The Dirac equation and the Lamb shift

Nuclear Physics B (Proc. Suppl.) 6 (1989) 251-254 North-Holland, Amsterdam

251

TIE DIRAC EQUATION AND T B LAB S~IFT

T.L. Gill and V.V. Zachary Department of Electrical Engineering, Howard University, Vashington, DC 20059

1.

I~TIOUUCTION

El = [V + mc2]t + c(~ • r)~o

A standard assertion is that quantum electrodynamics (qED) is an almost perfect theory which is in excellent agreement with experiments. Those with some (intimate) contact with the subject will also point out that they are not happy with the renormalization procedures and that a meaningful mathematical formulation is s t i l l required. A closer analysis reveals that qED

(1) Eio = [V- mc2]~o + c(~ • r)9.

The notation is standard (see Bethe and Salpeter2). By eliminating ~ in terms of ! and vice versa in (1) we obtain the following equations:

does not account for the complete spectrum of

Et = (V + mc2)t + c2(E + me 2 - V) -2

hydrogen. All but the Lamb s h i f t (and a major portion of hyperfine s p l i t t i n g ) must be

[(~. p)v](~ • ~)t + c2(E + mc2- v)-I

computed from the Pauli approximation to the Dirac equation and given as input. Historically, when Lamb and Retherford 1

(~ • !)2t,

confirmed suspicions that the 2Sl/2 state of

and

(2)

E@ = (V - mc2)@ + c2(E- mc2 - V)

hydrogen was shifted relative to the 2Pl/2 state, the Pauli approximation vas used to decide that the Dirac equation was not s u f f i c i e n t . In this paper we show that the Pauli equation does not contain all the essential information of the f u l l , completely separated Dirac equation. Ve point out that the term which is quadratic in the vector potential is small in a l l but s - s t a t e s , vhere it diverges. A simplistic analysis shows that this term can be used to account for the Lamb s h i f t . II.

SEPALATIONOF TIE EIGENVALUEP|OBLE|

Let us s t a r t d i r e c t l y with the stationary hydrogen-atom case for the Dirac equation in two-component form. Ve have: 0920-5632/89/$03.50 © Elsevier SciencePublishers B.V. (North-Holland PhysicsPublishing Division)

[(~ • p)V] (~ • ~)@ + c2(E- mc 2- V) -I

(~. !)2v.

(3)

Ve call Eqs. (2) and (3) the Slater equations. They were f i r s t discovered by Slater 3 (in a different form) at least as early as 1940, since one of his students used them to study cu+. Let us assume that V = -~ca/r, = ~ix r / r 3 Ca = e2/~c); then

(~. pv)Cf. !) = (pV) . , + i ~ . =(~v.~)-~(~v.~)+i~. - 1.e~ .

(pV.~).

(pVx !) (~vx~)

72L. Gill, W.W. Zachary / The Dirac equation attd the Lamb shift

252 Simple c a l c u l a t i o n y i e l d s

= (E - mc2

V)~,

(6)

and fi2dV a pV • p = - - - d ~ - f f ' pV • A = 0

rc~(~, r)2~

2, r ,2 + c ~-l~)

and pV

A "

=

~

tdVx2

e

t~]~

~(~I " {)) 2 r

(~l-

-ifi2dV

pV ~ p -

afic(1

[---~--'~ " r (~

- where fiL = r ~ p,

r) r2 L-

" ~)(~I

-

(1

r,-2 r

--~--~-~ c T r +

f " -~I

r)

2

-)]~:(E+mc

2-

V)~.

(7)

r

so t h a t

Let us f u r t h e r reduce the above e q u a t i o n s (o" • . .

PV)(o

.

• x)

.

= -1i 2 dV a

to t h e i r most elementary form so as to expose

~

a d d i t i o n a l e f f e c t s t h a t may appear because of ii 2 dV

"titdV~ 2

~(_~ • L) _ + -~J

+ ~

e (¢ • ~i) ~(

the exact s e p a r a t i o n .

Since ({ • ~)2 = 2

e

(~. -

(e'fi/c)~ • B, where B = V x A, u s i n g our

~)(e~ "

2

'f))"

(4)

r

Let us set mc2

+ E

mc

al = ~ - - '

2

-

identities,

E

a2 = ~ c

d e f i n i t i o n of h and some s t a n d a r d v e c t o r we have

'

B = ( e l " ~) V(1/r) + 4 r# 1 ~(r),

r l = --al' r2 = ~22 "

This leads (note t h a t a 1 z 2mc2#fic r I z r e / 2 = rp, r e = e2/mc 2) to

(E + mc 2

V) -1 = & ( 1

r__)-i + rl ,

(E - mc2

V) -1 = ~ a ( 1

- r__)-i r2

where 6(r) i s the n i r a c d e l t a f u n c t i o n .

B

may be f u r t h e r decomposed i n t o a symmetric and a d i p o l e p a r t (cf. Bethe and S a l p e t e r 2, p. I08), (5) 3

R e t u r n i n g to Eqs. (2) and (3), we use (4) and

~

r5

"

(5) to o b t a i n We may now w r i t e (S = ¢/2; fiS i s the spin rc2(¢ . r)2~

+c2(~)2~

operator) (a • r) 2

~2e2

r (a - r ) ( # i

(~" ~I

(I+ r___)-2 r1

a~c(1 + r ) r1 ~2e2 EL2~ a

:

: 2r

_

_

. (

3(s. -

-~e3 cr • r!

=

-

2efirSr/s

--~-L--~

r)(u. r) - ~-~r

)]~

r2

" #I)6(r)

~__~_S)]

.

(S)

253

T.L. Gill, W.W. Zachary / The Dirac equation and the Lamb shift

Ve now rearrange Eqs. (6) and (7) using (8) 2 ~

r

C1 + r_) ~ t

2r#o/~p + (l + r__)

rI

rI

{ 3(s. r)CT. _~) I S},

(_~• v.)(~

is.

4m2c 2 ÷

IL)2 {~2

_- 3I)6( ) [3(S-Ir)(I

r

_ r4

: [E- mc 2 -

=

Vii

!

(9)

with a s i m i l a r equation f o r ~. Ve have used e~ ~0 = ~ ' ~I = #p~' w h e r e ~ I is the proton me spin operator and #p = gp#O(~p ) with gp the proton g - f a c t o r (~5.585). Our main focus is the ! - equation (9). However, before f u r t h e r a n a l y s i s , l e t us return to the S l a t e r equations. I f we replace E + mc2 - V by 2mc2, we get: 2

v]t : ~

[E- mc 2 [ ÷

+

2/~0~p / ~ ( S

r5

• I)6(_r)

-

r)

+[

rI

+ 2#0/~p -

+ ~m9

fs,(s •

,H2 S • L m

2

• .~)t

[E- mv2- V]9 :

r 5-

(l÷

no other information in the Dirac equation. Equation (9) makes i t clear t h a t t h i s may not be the case. In f a c t , a b e t t e r but s t i l l approximate equation could be obtained by using equations (2)-(5) to get:

t.

This is the Pauli equation. It gives all the results of the Schrodinger equation in addition to the spin interaction terms. In general, it has been believed that there is

(I " S')]}9 (10)

--7-rJ

2 r r I f we set r(1- - -+- - - ~ and r)2 rl (1 + r l

equal to

unity, then eq. (9) becomes i d e n t i c a l to (10). In doing t h i s , however, we lose a l l knowledge about p r o p e r t i e s near the origin (s-states). H i s t o r i c a l l y i t is quite i n t e r e s t i n g to note t h a t S l a t e r 3 used the term (see (9))

r1 toaccount for the hyperfine splitting in hydrogen. He did not include the Dirac-delta term. Bethe and Salpeter 2 "squared" the Dirac equation and used the delta term to account for hyperfine splitting. This equation is now known as the Feynman-Gell-Mann equation. 4 In both books cited above, it is assumed that the term quadratic in the vector potential is small. To see that this is not the case, expand 2 to get:

~r2

p2 =

(I • L) + 4m#o/~p

+ 2mrl#2pI2 sin20

7"

254

T.L. Gill, W.W. Zachary / The Dirac equation and the Lamb shift

I t i s c l e a r t h a t the q u a d r a t i c term i s highly s i n g u l a r .

In f a c t , a simple

In c l o s i n g , we note t h a t t h e r e are systems (He4) where the expected value E[I] = O.In

c a l c u l a t i o n shows t h a t i t diverges in a l l s-states. states.

I t i s also very small in a l l o t h e r I f we use a c u t - o f f of 3.6r 1 with

s - s t a t e s the e l e c t r o n i s so c l o s e to the He4 nucleus t h a t i t i s not u n r e a l i s t i c to assume t h a t the e l e c t r o n sees the i n d i v i d u a l

the standard s o l u t i o n of the Schrodinger equation ( f o r hydrogen) we get a s h i f t of 2Sl/2_ r e l a t i v e to 2Pl/2 of 1057.841 EHz.

p a r t i c l e s in the nucleus. I t can be shown ~xr t h a t A = -~sin2~ - - 3 - - (where ~ i s a r

This i s c e r t a i n l y ad hoc and could be looked

constant) has the p r o p e r t i e s E[A] = O,

upon as u n j u s t i f i e d from a mathematical point of view.

On the o t h e r hand, t h i s approach i s

l e s s o f f e n s i v e (from a mathematical p o i n t of view) than using the Pauli equation to decide t h a t the Dirac equation does not account f o r the Lamb s h i f t .

CONCLUSION In l i g h t of the tremendous success of

ElY x A] : O, and E[p • A] : O, while E(A~ 2) = ~.

A has the p r o p e r t y t h a t i t i s a

3~ 7~ (minimum) maximum at ~ = (¼) , --4, t--4j,-~ . This observation further enhances the possibility that a simple eigenvalue mechanism in the Dirac equation accounts for the Lamb shift.

eigenvalue a n a l y s i s in chemistry, e n g i n e e r i n g , and p h y s i c s , i t i s s u r p r i s i n g t h a t no attempts have been made to completely

gEFgUlCgS

i n v e s t i g a t e the spectrum of hydrogen as due to an eigenvalue problem f o r some equations.

1.

N.E. Lamb, Jr. and R.C. Retherford, Phys. Rev. 72 (1947); W.E. Lamb, Jr., Rep. Prog. Phys. 14, 19 (1951).

2.

H.A. Bethe and E.A. Salpeter, quantum lechanics of One-and Two-Electron Atoms (Academic, New York, 1957).

3.

J.C. Slater, Quantum Theory of Atomic Structure, Vol. II McGraw-Hi11, New York, 1960); A.O. i11iams Phys. Rev. 58, 723 (1940).

The s t r o n g l y s i n g u l a r nature of the q u a d r a t i c term means t h a t i t i s improper both mathematically and p h y s i c a l l y to t r e a t the s p i n - r e l a t e d terms as a p e r t u r b a t i o n of the A = 0 s o l u t i o n s to the Dirac equation.

This

means t h a t the fundamental question of the

I

r e l a t i o n of the Dirac equation to the Lamb shift is still

open.

The complete ( c a r e f u l )

a n a l y s i s of (9) is c u r r e n t l y under way and the r e s u l t s ~ i l l be r e p o r t e d l a t e r .

4.

R.P. Feynman and M. G e l l - l a n n , Phys. Rev. 109, 193 (1958).