205782

120587119890minus(12)120578

21205792

(30)


119888119899= int

120587

minus120587

120595lowast

119899(120579 0) 120595 (120579 0) 119889120579 (31)


119888119899=

radic119899

21198992 (1198992)[

2radicΛ (0)120578

Λ (0) 1205811 (0) + 1205782(Λ (0) 120581

lowast

1(0) minus 120578

2

Λ (0) 1205811 (0) + 1205782)

119899

]

12

sdot 119890119894(119899+12)120574(0)

(32)




120595 (120579 0) =4radicΛ (0)

120587119890minus(12)Λ(0)(120579minus120573)

2

(33)


119899can also be


119888119899=

1

radic2119899119899

120573119899

[1205812(0) 2]

119899+12

sdot 119890(12)[11205812(0)minusΛ(0)]120573

2+119894(119899+12)120574(0)

(34)



120595 (120579 119905) =4radicΛ (119905)

120587radic

2

1205812(0)

exp [2120573

1205812(0)

sdot 119890119894[120574(0)minus120574(119905)]

(radicΛ (119905)120579 minus1

2

120573

1205812 (0)

119890119894[120574(0)minus120574(119905)]

)]

sdot 119890(12)[11205812(0)minusΛ(0)]120573

2

119890minus(12)Λ(119905)1205811(119905)120579

2

119890(1198942)[120574(0)minus120574(119905)]

(35)


1003816100381610038161003816120595 (120579 119905)10038161003816100381610038162= radic

Λ (119905)

120587

210038161003816100381610038161205812 (0)

1003816100381610038161003816


lowast(119905)]

radicΛ (119905))

2

]

sdot 119890[4|1205812(0)|

2minusΛ(0)]120573

2

(36)

where 119883(119905) = 119890119894[120574(0)minus120574(119905)]



119899= |119888119899|2 Thus using (34)

we have

119875119899=

1

2119899119899

1205732119899

[10038161003816100381610038161205812 (0)

1003816100381610038161003816 2]2119899+1

119890[2|1205812(0)|

2minusΛ(0)]120573

2

(37)



be

119899 =1205732

2 [10038161003816100381610038161205812 (0)

1003816100381610038161003816 2]3119890[4|1205812(0)|

2minusΛ(0)]120573

2

(38)


0102 0

20

40

60

t

001020304

0minus01

minus02

120579

|120595|2


0= 6 119892 = 1 and





Δ119899 =

1

4

10038161003816100381610038161205812 (0)1003816100381610038161003816

2

minus exp[( 4

10038161003816100381610038161205812 (0)10038161003816100381610038162minus Λ (0))120573

2]

1205734

[10038161003816100381610038161205812 (0)

100381610038161003816100381624]3

+1205732

2 [10038161003816100381610038161205812 (0)

1003816100381610038161003816 2]3

12

119890(12)[4|1205812(0)|

2minusΛ(0)]120573

2

(39)


[sum1198991198642



Δ119864 =ℏ120573

10038161003816100381610038161205812 (0)10038161003816100381610038162[

41205732

10038161003816100381610038161205812 (0)10038161003816100381610038162(ℏΛ (119905)

1198981198972 (119905)

10038161003816100381610038161205811 (119905)10038161003816100381610038162

+119898119892119897 (119905)

ℏΛ (119905))(

10038161003816100381610038161205812 (0)1003816100381610038161003816

2minus 119890(4|1205812(0)|

2minusΛ(0))120573

2

)

+8H (119905)

ℏ(

10038161003816100381610038161205812 (0)1003816100381610038161003816

2minus 1) + 2 (

10038161003816100381610038161205812 (0)1003816100381610038161003816 minus 1)]

12

sdot 119890[2|1205812(0)|

2minusΛ(0)2]120573

2

(40)


00501

015

02 0

10

20

30

40

50

t

0002004006008

0

Δn

120573

(a)

00005001

0015002

0025

ΔE

00501

015

02 0

10

20

30

40

50

t0

120573

(b)


0= 10 and 119892 = 1 Note that Δ119899


8 Conclusion

















Acknowledgments


References

















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







119864 = (119899 + 1)ℏ

2(ℏΛ (119905)

1198981198972 (119905)

10038161003816100381610038161205811 (119905)10038161003816100381610038162+119898119892119897 (119905)

ℏΛ (119905)) +H (119905) (29)




120595 (120579 0) =4radic1205782

120587119890minus(12)120578

21205792

(30)


119888119899= int

120587

minus120587

120595lowast

119899(120579 0) 120595 (120579 0) 119889120579 (31)


119888119899=

radic119899

21198992 (1198992)[

2radicΛ (0)120578

Λ (0) 1205811 (0) + 1205782(Λ (0) 120581

lowast

1(0) minus 120578

2

Λ (0) 1205811 (0) + 1205782)

119899

]

12

sdot 119890119894(119899+12)120574(0)

(32)




120595 (120579 0) =4radicΛ (0)

120587119890minus(12)Λ(0)(120579minus120573)

2

(33)


119899can also be


119888119899=

1

radic2119899119899

120573119899

[1205812(0) 2]

119899+12

sdot 119890(12)[11205812(0)minusΛ(0)]120573

2+119894(119899+12)120574(0)

(34)



120595 (120579 119905) =4radicΛ (119905)

120587radic

2

1205812(0)

exp [2120573

1205812(0)

sdot 119890119894[120574(0)minus120574(119905)]

(radicΛ (119905)120579 minus1

2

120573

1205812 (0)

119890119894[120574(0)minus120574(119905)]

)]

sdot 119890(12)[11205812(0)minusΛ(0)]120573

2

119890minus(12)Λ(119905)1205811(119905)120579

2

119890(1198942)[120574(0)minus120574(119905)]

(35)


1003816100381610038161003816120595 (120579 119905)10038161003816100381610038162= radic

Λ (119905)

120587

210038161003816100381610038161205812 (0)

1003816100381610038161003816


lowast(119905)]

radicΛ (119905))

2

]

sdot 119890[4|1205812(0)|

2minusΛ(0)]120573

2

(36)

where 119883(119905) = 119890119894[120574(0)minus120574(119905)]



119899= |119888119899|2 Thus using (34)

we have

119875119899=

1

2119899119899

1205732119899

[10038161003816100381610038161205812 (0)

1003816100381610038161003816 2]2119899+1

119890[2|1205812(0)|

2minusΛ(0)]120573

2

(37)



be

119899 =1205732

2 [10038161003816100381610038161205812 (0)

1003816100381610038161003816 2]3119890[4|1205812(0)|

2minusΛ(0)]120573

2

(38)


0102 0

20

40

60

t

001020304

0minus01

minus02

120579

|120595|2


0= 6 119892 = 1 and





Δ119899 =

1

4

10038161003816100381610038161205812 (0)1003816100381610038161003816

2

minus exp[( 4

10038161003816100381610038161205812 (0)10038161003816100381610038162minus Λ (0))120573

2]

1205734

[10038161003816100381610038161205812 (0)

100381610038161003816100381624]3

+1205732

2 [10038161003816100381610038161205812 (0)

1003816100381610038161003816 2]3

12

119890(12)[4|1205812(0)|

2minusΛ(0)]120573

2

(39)


[sum1198991198642



Δ119864 =ℏ120573

10038161003816100381610038161205812 (0)10038161003816100381610038162[

41205732

10038161003816100381610038161205812 (0)10038161003816100381610038162(ℏΛ (119905)

1198981198972 (119905)

10038161003816100381610038161205811 (119905)10038161003816100381610038162

+119898119892119897 (119905)

ℏΛ (119905))(

10038161003816100381610038161205812 (0)1003816100381610038161003816

2minus 119890(4|1205812(0)|

2minusΛ(0))120573

2

)

+8H (119905)

ℏ(

10038161003816100381610038161205812 (0)1003816100381610038161003816

2minus 1) + 2 (

10038161003816100381610038161205812 (0)1003816100381610038161003816 minus 1)]

12

sdot 119890[2|1205812(0)|

2minusΛ(0)2]120573

2

(40)


00501

015

02 0

10

20

30

40

50

t

0002004006008

0

Δn

120573

(a)

00005001

0015002

0025

ΔE

00501

015

02 0

10

20

30

40

50

t0

120573

(b)


0= 10 and 119892 = 1 Note that Δ119899


8 Conclusion

















Acknowledgments


References

















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




0102 0

20

40

60

t

001020304

0minus01

minus02

120579

|120595|2


0= 6 119892 = 1 and





Δ119899 =

1

4

10038161003816100381610038161205812 (0)1003816100381610038161003816

2

minus exp[( 4

10038161003816100381610038161205812 (0)10038161003816100381610038162minus Λ (0))120573

2]

1205734

[10038161003816100381610038161205812 (0)

100381610038161003816100381624]3

+1205732

2 [10038161003816100381610038161205812 (0)

1003816100381610038161003816 2]3

12

119890(12)[4|1205812(0)|

2minusΛ(0)]120573

2

(39)


[sum1198991198642



Δ119864 =ℏ120573

10038161003816100381610038161205812 (0)10038161003816100381610038162[

41205732

10038161003816100381610038161205812 (0)10038161003816100381610038162(ℏΛ (119905)

1198981198972 (119905)

10038161003816100381610038161205811 (119905)10038161003816100381610038162

+119898119892119897 (119905)

ℏΛ (119905))(

10038161003816100381610038161205812 (0)1003816100381610038161003816

2minus 119890(4|1205812(0)|

2minusΛ(0))120573

2

)

+8H (119905)

ℏ(

10038161003816100381610038161205812 (0)1003816100381610038161003816

2minus 1) + 2 (

10038161003816100381610038161205812 (0)1003816100381610038161003816 minus 1)]

12

sdot 119890[2|1205812(0)|

2minusΛ(0)2]120573

2

(40)


00501

015

02 0

10

20

30

40

50

t

0002004006008

0

Δn

120573

(a)

00005001

0015002

0025

ΔE

00501

015

02 0

10

20

30

40

50

t0

120573

(b)


0= 10 and 119892 = 1 Note that Δ119899


8 Conclusion

















Acknowledgments


References

















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics
















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



review article quantum analysis on time behavior of a … · 2019. 7. 30. · review article...

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