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L 1dV ~ 2m2; dr
ge dV U=--SoB+ (g - 2 o L - --S 2m 2m r dr
1) 1
(1-70)
If g = 2 this effect indeed reduces by half the spin-orbit coupling in agreement with observation. Dirac's theory (Chap. 2) leads naturally to this g value for an electron or muon. But of course other
CLASSICAL THEORY
spin i particles, such as the proton (g = 5.59) or the neutron (g = - 3.83), have very different g values due to internal structure. Let us return to the motion of a spinning particle in a constant magnetic field at small velocities. The velocity precesses with an angular frequency eB/m while for spin the corresponding frequency is geB/2m. In one period the relative phase will therefore be 2n(g/2 - l)eB/m. Bargmann, Michel, and Telegdi have obtained a relativistic description of this classical motion of spin valid in slowly varying external fields. We represent the spin degrees of freedom by a three-vector S in the rest frame of the particle. In covariant notations it is therefore described by a space-like four-vector S" orthogonal to the velocity u". We want to generalize the equation dS/dt = g(e/2m)S x B valid in the rest frame where S" = (O,S). To preserve the condition S' U = 0, we should haveS' U + S U = ousing dots for derivatives with respect to proper time. In the instantaneous rest frame, U = (1,0); therefore SO = S Ii and S" = [S' iI, (ge/2m)S x B]. To express this in any frame, we observe that F",S' reduces to (E' S, S x B) in the rest frame. Therefore the required equation is
S" =
P,S'
(~- 1)U"(S.F.PUP)
(1-71)
From (1-71) it is clear that if g = 2, Sand u move rigidly. This is not the case when g # 2. Thus we derive a method to measure the particle's magnetic anomaly g/2 - 1. Define in the laboratory plane with time axis f = (1,0) the four-vector L with L along u and L u = 0, that is, L = (u cosh <p - i)/sinh <p, cosh <p = U i = uO. Furthermore, let M be a unit fourvector in the (S, L) plane, orthogonal to u and L. We set S = L cos (J + M sin (J (see Fig. 1-2). Let us find from (1-71) the equation of motion for (J:
S = 8( -
L sin (J
+ M cos (J) + i
+ M sin
ge = - - (F L cos (J 2m
. . + F M sm (J) + -e ( g1 u(L' F Ucos (J + M . F u sm (J) -- )
= U'
(1-72)
Taking the scalar product with M and using L . M
0 we find
-8 +M i = f!...!... M'F.L
Now M
i = coth <pU' M = (e/m) coth <pM F' u, leading to
(1-73)
Figure 1-2 (a) Rest frame of a particle with spin S. (b) Motion in a pure magnetic field.
18 QUANTUM
FIELD THEORY
or explicitly
~ [(If. sinh 2 <p m 2
cosh 2 <p)
smh <p
~. M + (If. 2
coth <p(M x B)'
(1-74)
In the case of a pure magnetic field with the particle orbiting on a circle at the (proper) Larmor frequency w = eB/m, the mixed product (u x M B) equals sinh <pB. Thus
8 - 80 For a period
~, =
(~ (~ -
)w cosh
cosh <p
2n/w,
~8lperiod = (~ 3. Motion in a plane wave
1)2n cosh <p
(~ -
1)2n
(1-75)
Finally, we study the motion of a charged spinless particle in the electromagnetic field of a plane wave, which we assume linearly polarized for simplicity. The wave is characterized by its light-like propagation vector n" and polarization 8". These two vectors are such that
n2 =
n= 0
(1-76) = n' x:
(1-77)
The potential depends on an arbitrary function of the variable
A"(x)
= 8"/(~)
From (1-76) it follows that
a A =
0, while the field
F"V(x) = (n"8 VnV8")f'(~)
(1-78)
satisfies
F"lF 'v = F",F'v = n"n1'@2 F"vF 'v = 0
Therefore IE I = IB Iand E . B = O. SiI:1ce n"F"v = n.F"V = 0, the classical Lorentz equation
eFllvuv
(1-79)
leads to
= constant. Choosing coordinates such that at , = 0, x(O) = 0,
= u(O) 1',
The variable
can be used in place of, in (1-79) in such a way that
du" = -e ~ d~ m
f'(~)
U n"~-
n' u(O)
Multiplying by 8 and integrating gives
u . 8 = u(O)
+ - [J@ -
1(0)]
which can be inserted in
du"/d~,
leading to
CLASSICAL THEORY 19
du" _ ~
I (~) "
{n e' u(O) + (e/m)[J(~) - 1(0)] _ e } n' u(O) "
(1-80)
A last integration gives u@ as
u"(~) = u"(O)
+ -e
[J(~) - 1(0)] n" ~-) - e"
e' u(O) n . u(O
e + ~2 2m
[J(~) - 1(0)y - -
n" n . u(O)
(1-81)
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