IL NUOVO CIMENTO VOL. X X X I I I , N. 6 16 Settembre 1964

A Three-Dimensional WKB Approximation for the Dirac Equation.

M. ROSEN

U. 8. Naval Research L,~boratory - Washington, D. C. (')

(ricevuto il 20 Aprile 1964)

S u m m a r y , - - A three-dimensional semiclassica~ approxima, tion is devel- oped and used to describe the precession of the polarization of a Dirac particle in an electromagnetic field and the dynamical and kinematical origin of the precession clearly exhibited. The rate of change of the longi- tudinal polarization is calculated and shown to arise entirely from tha t change in orientation of the classical t ra jectory due to the influence of the electric field alone. The effect of the anomalous moment is exhibited by consideration of the Dirac equation modified by inclusion of a Paul i moment term.

1 . - I n t r o d u c t i o n .

I n r e c e n t y e a r s t h e r e has been a r e v i v a l of i n t e r e s t in d e s c r i b i n g t h e pre-

cess ion of t h e p o l a r i z a t i o n of p a r t i c l e s m o v i n g in an e l e c t r o n ~ g n e t i c field (1).

The v a r i o u s t r e a t m e n t s g e n e r a l l y t e n d to fa l l i n t o two g r o u p s (~ ) - - those t h a t

cons ide r t h e p r o b l e m f r o m a s t r i c t l y c lass ica l p o i n t of v i ew a n d those t h a t in-

(') This work was begun while the author was at the School of Physics, University of Minnesota, Minneapolis, Minn, where it was supported in par t by the U. S. Atomic Energy Commission, contract no. AT(l 1-1)-50.

(1) Extensive reference to the l i terature may be found in H. A. TOLHOEK: I~ev. Mod. Phys., 28, 277 (1956) and H. BACRY: Nuovo Cimento, 26, 1164 (1962). The la t ter lists classical papers only.

(2) Notable exceptions '~re TOLHOEK (1OC. cir.) who considers the small field l imit and S. I. RUBINOW and J. B. KELLER: Phys. Rev., 131, 2789 (1963), who use the WKB approximation also. Although the method of Rubinow and Keller, of course, parallels ours to some extent, our t rea tment places much more emphasis on the physics involved and hence is more straightforward and transparent .

1668 ~. ROSEX

vest [gate in detail the solutions of the Dira(: equation. In ali cases, the physi(.s tends to be obscured b y the details of the formal ism used. l:he semi-classical

approach described below obtains the equat ion of mot ion of the polariz~tion

direct ly f rom the Dirac equat ion without having to find e'~ I)licit solutions, thus

rendering the physics more t r ansparenL In Sect. 2, we describe the semi-

classical approximat ion , but consider in some detail only the lowest order.

In Sect. 3, an expr(~ssmn for the polarizat ion oper~,tor is ~btained, its equa-

t ion of motion is oe, ivcd and found to ~gree with t ha t obta ined classically (3),

and in Sect. 4, the ra te of change of the longitudinal polarizat ion is calculated.

Final ly in Sect. 5, we consider the Dirac equat ion modified b y the inclusion

of ~ Paul[ t e rm and calculate the effect of the anomalous magnet ic m o m e n t

on the precession of the polarizat ion ~nd on the ra te of change of the helicity.

2. - 7 h e ~ emi -c la s s i ca l a p p r o x i m a t i o n .

Consider the Dirac equat ion for a par t ic le of mass m, charge e and energy E

moving in an external e lect romagnet ic field,

(9..l) [ - - ir - - cA) + /~m + (e~ - - E)]yJ(r) = 0 .

Here ~. and fi are the usual Dirac operators , A and ~ the e lect romagnet ic po-

tent ia ls and we have t aken ~ = c = 1. We m a y write ~o(r) in the fo rm

(2.2) w(r) = u(r) exp [i S ( r ) ] ,

where the spinor u(r) satisfies the equat ion

[ a . n § tim § (e~ - - E ) ] u(r) = i a" Vu(r )

and = V S - - e A ( r ) .

The semi-classical ~pproximat ion is int roduced in the following way. We

assume tha t the rapid oscillations in tile wave function are contained in the

exponentfid t e rm and tha t u is a re la t ively slowly v~ryiug funct ion of r. This

enables us to define successive orders of approx imat ion to u(r) as follows:

(2.3) u = u 0 § . . . ,

(2.4) ( a ' ~ § tim + e~ - - E) Uo = 0 ,

(2.5) (ot.~ § tim + eq, - - E) un = i cx. Vu~_l .

(a) See R. H. GooD, Jr.: Phys. Rev., 125, 2112 ([962) or V. B,~m'~ta~N, L. MICHEl. and V. L. Tl.n.n(~D~: Phys. Rev. Lett., 2, 435 (1959).

A T H R E E - D I M E N S I O N A L WKB APPROXIMATION FOR TIIE DIRAC EQUATION 1669

The expansion, of course, may be semi-convergent even when the potentials

are well behaved. We shall restr ict our considerations to the lowest approx-

imation, in which case we need concern ourselves only with the equations

(2.4)

(2.6)

( 4 . ~ § tim + e ~ - - E) Uo = 0 ,

(4 . ~ ~- t im + eq~ - - E) ul = i 4 . Vu0.

I f we mult iply eq. (2.4) on the left by ( a ' n - 4 - ~ m - 4 - E - - e ( p ) , we find as condition for the existence of nontr ivial solutions

(2.7) ~ 2 + m 2 _ ( E - - e~) 2 = 0 .

This is just the classical t tami l ton-Jacobi equat ion for a charged part icle having momentum and veloci ty

(2.S) p o , = ~ , v o, = = / ( E - - eq~) ,

respectively; its solution defines a family of classical trajectories for the ine ident

particles. Note tha t in the lowest or (( classical ~) approximation, there is no effect on the trajectories of a spin interact ion with the f ie ld- - the effects of field gradients are contained only in the higher-order contr ibutions to the wave function. We are restr ict ing ourselves therefore to fields which, roughly, are slowly varying over a de Broglie wavelength of the particle.

In the same manner as above, we find as a solvability condition on eq. (2.6)

o r

(2.9)

where

(~ . ~ § t im § E - - eqJ) a . Vuo = 0 ,

{4 . f+ fig, 4. v} uo = 0 ,

f =- n / 2 ( E - ecf) , g = m / 2 ( E - - eq~)

and the braces denote the an t i -commuta tor of the operators included, e.g.

{A , B} = A B + B A .

We now have the following program for the calculation of uo: The eikonal

funct ion S is to be determined f rom eq. (2.7) together with appropriate boun-

dary conditions representing an initial beam of particles. One way to do this is to calculate the appropriate family of trajectories and then integrate along these trajectories to determine S. With S fixed eq. (2.4) restricts Uo at each point to a two-dimensional spinor space. Although this equat ion provides no

1670 M. t~OSr;N

relation between spinors at different points, it will be seen short ly tha t eq. (2.9), provides information about the change of u 0 along a t ra jectory. Thus if uo is specified on an initial surface cut t ing across the trajectories, our equations specify it everywhere.

We have mentioned, and in what follows we shall f requent ly deal with quantit ies assoei'~ted with the motion of a classical particle. I t is interest ing and necessary to indicate how the motion of a real particle, essentially quantum-

mechanical in nature, may be related to tha t of an associated classical par-

ticle. Under realistic conditions such tha t the three-dimensional W K B ap- proximat ion is v~lid i t should be possible to represent the real part icle by a

moving wave packet so constructed tha t i t moves wi thout rapid spreading.

The results obtained below would then apply to the polarization of the wave packet as well as to an associated classical particle. Briefly we may, e.g, express such a packet in the form

(2.10) cp(r) = z(r)Uo(r) exp [i S ( r ) ] ,

where z(r) is an appropriate ampli tude factor modulat ing the approximate wave function. The packet may then be used to define, e.g., the position of the classical particle:

rr = f lzI2u~uord3r.

The quantum-mechanical equations of motion then yield

(2.11) drr IZl~U*oOtuod3r= IZ]~u~OUO E~ecfd3r~vtoJ(E--ecf)=v.,,

where the peaking propert ies of Z are used to evaluate the quan t i ty in brackets (and where we have also used eq. (3.2) below). Similarly the mean kinetic

momentum is given by

(p -- cA) ...... ~ f ] )~ 12utouogd3r ~ ~ol

and

(2.12) d~oJdt ~_ e(-- V(vo, + vc, x curl Ao,).

This result can also be obtained direct ly by classical considerations. Le t Q(rcl ) be some quan t i ty depending only on the position of the associated classical

particle. Then the change in this quant i ty due to motion along the t ra jec tory is

(2.~3) dQ(ro:)/dt = (v c~" V)Q.

A THREE-DIMENSIONAL WKB APPROXIMATION FOR THE DIRAC :EQUATION" 1671

Both eqs. (2.11) and (2.12) follow direct ly f rom this equation. F r o m now on

we drop the subscr ipt cl denoting the associated classical part icle.

As i t t ravels along the classical t ra jec tory , the spin of the associated clas-

sical particle will preeess under the influence of the external field; we are

interested in calculating the ra te of precession. The spin direction is given b y

= u*o uo/luo I ,

and is, of course, a funct ion of the co-ordinates. To find, then, d<a}/dt one

needs to know duo/dt; this we obtain f rom eqs. (2.9) and (2.13),

�9 VUo duo (2.14) . . . . . . . . (-- V " f + fla "Vg -- i~. c u r l f ) u o .

E -- e~ dt

F rom the form of the first t e rm on the r ight and the fact t ha t i t contains no

Dirac operators, one might guess t ha t i t arises f rom the change in normali-

zat ion of the wave funct ion due to the conservat ion of flux. Indeed we find

d (u~uo) = -- 2 ( d i v f ) Qo uo.

d--~

This is just an expression of the conservat ion of part icle flux in the classical approximat ion , for

( no, u~ ~tuouo§ 2 divfQoUo = div E ~ e~

= div (Qo {or.f+ fig, a} Uo) -- div (u+oOtUo) = O.

I t is convenient , therefore, to in t roduce the normalized spinor

w = o/lUoL,

which also satisfies eq. (2.4), bu t whose (~ equat ion of mot ion ~> now has the

suggestive form

. dw (2.15) ~ ~ - = ( u . c u r l f + ifict.Vg)w =-- M w .

Note t h a t M is hermi t ian , so t ha t we have

d <a> (2.16) dt -- i < [ a , M ] > ,

1672

where for any oper,~tor O

M. ROSEN

<O> ~ w*Ow

and the bracke ts denote the c o m m u t a t o r of the operators included, e.g.,

[A, B] ---- A B - - B A .

3. - Precession of the polarization.

Consider eq. (2.16). We have

d a> -- i {[a, .curl f § iflc~.Vg]> (3.1) ~ < = a �9

In eva lua t ing the expression on the right, we shall need several relat ions and

identi t ies which will prove useful th roughout the rest of the paper. First ,

note t ha t since w satisfies eq. (2.4) and since also (a . f§ is Hermi t ian , i t

follows tha t for any opera tor O

(3.2) <O> = < { a . f + fig, O}>.

Also note the following identi t ies involving a, a and an a rb i t r a ry space vec-

tor m :

a(cx" m ) = a(a" m ) = m + i ( m Xa) ,

( a ' m ) a = (~" m ) a = m - - i ( m x a ) , (3.3)

a ( ~ ' m ) = m Q + i ( m x ~ ) ,

( a ' m ) a = m e - i ( m x a ) ,

where ~o is defined b y a = Ca. I t is now s t ra ight forward to show tha t

d e (<a x e u r l A> _L {v • (a xV9o)> ) (3.4) d-t <a> = (E -- e~)

where v ---- = / ( E - - eqg. One mus t be careful not to confuse this with the equat ion of mot ion of the

polar izat ion for, as DARWIN (4) and MOTT (s) pointed out quite early, the po-

lar izat ion is not given b y <a> bu t ra ther b y the expecta t ion value of a in the

(a) C. G. DARWIN: t)'roc. ]~oy. ,~oc. (London), A i20, 621 (1928). (5) N. F. ~/[OTT: Proe. Roy. Soc. (London), A 124, 425 (1929).

r

A T I I I { E I , ; - I ) I M E N S I ( ) N A I , \VKB A I ' I ' R ( ) X I ' q A T I O X I"()R T i l l , : I ) I R A C E Q U A T I O N 1673

rest s/tstem o] the l)irac particle. If we denotc the polarization ve(,tor by {T) ,

then

(3.5) ~T, + = ~ , , a ~ , : l ! ~ , ~ ! ~ ,

where ~, is the res t - sys tem wave function. We ('an cxpress the polar izat ion

as the expeeta t ion value of an operqtor in the labora tory system, however,

t)y using the fact tha t ~ve is related to the labora tory wave fun(.tion, yJ, by an

inst ,mtaneous Lorentz t rans format ion .

: A~o..

Good and ROSE (';) have shown tha t for ~ free l)artiele, A can be wr i t t en in

the very interest ing form

(3.6) , 1 - , : E-~ e x p [ i ~ q ,

where E , = ( p : ~ m U '-' ~md where e x p [ i F ] is the Foldy- \Vouthuysen t ransfor-

mat ion, given by

m + E,,+ fia.p ( ' X l ) [ i F ] [2E,,(E,, i-1)]~ "

With slight nmdification, however, their a rguments also hold for a p~rrticel

in an external field and, with the subst i tu t ion

p ~ p - - e A

eq. (3.6) for 21 is still a wdid one. We thus obtain

(3.7) ~(T) - - y)+E~' exp [ - - i F l a exp [iF]~vfiptE;~.

Now to lowest order in the W K B approx imat ion outlined in Sect. 2, we nmy

replace E~ by (E--eq;), (p - -eA) by n and y, by no.

(3.8) ( T ) = w + T w ,

whel'c

( 3 . 9 ) T = a - ~;/3(~ x~) (~ x (a x~)) . E - eft ( E - eq~)(E- e~. -- m)

We then find

(~) R. H. GOOD, Jr. and M. E. RosE: Xuovo Cimento, 24, 864 (1962).

1 0 6 - I I Nuovo Cimento.

1674 x[. ]:os i.:,,-

Using eqs. (3.2) and (3.3), i t is easy to show tha t

(3.10) (,T~ = E - - Gv <a> 1 m m(E eq + m) <(a .= )r~> .

We shall call T the polarizat ion operator . I n the rest system, of course, i t reduces to a. 2qote tha t

(3.11) T ' ~ = a.rc ,

(while the component of the spill normal to the veloci ty is not invar ia l l t under the Lorentz tra.nsformation), and hence

m < T . ~ > r c (3.12) <a? ..... (T>-~-

(E eq) ( E - - e</~J(E-- e 7 + m) "

Using eqs. (3.10), (3.1l), and (3.12), one finds in a s t ra ight forward manlmr

(3.13) d ( T > .... e - ( T X c l l r l A - ( E eq, ) T x ( v X V q ~ ) } . , it ~: = ~ E 7 ~ ~ .,

The physical meaning of eq. (3.13) is nlade more apparen t when we note tha t aceordino' to eq. (2.7)

E - - c q ~ = m(1--r~-) - ~ = m 7 ,

so tha t eq. (3.13) m a y be wr i t t en

d <T> e (3.14) dt - - m 7 -- - - (T> • ((.url A - ~ - ~ v x V~) .

The ra te of precession is therefore given b y

(3.15) to = ~ ,curl A - y # l v • �9

This precession of the polarizat ion has its origins in two sources, one dyua-

mical the other kinematical . To see this, consider the last t e rm in eq. (3.15).

We wri te

(3.1~I) Y - ~ - ' ( v • Y+IVXVq ~ = y ( v • 7~ I

F r o m eq. (2.12), however, we have

d eV~ = e(v • Clu'l A) - ~ ( m ~ v ) ,

A 'I ' III{], :E-I)IMI,~NSII)NAI, ~VK}{ AI ' Iq~(}XIMATII)N I,'t~R Ti l l , ; I ) IRA( ' I,;QUA'I'ION 1675

so tha t in the last t e r m of eq. (3.16) (evYV~f) may 1)e repla(,ed by

(3.17) e(v • Vq) e[v • (v • curl A)] - - my(v • v ) .

( 'ombinin~' eqs. (3.15), (3.16), and (3.17) we find

(3.1S)

w h o r e

e H' Y~ (vx /~ ) , r - - - - - "

m?, ~p ~ 1

y H ' y c u r l A 1 ~ ~, . . . . . (v.(,url A)v ~- (v : V ~ ) ]

and is the magnet ic field seen in the rest frame. The first t e rm in eq. (3.18),

dynamical in origin~ is the La rmor precession of the spin in the magnet ic

f e ld H ' ; the second term, clearly k inemat ica l in origin (note tha t it depends

explicit ly nei ther on charge nor mass nor external fields) is jus t the Thomas

l)recession. Equa t ion (3.15) for the veloci ty of precession is the same as tha t

obtained by classical method.~.

4. - Rate of change of the longitudinal polarization.

The longitudinal 1)olarization, or helicity, is defined as ~T.n;,, where

F rom eq. (3.11), we have

(4.1) d~ " ~ ( a . n > ~ <a> .n + a ' - ~ .

There are therefore two contr ibut ions to the change in the longitudinal po-

l a r i z a t i o n - t h e ro ta t ion of the spin and the change in or ienta t ion of the t ra-

jectory. F rom (3.4), we obtain

l t is s t ra ight forward to show t h a t

dn (4.3) L , o . - - , ~ = - "

\ dt / E --

((n • (.url A). n>.

E - - eq' eo~ ( ( a • .-~!2 ( [ a • •

11176 M. R()S I':N

so theft we obtain, with the help of eq. (3.12),

(4.4) d o t~tt d - t ( T ' n > = i~ x I ~ ( [ T X ( V 7 •

Thus it is only the influence of the external electric field on the classical tra-

jec tory t ha t causes a change in the longitudinal polarization. Indeed we have

(4.5) (d.) d ( T . n ~ (T> . ~ , , dt

where (dn/dt)o~ is the rate of change of or ientat ion of the tr~tjeetory due to

the electric field alone. I t is interest ing to note tha t in the ext reme relativistic linfit, (.hara(.terized

by ), >>1, the rate of change of the longitudinal polarizat ion becomes very

small s imply because of the increased difficulty in deflecting the t ra jec tory .

.~[oreover, if we let m - + 0 in such a way tha t m y = ] ~ ! remains finite, we

obtain the well-known result tha t , for rn~ssless Dirac particles, the helicity

is conserved.

5. - D irae equat ion w i t h P a u l i term.

I t is well known tha t the electron behaves as though its magnet ic m o m e n t

were larger than t ha t predicted by the l)irac equat ion by a factor (1 +G), where G is a small constant . For not too high energies this m a y be taken into ac-

count by explicit ly a t t r ibu t ing an :momalous intrinsic momen t to the electron and adding an appropria.te relat ivis t ical ly invar ian t t e rm to the Dira(. equat ion

which describes the in teract ion between this momen t and the field. The mo-

dified equat ion is (7)

(5.1) ot . (p - -eA)~ fi m-2~cT ' ( ' u l ' lA 2m ,

We proceed as in Sect. 2 and write

~f = u exp[ iS ] .

where u satisfies the equat ion

ea /3a. curl A + ~/~, ' , 'Vq~ u . (5.2) [a.~ + flm + eq -- E]u= ia 'V + 2n~

(7) H. A. BETIIE and E. F. SALPLTER: Qua~dum .lleehauies o] One-and. Two.Electron Atoms (New York, 1957), p. 50.

A THREE-DIMENSIONAL ~ V K B APFROXIMATION FOR TIlE DIRAC EQUATION 1 6 7 7

As before, we assume the r.h.s, is small relative to the 1.h.s. and define suc-

(.essive orders of approximation as follows:

( a . ~ 4- t im + ~q~ - - E) Uo = 0 ,

( eG l ieG ) (e*.n -.- flm T eq -- E)u~ icx.V ~- )~ fla'('ur A + ~ fl~'Vq~ u,-l .

In the lowest approximation we need consider only the two equations

(=.n + tim + eq~ - - E) Uo = 0 (5.3)

and

leG ] eG fla. curl A + fla'liT~ Uo (5.4) ( ~ . = + f l m ~ - e ~ - - E ) u , = i = ' V + 2 m ~ "

From eq. (5.3), we obtain as before

(5.5) ~'-' + m" - - (E - - e~) 2 = 0 ,

i.e., the classical trajectories are unaffected by the anomalous moment. As

before, therefore, we may again associate • and g/(E--e~) with the mo-

mentum and velocity, respectively, of an associated classical particle and

introduce a time derivative along a classical t ra jectory

(5.6) dQ(r) (~ 'V) dt = (E--eq~) Q( r ) .

Also in the same manner us in Sect. 2, we find us a solvability condition on

eq. (5.4)

l c(; ieGfle,.Vq~}uo_ 0 (5.7) r + fig, i='V 4- ~ f io . cu r lA 4- 2--m "

In t roducing the normalized spinor W=Uo/lUot, we find tha t

. d w (5.8) ~ - = M'w,

dt

where the hermit ian operator M' is given by

(5.0) ieG riot. curl A • f -- eG M' ---- a -cur l f + i[3a " V g - m- ~- ga .cur l A +

c(/ + - - ~a .V~ • m

1 6 7 8 M. ROSEN

We are now in a position to reconsider the precession of the polarization. It is convenient first to separate M' into two terms

c(; (5.10) 3 1 ' = M + ~ M1,

where M~, which describes the effect of the anomalous moment , is given by

(5.1 l) M~ = fla'(Vq. ~:f) - - i f l a " (curl A • - - ga . eu r l A ,

and M, of course, is just the operator obtained in Sect. 2. We have, therefore,

_ / d r ~ d<T> = ~([T.M)>~\~-/ (5.12) (It ' '

r (,

(T> • ]curl A § my [

ieG 7 (v • - - - <[T, M, ] ; ,

where w e have used the results of Sect. 3. .~[aking use of the same te(.hniques

as in the previous Sections, it is s t raightforward to show that

leg ( 5 . 1 3 ) -

Yn

c(; ( [T, M,]> = ~yy (T> x y ( ( . u r l A - - y

y + l v(v.(.url A) (V(/ :>:v)),

(' :r

(5.14) - {T> >~H'. my

H ' again is the magnetic field seen in the rest frame. I~Ience the rate of pre-

(,ession is given by

(5.15) t O - - (1 + ~; )H'- - - - ( v • my 7 + 1

The in terpre ta t ion of the various terms is the same as for eq. (3.18). l~ote

tha t the magnet ic moment is larger by a factor (1 § than t h a t obtained

front the unmodified Dirac equation. The rate of change of the longitudinal polarization is now easily obtained.

We have

d (In ie~ (5.1fi) --(lt(T'n. . . . . . i { ( T , M j > . n ~ {T> dt m ([T, MII)>'n

C(J m em (lIT / (Wr • -f -~y(~T> x H ' ) ' n . - - - - i ,)

A T I I R E E - D I M E N S I O N A L WKB A P P R O X I M A T I O N FOR T I I E D I R A C E Q U A T I O N 1679

Thus the precession of the anomalous moment about the magnet ic field does

contribute to the rate at which tile longitudinal polarization changes. The

contribution arising from tim precession of the nonanomalous nloment was,

as seen in Sect. 4, exactly compensated by the precession of tile classical

t rajectory in the magnet ic field. Expressing eq. (5.16) in terms of the labo-

ra tory fields, we find

d cm c(; ( [ T • • v)J.n> ~<T 'n? ~ <[T•215 'v~-~

The first term is the dominat ing one, the second bein,z of order v"(~/c"- relative

to it.

$ * $

It is a pleasure to acknowledge several impor tant and il luminating sug-

gestions by Professor D. R. YENNIE.

I I I A S S U N T ( ) (*)

Si sviluppa un'approssimazione semielassiea tridimensionale e la si usa per deseri- vere la preeessione della polarizzazione di una partieella di Dirae in un eampo elettro- magnetieo e le manifeste origini dinamiehe e einematiehe di questa preeessione. Si ealeola la veloeit,~ di variazione della polarizzazione longitudinale e si dimostra ehe deriva interamente da quel eainbiamento di orientazione della, traiettoria elassiea dovuta all'influenza del solo eampo elettrieo. L'effetto del momento anomalo si spiega prendendo in eonsiderazione l'equazione di Dirae modifiea~a ineludendovi un termine col momento di Pauli.

(*) T raduz ione a cura del la Redaz ione .

i