quantum theory and classical, nonlinear electronics

Physica 20D (1986) 285-302 North-Holland, Amsterdam

QUANTUM THEORY AND CLASSICAL, NONLINEAR ELECTRONICS

Dale M. GRIMES Department of Electrical Engineering, The Pennsylvania State University, University Park, PA 16802, USA

Received 5 March 1985

The present view of quantum theory considers electrons to be point charges or waves, and ignores radiative near fields. It was shown previously that a first order (near held) reactive radiation reaction force acts to rend a small charge within a potential well into a non-radiating current configuration. It is shown here that, through the near field, a continuum charge density responds to an applied radiation field nonlinearly; the nonlinearity results in radiation with quantum properties, including full directivity and quantized angular momentum. During radiation exchange large near fields act nonlinearly to expand atomic radii from ground state to or beyond excited state values. Necessary supporting atomic current magnitudes do not appear to be excessive. Results are the S&r&linger wave equation as a non-mechanistic descriptor of system- and time-average kinematic values, and quantum radiation; thus quantum theory appears as a special case of classical theory applied to nonlinear, atomic-level systems.

1. Introduction

There are reasons to believe that the present version of quantum theory does not contain all needed answers to atomic level phenomena. For example, the Manley-Rowe equations correctly predict power-frequency relationships in quantum radiation [l, 21, yet they predict only the trivial case when dealing with linear media, and quantum energy exchanges are described by the (linear) Schriidinger wave equation. Also, for small size- to-wavelength ratios in man-made antennas the behavior of radiators is dominated by near-field reactive energy [3, 41, which is ignored in quantum radiators. Are explanations possible? During the first two decades of this century much effort was devoted to seeking classical electromagnetic solutions of atomic phenomena. These efforts reached their apex, perhaps, with Schott [5], who postulated small, spherical electrons about a point nucleus and showed that a charged sphere in con- fined motion will radiate power unless vibration occurs with a period whose wavelength corre- sponds to approximately the size of the sphere,

and that its translatory motion and its spin couple in an inseparable way. It was concluded that classical theory is inapplicable within atomic di- mensions, and the uncertainty principle implies that processes occurring within time uncertainty cannot be examined in detail [6-91.

Even though there is a much greater wealth of analytical techniques and experimental informa- tion available now than when these ideas were formulated, still they have been taken so literally that relatively few further attempts to apply classical laws within the forbidden range have been published since.

This paper is an attempt to describe atomic stability and detailed radiative processes on the basis of classical theory. It is not an attempt at an improved quasi-classical view of any particular quantum phenomena. It postulates that an atomic electron is best modelled neither as a small sphere nor as a wave but as a spatial region whose shape and form is extensive in, and adaptive to, local force fields. It further postulates that the classical

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286 D.M. Grimes/Quantum theory and classical, nonlinear electronics

equations are obeyed in atomic environments and that a mechanistic description of quantum radiators is possible.

Results include that: 1) Although the reactive radiation reaction force

is a first order effect and, in the main, not describ- able using perturbation techniques, field nulls exist for certain radiation combinations at which the total reactive reaction field vanishes; i.e. reso- nances occur. Radiative transitions occur at a field resonance.

2) Quantized radiation is the result of a continuum, nonlinear electronic process. The Manley-Rowe equations are directly applicable and nontrivial in such processes.

3) The S&r&linger time-independent wave equation follows as a derived result and therefore, by implication, all results predicted by that equation derive from the classical physics of nonlinear systems.

In this view earlier work misestimated the role and scope of electromagnetic reactive power in quantized energy exchanges and incorrectly modelled static and dynamic electronic charge distributions and, for these reasons, the basic nonlinearities were missed. A conclusion is that quantum theory may be viewed as a special case of classical theory applied to nonlinear, adaptive systems.

2. Fields, Poynting vectors, and radiation energy

Although by the uncertainty principle we can know little of detailed events that occur during quantized radiation exchanges, transition probabilities for strong transitions are easily calculated using perturbation techniques and electric dipolar transition elements [lo, 111. The calculation pre- dicts correct results. In the abstract this seems somewhat strange since electromagnetic near fields are ignored and, for man-made antennas, they dominate antenna behavior at wavelengths much greater than the maximum dimension of the radiator [3]. Examples are optical radiation exchange by atoms and gamma radiation exchange by nuclei.

Also quantum radiation is directed and carries quantized values of angular momentum, not at all like classical dipole radiation. It has been gener- ally agreed that these profound differences make classical expectations and observations incompati- ble.

In contrast, it is shown here that if a radiating object occupies a volume whose size and shape are dependent upon local force fields, even if the object’s largest dimension is much less than a wavelength, when near field terms are included and their reactive energy is put equal to zero the result is that the classical electromagnetic equations correctly predict atomic stability, directed radiation exchanges, and quantized values of radiative angular momentum.

The analytical procedure begins with a general multipolar field expansion and, from it, finds the real and reactive power flowing through the smallest spherical Gaussian surface surrounding the radiator. The Poynting vector and, from it, the real and reactive powers are calculated. System char- acteristics are noted at root values of reactive energy. For the radius-to-wavelength ratio of atomic optical radiation the radiative Q (the reactive to real power ratio) for dipolar radiation may be as high as 109; its real radiation reaction force is small. The attractive Coulomb force is the same order of magnitude as the reactive radiation reaction force, whose root values determine the expansion parameters and they, in turn, describe unique Kirchhoff sources on the Gaussian surface [12], but not unique interior sources.

The interior solution of a mobile, turbulent, radiating charged cloud interacting with its own radiation field is a nonlinear one and requires a solution of the full Hehnholtz equation. We are, at best, ill equipped to solve such a problem. Instead, we relate the strength and symmetry of the Kirchhoff sources to interior forces. Fortunately system averages of the interior charge and current distributions may be addressed separately from the above. Schott [5] showed that closed currents couple with their own spin to produce motions fundamentally different from those of classical

D. M. Grimes/ Quantum theory and classical, nonlinear electronics 281

mechanics, and result in stable, repetitive, non-stationary currents; as shown in appendix B, stable solutions lead to the time-independent Schriidi- nger equation as a non-mechanistic descriptor of system- and time-average density values; from that equation comes the quantum theory.

A radiating region is centered at the origin and suppressed eiot time dependence is used. k represents the wave number and u equals the product of k and the radial distance from the origin. j,(u), y,(a), and h,(o) represent, respectively, spherical Bessel, Neumann, and Hankel functions of the second kind. Derivative functions J,(u), Y,(u), and H,(u) are defined by

H,(u) =; -&oh,). The field coefficients are represented by F(I, a) and G( 1, a), associated Legendre functions of order I and degree m by P,m(cos8), the impedance of free space by n, and i* = j* = -1; the double notation makes convenient later separation of spatial rotation from time delay effects. In these terms, the general multipolar expansion of the continuum electromagnetic equations is [13]

E,= i f i i-‘F(I, m)l(l+ 1) I=-lm=O

x h,(u) ,-P;t( cos 6) e-jm*,

vH,= -ij E i i-‘G(I, m)l(l+ 1) I-lm=0

x ht$J) -Pp;t( cos 0) e-jm*,

E8 = 5 2 i-’ (2)

I==lm=O

x iF(I, m)H,T- G(l,m)h,s e I

-jm+ ,

,H+= f i i-’ I-lm-0

WY mP;” F(1, m)h,w-iG(l,m)H,s e

I -jm* ,

-E+= E i ji-'

/=-lm=O

X mP;”

iF(I, m)H,-ggg- G(I,m)h,$ e I

-h+ ,

qHe= c i ji-’ I-lm=O

F(I,m)h,m-iG(I,m)H,$ e mP;” 1 -jm+ .

The vector Kirchhoff integral equation for electromagnetic fields produced by a closed, radiating source assures the existence, for each radiation pattern, of a unique surface source [13]; it de- termines the field coefficients F and G. They, in turn, uniquely determine all kinematic properties of the source.

The angular fields of (2) are used to find the radial component of the Poynting vector, N, in the form

N = No + Nlcos2wt + N, sin2wt,

where

(3)

2No = Re( E,H,* - E+H,*),

2Ni = Re( E,H+ - E+H,),

2N2 = Im( E,H+ - E,H,).

(4)

IQ. (3) may be written as

N=N,[l+cos\C,]+N,sinIC,,

where

# = 2wt + qe, r#B),

N4 = [ Ni* + N2* - NO*] “*,

tanS= (N,N,-N,,N,)/(N,N,+N,N,).

Although the integral of (4) over the smallest Gaussian surface surrounding the radiator yields the instantaneous power, it is the surface integral of the time-average and phase-quadrature compo-

288 D. hf. Grimes/Quantum theory and classical, nonlinear electronics

nents that express the real and reactive power, respectively designated by P and R. Using (4) they are given by

P = a2 /

N,dS2/k2,

R = a2 I

N,dO/k2.

(5)

(6)

The radiative Q is equal to the ratio of the magnitude of the reactive to real power, and therefore is given by

Q = PI/P. (7)

It is convenient to form the function

N,, - iN, = (E,H,* - E,*H,)/2. (8)

Comparison of NJ and N4 shows that

N,2-N~=Im(E,H,*)Im(E,H+*). (9)

Eq. (2) describes circular or linear polarization, respectively, as j is put equal to i or as its imaginary part is discarded. Upon choosing the direction of maximum power density to be the +z axis, where all angular field terms for which m # 1 are equal to zero, the parameter set simplifies to

P(I, m) = P,S(m,l),

G(I, m) = G,S(m,l). (IO)

Delta represents Kronecker’s delta function. It is convenient to define

S, = P:/sin 8,

T) = d P,‘/d& 01)

A special case previously analyzed in linear antenna size-wavelength studies is for electric multipolar radiation only, i.e., G, = 0. For this case N3 = N4 and, for each modal value,

2qN,,=F~(S~sin2++ TFcos2(p)(jlY,-y,J,),

2~N4,=F~(S~sin2~+ TFcos2(p)(j,J,+y,Y,),

(12)

where all functions are evaluated at r = a; i.e., on the Gaussian surface. Eq. (12) leads to

Under the common

plifies to

constraint (I e 1, (13) sim-

Q,= ;( (21;;)!!)2

(14)

and (14) describes a precipitous increase of Q with I. Since the nascent value of u is about l/1250 for atomic radiation

Q, 5: 2 x 109. 05)

A second special case is when both electric and magnetic moments are present with a single order 1. Then the square of the reactive Poynting vector may be written

4q2N,/ = [(P’F- G;)2(S:- T:)‘cos~$J

+(F,?S;- G;T,Z)Z

-2(Ff-G;)($-T:)(F,%f+G:T:)

Xcos*+l(.&JI+~~?)~. (16)

The real power density is proportional to

N,,- (F;+ G;). (17)

Since both (16) and (17) are stable with respect to parameter fluctuations, it follows that

F,G,(F;-G;)[(S;+T;)2+2S;Tf] =O. (18)

(18) is satisfied and the reactive energy minimized when

G,= -F,. (19)

D. M. Grimes/Quantum theory and classical, nonlinear electronics 289

Substituting (19) into (16) yields

2nN,,= F?(C - S:)W/+r,r,). (20)

Integrating N4 and NO over the Gaussian surface then solving for Q, results in

(21)

The ratio of (21) to (13) is equal to

12-l-1

I I 12+1 * (24

(22) shows that the radiative Q is decreased by half when magnetic and electric dipolar radiation are optimally mixed.

For the general case we postulate that the source is sufficiently adaptive to minimize the reactive energy by evolving to produce equality (19) and it, in turn, leads to the simplified angular field expres-

sions

EB = E i-‘F,(iH,T, + h,S,) cos +, I=1

qH+= f i-‘F,(h,T,+iH,S,)cos+, I=1

E+ = -qH+ tan+,

qHB = EB tan+.

(23)

Eqs. (9) and (23) show that (8) is an appropriate Poynting vector, and from (8) and (23)

2nN4= E E i”-‘F,F,[((.LY,-.M) I-1 n-=1

+iw/+Y*~))~T,

+W~~-YJJ -i(~~J,+.09PJn

+ (L4_i” +.wJ + i(_hvn -AY,))S,T,

+((J,Jn + Y,Y,) +i(W-JJ/))SJ/I. (24)

Q values comparable with the previous special case are dependent upon surface integrals of (24), and are given in the next section.

3. Kinematic parameters and the nonlinearity

The kinematic parameters of interest are real power P, reactive power R, z-directed power P, (proportional to the radiated momentum), and rate of change of z-directed angular momentum L,. The kinematic parameters are calculated by integrating the appropriate Poynting vectors, weighted as needed, over Gaussian surfaces of different radii. The radius is the smallest which just contains the radiator for R, is many times larger than a wavelength for P, and L,, and any value equal to or between the two extremes for P,

P = a2 /

dS2NO/k2, (25)

R = a2 J

dL?N,/k2, (26)

Pz=a2 dJZN,,cos8/k2. / (27)

L,=d dS2N+/k3, J (28)

290 D.M. Grimes/ Quantum theory and classical, nonlinear electronics

Table I Some useful integrals [24]

iWsinOdB [s,s, + T,T,] 3 *a(/. m) (1.1)

~~sint9dB[s,T,+s,,,~]-0 (1.2)

&*sint9fos8de[s,sW+ T,T,]

2/y/+1)(1+2)* 2(1- 1yq/- 1)2&[_ l,m) - (2/+ 1)(2/+ 3) Q’+ l* ml+ (2/- 1)(21+ 1) (1.3)

~~sineoosBdB[s,T,+s*T,]=~8(l,m) (1.4)

~“sid~dtJs,J,- ;:‘,‘:~$(~~~~8(/+ l,m)- ~~~::;(:1~~~B(I-l,rn) (1.5)

~*sinBdB((~~z-(s,~z]-~(12-I-l) (1.6)

[sined@ [s,s,_,, -T,T,_,,]-2n(n+l), whereO<s<l/2and2s=I-n (1.7)

sinode [s,T,_*,_, - s,_,,_,T,]=2n(n+l), whereO<s<1/2and(2s+l)=I-n. (1.8)

where

A$ = Re (E,H#* - E,*H,)/2.

Incorporation of (23) into the expressions for the various flows:

p,2r &f 12(1+ 1)’

rlk2 ,wl (21+1) ’

(29)

power flow expressions and use of table I results in the following

~2fI(I+1)(12-I-l)(iJ+yy,)

(21+1) I’ ’

- i F,F,2(1-- n)(l - n + l)( j,J, + y,Y, +i*J, + Y”Y,)( - l)(‘--n)‘2

I-1

- c F,F”2(1-n)(l-n+1)(-j,j”-y,y”+J,J,+ r,Y”)(-l)~‘-“+“‘2],

(I-nn)O

pz = 3 ,fl -[ F/-lF/ 2(f - 1)21(1+ 1)2 I(1 + 1) (21- 1)(21+ 1) + FF (21+ 1) I *

Discarding imaginary values of j results in L, = 0; putting j = i results in L, = P/o, that is

L,=O, orP/w.

(30)

(31)

(32)

(33)

D.M. Grimes/Quantum theory and classical, nonlinear electronics 291

Using eq. (A.l) and introducing new alphabetical parameters by the definitions

where q represents any integer, and

C,(u) = dA,/da - B,,

D,(a) = A,+ dB,/du,

leads to the equalities

(34)

(35)

(36)

(37)

u’( j,J, +y,Y,+j,J,+y,,Y,) =A,C,,+B,D,+A,,C,+B,D,, (38)

u’( -j,j, - y,y, + J,J, + Y,Y,) = -A,A, - B,B,, + C,C,, + D,D,,.

For the special case where only modal numbers I = 1 and I= 2 are present, the functions are given by

A&, + B,D, = - l/u3,

A,& + B2D2 = -3/u3 - 18/u’- A,A, - BIB2 + C,C, + DID2 = -6/u3 + 6/u5. (39)

Combining (31) and (39) and taking the ratio of (31) to (30) shows that

Q(u) = 12F12/3u3 - 18F,*(l/u3 + 6/u5)/5 - 12FiF&u3 - l/us)1/[4F12/3 + 36Fz2/5]. w

Eq. (40) has real roots at values of u which are dependent upon the ratio FdF,. For u small the ratio

FdF, = 5/9 - 16u2/27 (41)

-28

.- 16 Ln Q

.- 12 Q =I/125

I- 8

F2/FI -- 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 t 16 I I I I 11 1

Fig. 1. Radiative Q plotted as a function of the ratio of quadrupolar strength F’ to dipolar strength 4, for (I * 0.008. When ratio fluctuations to less than 0.21 occur the system returns to ratio zero. When ratio fluctuations to greater than 0.21 occur the system moves to 0.55. Radiation as described herein occurs near the 0.55 ratio.

292 D. M. Grimes / Quantum theory and classical, nonlinear electronics

is an approximate root. Even with a = l/1250, as discussed above, the ratio (41) in (40) shows that real power may exit the system with no reactive power penalty, see fig. 1. In other words, radiation with a Q of 2 x lo9 in electric dipolar radiation and lo9 in optimaly combined electric and magnetic dipolar radiation produces real power with no reactive power at all when resonate in this way.

Since the reactive energy of correctly weighted values of dipolar and quadrupolar modes sum to zero, energy minimization alone requires no higher-order modes. The directivity, defined to be equal to the ratio P,/P, is found from (41), (32), and (30) to be equal to 2/3. The same equations show that the coefficient recurrsion relationship

21+1 F’=I(I+l)’ (42)

with I increasing without limit uniquely describes full directivity. Since quantum radiation is fully directed, (42) seems worthy of detailed study.

An incoming plane wave contains driving fields of magnitude j,(a) and (42) requires the source modes to be of magnitude y,(a). The ratio of the two for small sources is (see (14))

y, [(2f + 1)!!12

jl= (21-t l)a2’+1. (43)

To study whether or not (42) is naturally met in radiating quantized systems, consider the fields near the source. From eqs. (lo), (19), and (42), the needed source fields, for linear polarization, are

E, = ici-‘(21+ 1) :Z’f( cosB)cos+, vH,= E, tan+, I

E, = ici-’ I

(H,T,- ih,S,)coscp, qH,= E, tan+,

E+= - xi-’ (h,T,+iH,S,)sin+, qH+= -E+cot$. I

Substitution of (A.l) into (44) shows the field to be proportional to negative powers of (I and, for every power of (I, all modal contributions are in phase; this in-phase quality is necessary for the fields to add regeneratively. Field evaluation requires knowledge of the sums

Z1(u,O) = i i-/(21+ l)h,P:, I-1

Z,(U,~) = i i-‘( $-&)hJ,, I-1

(45)

Z,(u, 0) = i i-f $-$)h,S,, l-1

D. M. Grimes/ Quantum theory and classical, nonlinear electronics 293

from which the fields are calculated by the equalities

aE,= iIicos+,

(46)

The sums of (45) may be done for the three polar angles 0, n/2, and s. Selected sums are developed in appendix A and results are listed in table II. Corresponding field expressions are listed in table III. Numerical radial field values are listed in table IV for several polar angles.

In a way unique to recurrsion relationship (42), the fields break into three types of quite different character, as may be seen in tables III and IV.

The first field type occurs in all three fields near the z axis. The angular fields may be seen in table III, with reference to all except the 2ePi” term on the positive axis. The radial field may be noted from the 5” column of table IV. The fields are phased similarly as functions of L, with the radial field nearly L2 sin8 times the angular ones, and with mirror symmetry through the t axis. They are in the correct direction and of the correct phase to produce multipolar moments with the spacing and phase needed for (42), see fig. 2,

Table II. See appendix A Approximate sum values over j(u), y(u), and h(u) at the three polar angles B = 0, r/2 and a

e=o e=n/2 e=a

j(u) 0

(e-” - sin u/u)/2

(e-‘” - sinu/u)/2

-iu

(1 - sin u/u)

- i(1 - cos u)/u

0

(eiO - sin u/u)/2

_(,iu _ sin u/u)/2

.I’(o) I” I 0 iL!! (2L + l)!! (L-l)!! uL+l ’ Lodd 0

(ie-‘” + cos u/u)/2 coSu/u- -9, L even _ i- L (‘:,~~I!!

(ie-I0 + cos u/u)/2

_i_L (2tu$,‘!! i[sinu/o+ H-1, L even

( -ieiO+ cosu/u)/2

_ iL (22~~,‘I!!

(ie’“- cosu/u)/2

+iL (‘,“,tt’,‘!!

h(u) 4 0 -iu+ L!!(2 L + l)!! (L _ l)wuL+ ’ L odd 0

12

[(l - i/(2u))Jemi0

i’-I~ 2L+l ( !! +

2uL+’

(1 - ie-‘O/u) + i(L- 1)!!(2L+ l)!! iL+‘(2L+1)

, L even - ie-‘“/(2u) + L!!uL+’ 2uL+’

(1 - i/(20)) e-”

+ ildL(2L+ l)!!

2uL+’

i(e-” - 1)/u + (L-2)!!(2L+l)!!, Lodd iL+‘(2L+ l)!!

(L + l)!!uL+’ ie-‘“/(2u) -

2uL+’

294 D.M. Grimes/ Quantum theory and classical, nonlinear electronics

Table III

Approximate field values for polar angles equal to 0 = 0, n/2, and ST. o = odd, e = even

e=o e=n/2

E, 0

E8 Ze-‘” + imLL(2L+ l)!!

zol.+2 I

cos $I

E+ -EB tan+

qH,= tan+& ~He==tan$&,, qH+= -cot#E,

Table IV Zenith-angle dependence of radial field, eq. (42) applicable

e Y- L 0

1 0 -0.2615 - 0.7765 - 1.5000 - 2.1213 - 2.5981 - 2.8978 3 0 3.3643 9.1837 12.9375 9.0156 - 0.3248 - 9.6430 5 0 - 10.6362 - 24.0120 - 10.9102 19.9537 20.8861 - 12.7591 I 0 24.1183 41.3573 - 14.8579 - 15.4293 46.5491 - 0.3917 9 0 - 44.3256 - 49.4847 44.8548 - 38.7503 34.6412 + 37.0506

11 0 72.2562 39.8483 - 29.1287 28.7950 - 29.6774 99.7909

13 0 - 107.2084 - 5.5842 - 34.4843 67.2611 - 90.8173 174.4559 15 0 148.8570 - 47.6148 87.8557 - 38.7371 - 65.1326 235.3001 17 0 - 195.1436 107.3060 - 52.3514 - 94.6793 55.7626 251.4947

2 0 - 1.3024 - 3.7500 - 6.4952 - 7.5000 - 6.4952 - 3.7500 4 0 6.4079 16.1124 15.4261 - 1.8750 - 18.6737 - 17.9874

6 0 - 16.4905 - 32.8275 - 0.2740 27.9844 - 12.2089 - 44.7624

8 0 33.3705 47.3162 - 32.0408 11.8477 28.6165 - 77.0041

10 0 - 57.2876 - 47.2823 45.0208 - 48.7786 70.7561 - 98.6457 12 0 88.9248 25.8959 -0.5166 - 18.9644 52.3946 - 89.8462

14 0 - 127.2427 19.4164 - 67.7498 78.9564 - 38.7376 - 35.7447

16 0 171.5747 - 71.7649 83.7320 32.9165 - 121.1001 64.5682

5 15 30 @=O”

45 60 75 85 90

- 2.9886 - 3.0000 - 13.0513 - 13.5000 - 31.4377 - 34.1250 - 57.6068 - 66.9375 - 89.3754 - 113.6953

- 123.0424 - 175.9570 - 153.5793 - 255.1377 - 114.9032 - 352.5425 - 180.2288 - 469.3889

-1.3024 0 - 7.0591 0

-21.2039 0 -47.8825 0 -90.9606 0

- 153.5751 0 - 231.7322 0 - 343.9701 0

and their magnitudes increase as

L(2L + l)!! zaL+2 '

with increasing maximum modal number L. The second field type occurs in the three fields in the equatorial plane. With regard to the near field

terms, in the radial case the odd and even modal numbers are respectively symmetric and antisymmetric with respect to the z coordinate. Odd-numbered modes are in phase, and n/2 out from the even-numbered ones. They are of the proper phase to drive the moments generated by the fields of the first type; odd modes are driven in the same phase in the upper and lower hemispheres while even modes are not. Field magnitude increases as

L!! (2L + l)!! (L-l)!! aL+z ’

D.M. Grimes/Quantum theory and classical, nonlinear electronics 295

-i j;+i =

2 -I ’

T *+I

+I* *-I

Z Z +i l m-i +I* *-I

-2i l I l +2i -3*

I

l +3 +i* l -i +3* . -3

A=3 -I* *+I Q=4

-i l i l + i _

Z ,I l t l + I

+4i l l - 4i +5* 9-5 -6i. ~‘64; -10 l l + IO +4i l

;T* l + 5

-i . l t i . I Q=5 Q$-

Fig. 2. Electric multipolar charge configuration which, when combined with similarly phased magnetic multipolar sources of magnitude given by (42), produce the fields described in this paper.

with increasing maximum modal number L.

The third field type occurs in the angular fields on the +z axis, and in the radial and azimuthal angular fields in the equatorial plane. Their magnitudes are independent of radius. They describe a z-directed power density of magnitude two on the +z axis and one half through the equatorial plane. This, in turn, requires that power emanate into the lower hemisphere, become z-directed and pass up through the equator, becoming fully z-directed.

A detailed numerical analysis of Q(u) versus maximum modal number L may be combined with knowledge of the fields for table III to show that emission of an electromagnetic wave train by a small, adaptive source grows regeneratively and nonlinearly by the following series of events.

Radiation initiation is uncertain and only occurs after the ratio FJF, has somehow exceeded 2/9, see fig. 1. Energy mini&a tion then requires the field coefficients to shift to ratio (41), as radiation starts. Change continues to occur, however, because F3 is driven in phase by the two lower coefficient fields. Where M, and N, are coefficient-independent terms with M B N, the extension of reactive energy equation (40) to include F3 terms as calculated from (31) and (38) using I = 3 and n = all integers equal to or less than 3 shows that the leading additional reactive energy terms may be written

(47)

where all parameters are positive. Energy minimization requires F3 to increase and approach the ratio

FYF2 = NJ%. (48)

Freed from the constraint of (41), F2 increases until the equality

FJF, = 5/9 (49)

is approached. As this occurs, Fa is driven in phase by the three lower coefficient fields. This generates an additional reactive energy, whose coefficients are calculable similarly to F3 but maximum order 4,

[ -M4Fd2 + N,F,F3]/a9 (50)

296 D.M. Grimes/Quantum theory and classical, nonlinear electronics

and (50), in turn, permits FJ to increase beyond the limits of (48). F4 and F3 then increase at the fixed ratio

&/FJ = K/K (51)

until the equality

FJF, = l/10 (52)

is approached. As this occurs, F, is driven in phase by the four lower coefficient fields. This generates an additional reactive energy, whose coefficients are calculable similarly to F; but maximum order 5,

[ -MsF; + NsF5F4]/d1 (53)

and (53), in turn, permits F4 to increase beyond the limits of (51); etc. This process continues with no field-imposed upper limit on L and ah coefficients approaching (42) as an

asymptotic upper limit until the available energy has been radiated, or until a source-dependent restriction is imposed.

4. Discussion

Are the foregoing arguments applicable to atomic level phenomena? Of course a theory can only be disproven or made to seem plausible, never proven. The following items support the theory presented here as a plausible one.

A) A plausible theory based upon classical equations must explain the existence of stable atomic states; a sufficient postulatory base is electrons which are adaptable spatial arrays of charge, rather than point charges or waves. It was shown previously that the radiative reaction force upon a point charge oscillating collinearly with the z axis and generating only electric dipolar radiation is described by the equation [4]

where the third term on the right is equal to the (real) radiation reactive force on a free charge. z is the position of the charge, e its magnitude, and a is the radius of the smallest Gaussian surface which just contains the charge and its orbits. The first term has magnitude equivalent to the attractive Coulomb force; it is non-dissipative and three

orders of magnitude larger than the real radiation reaction force. Energy is minimized when the charge is rendered into a non-radiative configuration. Non-radiating configurations are stable and, as shown in appendix B, stability leads to the Schriiinger wave equation as a non-mechanistic descriptor of system- and time-average values. (A partial justification of the impedance technique used in the development of (54) may be found in Shelkunoff [14].)

B) A plausible theory based upon classical equations must explain the kinematic parameters of quantum radiation. By item A the S&r&linger wave equation with its quantized source energies apply. As shown in this paper reactive radiation energy minimization by an adaptable and deform- able radiative source drives system nonlinearities whose outputs, together with source energy states, predict the observed kinematic properties of quantum radiation.

C) A plausible theory should provide a reason for the probabilistic nature of spontaneous decay and field-induced transitions [15]. The quadrupolar-to-dipolar coefficient ratio needed for onset of radiation is a configurational (non-energy) thres-

D.M. Grimes/ Quantum theory and classical, nonlinear electronics 297

hold condition that would eliminate all such radiation were there no other factors. Although details depend upon interior processes, electronic orbital-spin coupling, as described by Schott, results in continuously variable charge and current structures and they, in turn, by the model of this paper, have occurence probabilities. Since for radiation exchange to occur the ratio must exceed 2/9, and since the quadrupolar-to-dipolar coefficient ratio is proportional to u on the Gaussian surface, the smaller u the smaller the expected transitional probability.

D) A plausible theory should explain the energy-frequency relationship of quantum fields [l, 2, 161. By the wave equation, a quantized system may contain an occupied eigenstate s of energy Es and frequency ws, and an unoccupied state t of energy E, and frequency u,. If the system is driven by a plane wave of frequency w, a value equal to the frequency difference of the two states, i.e.

a transition may be initiated. During transitions, the Manley-Rowe equations

P P I=-2 @t *s ’

P (55)

-= u *;,

s

apply where the upper or lower sign applies, respectively, if 0, is larger or smaller than u,. Posi- tive powers represent power from the eigenstate or the far field, as indicated. Using the eigenstate energy-frequency relationship E, = ho,, time in- tegration of (55) gives

E, = hq,

E= The,

the Einstein energy-frequency relation of quan- Since similar summations are not available over tized fields. In the past a difficulty has been that, Neumann functions the radiation pattern is not although (55) may remain valid for linear systems, known in detail. They do, however, describe fully since frequency mixing cannot occur, the results directed radiation.

are trivially equal to zero on both sides of the equation. The present model, however, shows the transition to be a nonlinear one; power flows are nonzero and (56) is an expected result.

E) A plausible theory should not require singu- larly large values of atomic, surfacial currents. From quantum theory, the expectation value of an atomic radius is inversely proportional to its ion- ization energy. Although the radius obtained during energy exchange is unknown, presumably the post-absorption radius cannot exceed that reached during absorption, and during absorption to ion- ization by the Schriidinger equation the effective radius increases without limit. Since even for certain stabilized atoms the ratio of excited- to ground-state radii may exceed 100, the ratio of the Gaussian surface radius to an optical wavelength may not be small; although (13) is always valid, (14) may not be. If not, (13) shows that the otherwise precipitous increase in surface current with modal number doesn’t occur and the surfacial currents necessary to support (42) don’t increase precipitously with increasing modal number. It also suggests that transitioning atoms un- dergo nova-like expansions during energy exchanges.

F) A plausible theory should address the directivity and physical extent of quantum radiation. A common question is whether a photon passes out- wards tending toward infinite dilution or whether it flies, needle-like, in one direction only [17, 181. Dirac addressed a similar question when he concluded that mixed states occur within an inter- ferometer [19].

Field sums over spherical Bessel functions may be done directly, and their portion of (44) goes to

E, = sinde-i’JmS@e-i+,

E, = COSee-io==@e-i+, (57)

E+ = _ie-iOCOS#e-i”.

298 D. M. Grimes/ Quantum theory and classical, nonlinear electronics

The fields of table III show the average Poynt- ing vector to be independent of distance from the source and of value 2, l/2, and 0, respectively, on the + z axis, the equatorial plane, and the -z axis. Table III fields show that power emanates into both hemispheres, then the lower-hemisphere power turns and passes up through the equator with an intensity term independent of distance from the source (out to a distance determined by the length of the wave train.) This implies a negative answer to needlslike photons, for, were the wave train long enough, the full breadth of upper half space would be occupied by it.

A quite different view of the same phenomenon, with a similar conclusion, may be seen from antenna aperture theory [20]. Antenna aperture A is the ratio of power absorbed by a receiving antenna to incoming power density; it equals the lateral

area from which received power is extracted. For example, the aperture of small, isotropic antennas is

A = X2/(4a). (58)

The effective aperture of a directive antenna is increased by its gain G and G is limitlessly large for fully directed radiation. The area from which power is drawn by such a receiver is equal to the product GA and, therefore, the lateral area from which power is drawn is limited only by the transition time of the receiver. By reciprocity, the same system will transmit to a lateral area limited only by the length of the wave train.

Thus a photon’s radiation occupies a right, circular volume of approximately equal radius and height. These results support interferometric phase cancellation experiments.

Appendix A

So long as the order, 1, of the radial functions is non-infinite, they are correctly represented by the series

PI

(A.1)

Although (A.l) is convenient for most purposes, it may not be used to evaluate sums I(a, 0) with an infinite upper limit.

That portion of I(a, 8) over Bessel functions may be achieved for 0 = 0, n/2, and s either by direct summation using the expansion [21]

(-1) = j&J) = Sc0 (2S)!!(211 ;,“I l)!! ’ (A-2)

or the equality

exp(-iacos8)= c(-i)‘(2f+l)j,(a)P,(cosB). 1-o

(A-3)

Their evaluation is straight-forward and will not be detailed here. Results are listed in table II. The portion of the sums over Neumann functions may be achieved for 8 = 0, s/2, and n using the

expansion [21]

J&J) = - c

‘-I (21- 2s - l)!! ezs_,_i _ m s Zs+l-1 (-1) *

s_o W! $go (21+ 2S)!!(2S - l)!! * (A.4

D. M. Grimes/ Quantum theory and classical, nonlinear electronics

For sum evaluation, note the following equalities:

I,“(a,O) = 1[‘(6,7r) = 0,

1~(u,o)=I~*(u,7r)=1~(u,o)= +*(u,r).

This leaves I;‘( [I, 0) and the B = a/2 values to be calculated in detail. On the z axis

299

(A-5)

1 L li’(u,O) = - LFr z C (-i)‘(21+ 1)

‘-1 (2& 2s _ l)llu2”-‘-’ + 5 C

( _l)“u2s+~-i * l-1 [ s-0 (2s)!! $_a (2s - 1)!!(21+ 2s)!! 1 For even values of 1, odd powers of u appear in the series, and vice versa. Within the bracket of (A.6), integer powers of u from -(I + 1) to + (I - 3) occur in the first term, and from (I- 1) to cc in the second; to evaluate (A.6), first include the I = 0 term and then subtract it from the completed result.

That portion of the first sum of (A.6) with negative powers of u may be written

-fLil Ii i-’ (&n+l)!!u” 3 w+ w+ n - w qn + 1 zrn + Q

3

n-l I-n-l

where m is any integer. The sum over I may be done, after which the double sum goes to

ieL(L + n)!! -- : I$ (L-n+l)!!a”’

For power q 2 0, the sum starting with I = 0 may be written

which is equal to

‘,-in.

2

(A.7)

(A-8)

After subtracting the I = 0 term, the completed sum is therefore approximately equal to

In the equatorial plane,

I;I(u, 7r/2) = iLliml i(21+ I) (/!$I ‘-l (21- 2s - l)!! u2S_,_1 + E c

4 I’, ( s-0

(2s)!! s-0

s 2x+/-l t-0 0

1 (21+ 2s)!!(2s - l)!! -

(A.lO)

300 D. M. Grimes/Quantum theory and classical, nonlinear electronics

Since I is odd, all powers of (I are even. The coefficient of 04 is

L I!! (,-q-2)!! 9+1 (- 1)(9-‘+1)‘2

i9;s(21+l)(1-1)!! (I+q+l)!! +i c (2l+l)(,f!i)!! (f+q+l)!!(q-l)!!’ lo lcl

This sum is equal to zero for each positive value of 4. For q = 0, the coefficient is approximately equal to L and for q c 0, putting n = -4, it may be written

$ri (21+1)1!!(f+n-2)!! pi L!! (2L+l)!!, Lodd ne ,o (I-l)!!(l-n+l)!!a” (L-l)!! eL+l ’

and, therefore,

I;‘( u, a/2) = i L!! (2L+l)!!, Lodd

(L-l)!! CTL+l . (A.ll)

The second equatorial sum is

I;‘(u, R/2) = - LtiW i(21+ 1) (I- l)!! (-1)“u2s+‘-*

I,, l? (26 2s - l)!! u2s_,_1 + E

. -b 0 . . [ s-0 (2s)!! s=. (21+ 2s)!!(2s - l)!!

le 1 (A.12)

Since I is even, all powers of u are odd. The coefficient of 04 is

_ i (21+1)(1-1)!!(1-q-2)!! +(_l)&_‘),29~l (21+1)(1-1)!!(-1)“2

9,:3 I!!( 1+ q + l)!!

P, I!!(l+q+l)!!(q-l)!! .

The coefficient of each positive power of u sums to zero. For CJ < 0, the series is equal to

I;‘(u,a/2) =y- “i’ i (21+ 1)(1- l)!! (1+ n - 2)!!

no le u”l!! (1- n + l)!! ’

which may be approximated by

z;‘(u,a/2) z--g-- cosu (L-l)!! (2L+l)!! ) L even

L!! uL+l

The third equatorial sum is

(A.13)

(l-2)!! GGM2) =iL’i”, I?(21+1) (,+ l)!!

O3 (21- 2s - l)Hu2”-‘-1 + 2 ( _1)Su2s+l-1

* 4 :, [

C S-O

(2s)!! s=. (21+ 2s)!!(2s - l)!! I

(A.14)

D.M. Grimes/ Quantum theory and classical, nonlinear electronics

Since 1 is odd, only even powers of u appear. The coefficient of u9 is

i; (21+1)(1-2)!!(1-q-2)!! 9+1 (21+1)(1-2)!!(-1)“-“‘2

a+3 (I+ l)!!(l+q+ l)!! + i(-1)9’2 F (I+ 1)!!(1+ q+ l)!!(q- r)!! *

.I0 lo

For q r 0, the series sums to i( - 1)912uQ/(q + l)! which, in

.SiIlU l-

u .

For negative powers of u, the series is equal to

turn, sums over q to

iLi1t (21+1)(1-2)!!(1+n-2)!! =i(L-2)!! (2L+l)!!, Lodd

ne lo u”(r+ 1)!!(1- n + l)!! (L+l)!! aL.+’ .

The sum may be written

Z;‘(u,77/2) =iy + i(t - 2)!! (2L + l)!!

L!! uL+l .

Expressions for all sums are tabulated in table II.

Appendix B

Many different postulate sets yield the Schrii- dinger wave equation as a derived result. Bohm emphasized that, if the derivation is based upon underlying hidden variables, the proper wave in- terpretation is dependent upon the nature of the variables [22, 231. This appendix shows that the adaptive electron model yields the wave equation and, therefore, it is appropriate to fit interpreta- tions to the model.

Non-stationary, repetitive electronic motion is anticipated with measurable average values of kinematic parameters. Since a detailed description of extended, moving charges coupled with their own spins is beyond our present capability, we seek instead a statistical description which pro- vides the proper system- and time-average values of kinematic parameters. For this purpose, it is postulated that average state values transform lin- early between different characterizing sets of coordinate variables; a Fourier transform is utilized to define the transformation between them.

301

(AX)

An electron trapped by an electrostatic potential field V(r) will produce time-average charge and current densities at each spatial coordinate. It is convenient to re-state the spatially described current density as the charge density between current-density coordinate positions p and p + dp. The time-average charge densities as described using the different coordinates sets are represented, respectively, by p(r) and p(p). Since both densities are constrained to be negative real, new constraint-free functions U(r) and W(p) are defined by

U*(W(r) = -p(r),

W”(P)W(P) = -P(P),

from which it follows that

j U*(r)U(r)dr3 = 1,

jB’*(p)@)dp3 = 1,

(B.1)

(B.2)

302 D. M. Grimes/ Quantum theory and classical, nonlinear electronics

where the integrals are over all coordinate positions.

Since U(r) and W(p) are but different methods of describing the same dynamic charge-current distribution, they are interdependent. A general linear relationship between them is of the form

w = L[WP)l, W(P) = L-w(~)l, (B.3)

where L represents a linear operator and L- ’ its inverse; a general linear transformation is the Fourier integral one

U(r) = (&)“‘jW(p)exp( q)dp3,

W(P) = (&)“*jU(r)exp( s)dr’,

(B.4)

where K is a unit-determining constant whose value is to be determined.

If the normalized mass and charge densities are equal, the low speed kinetic energy E is given by

E= $jW*(p)Wb)p*dp’

+ V(r)U*(r)U(r)dr3. j (B.5)

Substitution of (B.4) into (B.5), making two partial integrations, and incorporating (B.2) into the result yields (B.6)

jU*(r)dr’ [ - &VW(r) + [V(r) -E]Ll(r)]

= 0, (B-6)

where ft = mK/e. Eq. (B.6) is satisfied if the in- tegrand itself is put equal to zero, and the result is the Schrijdinger wave equation as a time-independent, non-mechanistic descriptor of system- and time-average values of the kinematic variables.

References

(11 M.T. Weiss, Proc. IRE 45 (1957) 1012. [Z] D.M. Grimes, Proc. IEEE 53 (1965) 1663. [3] L.J. Chu, J. Appl. Phys. 19 (1948) 1163. [4) D.M. Grimes, J. Appl. Phys. 49 (1978) 4341. (51 G.A. Schott, Phil. Mag. 36 (1918) 243. [6] N. Bohr, in: Albert Einstein, Philosopher-Scientist, P.A.

SchIipp, ed. (Harper Torchbooks, New York, 1959) p. 201. [7] W. Heitler, in ref. 6, p. 181. [8] N. Bohr, Phil. Mag. 26 (1913) 1; also in: World of the

Atom, H.A. Boorse and L. Motz, eds. (Basic Books, New York, 1966) p. 751.

[9] WK. Heisenberg, in: World of the Atom, ref. 8, p. 1107. [lo] W. Heitler, The Quantum Theory of Radiation, 3rd ed.

(Oxford Univ. Press, London, 1960) p. 175. [ll] P.A.M. Dirac. The Princinles of Ouantum Mechanics. 4th

ed. (Oxford Univ. Press, bxford, 1958) chap. VII.

WI

[I31

(141

t151

P61 P71 1181 P91 PO1 WI

J.D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975) section 9.9. W.K.H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed. (Addison-Wesley, Reading, 1962) p. 232. S.A. Schelkunoff, Electromagnetic Waves, chap. 12 (Van Nostrand, New York, 1943). A. Pais, The Science and Life of Albert Einstein (Oxford Univ. Press, London, 1982) p. 411. P.A.M. Dirac, ref. 11, p. 1. M. Planck, in: World of the Atom, ref. 8, p. 501. A. Pais, ref. 15, chap. 21. P.A.M. Dirac, ref. 11, sec. 3. J.D. Kraus, Antennas (McGraw-Hill, New York, 1950). M. Abramowitz and LA. Stegun, Handbook of Mathe- matical Functions, Applied Mathematics Series 55, (U.S. Department of Commerce, U.S. Government Printing Office, Washington, 1964).

WI D. Bohm, Phys. Rev. 85 (1952) 166. [231 D. Bohm, Phys. Rev. 85 (1952) 180. 1241 D.M. Grimes, J. Math. Phys. 23 (1982) 897.

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