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+ u' tT + itT (a
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x b) we rewrite
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where tT stand for the Pauli matrices. Using the identity tT a tT b = (a' b)! (1-58) as
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d e (E+iBT U + U - _ t T E-iB) ~u=~ -_ t dr m 2 - - 2
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(1-60)
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from which it follows that !l(r) = exp ( : E
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If 0 stands for the complex three-vector E
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~ iB. tT)!l(o)exp ( :
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~ iB. tT)
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(e/2m)(02)'/2 we find that
er exp [ ~ (E ~ In the frame where !l(O) =! and parameter a is real,
+ iB)' tT ]
~(O)
= cosh (ar)!
o tT + sinh (ar) 2Ti2
= 0 we obtain in the case where E' B = 0, i.e., when the
CLASSICAL THEORY
Figure 1-1 Motion in a constant uniform field. (a) E = O. (b) B = 0, E' Vo =
per)
2 [ cosh (ar)
+ sinh 2 (ar) E2 +-B2] I -02
(0) 0
E + [ 2 sinh (ar) cosh (ar) 2Ti2 + 2 sinh 2 (ar) EXB] iT --2-
~(r) =
+ einh4~ar) 2a
~)(1 + E2 :2 B2)]
+ sinh (2ar) 2a
(0 2)1/2
(1-61)
+ [COSh (2ar) -
1 _E_
2ar E x B] .
In the limiting case where 0 2 = 0, that is, E . B = 0, E2 - B2 = 0, we find
u(r) = r -
+ -2- E
e 2 r2
Note that the velocity grows faster in the E x B direction. If E . B vanishes but E2 - B2 # 0 we can in fact pick a frame where either E or B vanishes. If E = 0 the particle describes a helix with an axis parallel to B, at constant angular velocity w = (eB/m)~ If B vanishes and Vo' E = 0 the trajectory is a "catenary" in the (vo, E) plane with its concavity in the eE direction (this catenary reduces to a parabola in the nonrelativistic limit) (see Fig. 1-1).
2. Gyromagnetic ratio. Thomas precession. Bargmann-Michel-Telegdi equation We introduce the concept of intrinsic magnetic moment and gyromagnetic ratio in this classical picture. This is a subject of controversy if one requires a consistent theory. Suffice it to say that we simply consider here a useful limiting situation of a complete quantum treatment. Recall that an elementary current loop is equivalent to a magnetic moment dp = id~, where i is the current and dT. = 0 d"L stands for a vector normal to the plane of the loop of magnitude equal to the area. When the current is generated by a nonrelativistic orbiting charge, i = ev/2nr: e p=-L 2m
(1-62)
where L = r x p is the orbital angular momentum. Equivalently in a homogeneous external magnetic induction B the interaction part of the Lagrange function can be given the form
Lint
= eA . V = :2 (B x r) . V =
p' B
(1-63)
QUANTUM FIELD THEORY
F:rom the Lorentz force law
dL e - = r x (ev x B) = - L x B
dt 2m
(1-64)
we learn that both the angular momentum and the magnetic moment precess around the magnetic field with the classical Larmor frequency w = eB/2m. In 1926 Uhlenbeck and Goudsmit introduced the idea of electron spin (intrinsic angular momentum) and magnetic moment, with p = g(e/2m)S (I S 1= h/2; in modern language, spin i) and g (reduced gyromagnetic ratio) equal to 2, to reproduce the Zeeman splitting. Unfortunately this same value, g = 2, seemed to produce, for an electron moving in a central potential, a spin-orbit coupling twice as large as the one required by the fine structure of hydrogen. Let us assume that
dS dt
-=pxBrest
(1-65)
holds in the rest frame of the electron. If E and B are the fields in the laboratory frame where the electron has velocity v, then from (1-35) Brest = B - V x E + O(V2). Therefore, in the small velocity approximation the magnetic interaction energy due to spin would be
U'= -po(B-v xE)
If the electric field of an atomic nucleus is taken to derive from a spherical mean potential
r dV eE = - - r dr
(1-66)
(1-67)
the above energy reads
ge U' = - - SoB 2m
2m 2
1 dV SoL - r dr
(1-68)
The ('atch, as first shown by Thomas, comes from the incorrect treatment of Lorentz transformations due to rotational motion. In other words, a pure boost from the laboratory inertial frame leads to a frame rotating with angular velocity WT so that the correct energy must be written
U = U'
-SoWT
(1-69)
This is a typical relativistic effect which can be obtained by considering the product of the two boosts of velocities - v and v + (iv. Here (iv = v Dt. As any Lorentz transformation this product can be factorized into a boost of velocity ~v given by 1 ~v = - - -
fi-=7
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