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We note a strong resemblance of the interaction part Hint with the hamiltonian of a classical particle in an external field - ev A + eAo, in agreement with the interpretation of ct as a velocity operator. In the Heisenberg representation, an operator (!) satisfies the equation of motion
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d dt (!)(t)
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Thus, here, the position operator r and the gauge-invariant momentum eA satisfy
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1t == P -
dr dt = i[H, r] =
d1t dt
i[H, 1tJ
eat = e(E
(2-65)
+ ct x
THE DIRAC EQUATION
with
oA _ VAo
B = curl A The second equation is the operator version of the Lorentz force equation. In view ofthe paradoxes encountered in the preceding subsection, the interpretation of rand 1t as the position and momentum is, however, limited. To study the physical implications of these equations, we consider their nonrelativistic limit. We write !/J = (;) and use the representation f3 =
_ ~),
~ = (~ ~). Equation (2-64) leads to
.O<p
at =
(1 1tX (1.1t<p
+ eA <p + m<p
(2-66)
. OX
at =
+ eAox -
In the nonrelativistic limit, the large energy m is the driving term in (2-66). We introduce the slowly varying functions of time <I> and X:
= e - imt<l>
X = e-imtX
(2-67)
These spinors satisfy
at = (1' 1tX + eA <l> ax = (1' 1t<l> + eA Ox iat
0<1>
(2-68)
If we assume eAo 2m the second equation is solved approximately as
(1.1t - - <I> <I> 2m
and the first one is the Pauli equation
i 0<1>
[((1' 1t
+ eAoJ<I>
(2-69)
This justifies the use of the terms large and small components for <p and X (or <I> and X respectively). As for Eq. (2-69), it is a generalization to spinors of the Schrodinger equation in an electromagnetic field. After simple algebraic manipulations,
QUANTUM FIELD TH~ORY
we may rewrite it as
0<1> i-=
[(p - eA --u"B+eAO] e
2m 2m
(2-70)
The only spin dependence is through the magnetic interaction u" B. Restoring the factors hand e,
H magn
---u"B= -,u-B 2me
(2-71)
where the magnetic moment Jl is defined as
== ~ hu = 2 (_e
(2-72)
The spin operator S is hu/.2" According to the definition of Sec. 1-1-3, the gyromagnetic ratio g is 2. This is a nontrivial prediction of the Dirac theory, derived within the nonrelativistic context of the Pauli equation. The actual value of IJlI equals
eh IJlI = -2- = 5.79 X me
10- 9 eV/G
Radiative corrections affect the experimentally measured value of g by a tiny amount, which we shall study later.
The Pauli equation (2-70) may be further reduced, if we consider a uniform magnetic field B = curl A, with the choice A = -!B x r, and neglect the quadratic term in A (weak field approximation). We obtain
a<l>
---(L+2S)"B <I>
e 2m
where L = r x p is the orbital angular momentum operator. The reader is invited to derive a complete set of solutions.
The previous study may, in fact, also be performed on the quadratic form of the Dirac equation, i.e., without assuming the nonrelativistic approximation. Starting from (2-62), we mUltiply it by the operator (i9 - e,4. + m). This yields
[(i9 - e,4.)2 - m 2] l/J =
{(i0 [(i0 llV
eA)2
+ ;i u llV [iOll -
eA Il , iO v - eAv] - m2}l/J
eA -
~ u llV Fllv -
m2]l/J = 0
(2-73)
Therefore the spin-dependent term reads
e u -g2.2 Fllv = -g (e) (ioc"E+u"B) 2.
THE DIRAC EQUATION
in the usual representation, and again corresponds to the value g = 2 for the gyromagnetic ratio. Notice that this value is a consequence of the minimal prescription (2-61). We could have written a nonminimal equation:
[ (i~ - ejt) - m I:!g +2
4m a llV Fllv ljJ = 0
(2-74)
which would .lead to g = 2 + I:!g. Such an ,equation has to be used to study the behavior in weak fields of particles with g factors very different from 2.
It is interesting to determine the energy levels in a uniform magnetic field. Assume the field B along
the z axis. The potential vector A may be chosen such that AO = AX = AZ = 0, AY = Bx. For a stationary solution of energy E,
t/I = e- iE, ( ; ) . Eq. (2-66) reads
(E - m)<p = (T' (p - eA)x
+ m)x =
(T'(p - eA)<p
Eliminating X yields an equation for <p:
(E2 - m 2)<p = [(T' (p - eA)Y <p = [(p - eA)2 - e(T' B] <p
+ e 2B 2x 2 -
eB(u z
+ 2xpy)]<p
This is the hamiltonian of an harmonic oscillator. Since py, Pz, and hand side, we seek solutions of the form
<p(x)
commute with the right-
ei(P,y+p,z)f(x)
where f(x) satisfies (2-75) Assuming the sign of B such that eB > 0, we introduce the auxiliary variables
thereby reducing (2-75) to
If f is an eigenvector of 6 z with eigenvalue a =
1,
then fa satisfies
(~ )
for a
0 ) for a f-l
(~22 ~2 )fam =
-(a + a)fam
QUANTUM FIELD THEORY
The solution which vanishes at infinity is expressed in terms of a Hermite polynomial
Hn(~):
provided a + ex = 2n
e-<'/2 Hn(~)
+ 1, n integer, n = 0, 1,2, .... Therefore, the energy levels are given by E2 = m2 + P; + eB(2n + 1 - ex)
(2-76)
and the corresponding wave functions may be easily written. These levels have both a discrete (n, ex = -1; n + 1, ex = 1) and a continuous degeneracy (in Py). The latter may be reduced to a discrete one if we consider a particle in a finite box. Equation (2-76) gives a relativistic generalization of the Landau levels. The value g = 2 is such that the spectrum extends down to E2 = m2. As a second example let us study the case of a Dirac particle in an electromagnetic plane wave, generalizing thereby the classical treatment of Chap. 1. The plane wave, which is supposed to be linearly polarized, is characterized by its propagation vector n" (n 2 = 0) and its polarization vector e" (e 2 = -1, e n = 0). We write A" = e"fW where ~ == n x and a"A, = n"A~ with A~ == e,f'(~). The quadratic equation (2-73) takes the form
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