<< . .

. 94
( : 97)

. . >>

then there is a unitary transformation U which relates the bM s and aM s by eqn (C.86).
The proof of this claim depends on the argument in Section 2.1.2-A showing that a
Hilbert space in which eqn (C.85) holds is spanned by the number states, which we
will now call |n; a , satisfying

a† aM |n; a = nM |n; a , n = (n1 , n2 , . . .) . (C.89)

This argument applies equally well to the bM s, so there is also a basis of states, |n; b ,
b† bM |n; b = nM |n; b . (C.90)

It is easy to check that the operator U , de¬ned by

|n; b n; a| ,
U= (C.91)

is unitary, and that

U aM U † = |m ; b m ; a |aM | m; a m; b|

|m ’ 1M ; b
= mM m; b| , (C.92)

where m ’ 1M signi¬es (m1 , m2 , . . . , mM ’ 1, . . .). Calculating the general matrix el-
ement of U aM U † in the |n; b basis yields

n; b U aM U † n ; b = δn,n ’1M nM = n; b |bM | n ; b ; (C.93)

therefore, this U satis¬es eqn (C.86).
¾ Quantum theory

C.5 Angular momentum in quantum mechanics
In classical mechanics, the angular momentum of a particle (relative to the origin of
coordinates) is r—p, where p is the momentum. In quantum mechanics (Bransden and
Joachain, 1989, Chap. 6) this becomes the operator L = r — (’i ∇), which satis¬es
the angular momentum commutation relations

[Li , Lj ] = i ijk Lk . (C.94)

Because of its relation to the classical angular momentum used to describe orbits, L
is called the orbital angular momentum. This operator is also related to spatial
rotations, r ’ r = R (n, ‘) r, where R (n, ‘) is a 3 — 3 orthogonal matrix (RT R =
RRT = 1), n is a unit vector de¬ning the axis of rotation, and ‘ is the angle of rotation
around the axis. For small ‘ one can show that

δrj = rj ’ rj = δri = ‘ ijk nj rk . (C.95)

By de¬nition, a vector V transforms like r under rotations.
A scalar wave function ψ (r) transforms according to ψ (r) = U (n, ‘) ψ (r), where
the unitary operator U (n, ‘) is given by
U (n, ‘) = exp ’ ‘n · L . (C.96)

Thus L is the generator of spatial rotations.
The corresponding transformation for an operator O is O = U (R) OU † (R). Ex-
panding to ¬rst order for small ‘ gives the in¬nitesimal transformation
δO = O ’ O = ‘ [O, n · L] . (C.97)

Combining eqn (C.95) with eqn (C.97) yields [Li , rj ] = i ijk rk ; therefore every vector
operator V satis¬es
[Li , Vj ] = i ijk Vk . (C.98)
The in¬nitesimal rotation formula for an operator which is a vector ¬eld, V = V (r),
contains additional terms due to the argument r:

[Li , Vj (r)] = i {(r — ∇)i Vj (r) + ijk Vk (r)} . (C.99)

Now let us suppose that L is an operator satisfying eqn (C.98) for any choice of V;
then choosing V = L yields eqn (C.94). Therefore any operator L satisfying eqn (C.98)
for all V is the generator of spatial rotations.
In quantum mechanics, there is another kind of angular momentum, called spin,
which has no classical analogue. Particles (or other systems) with spin are described
by n-tuples of wave functions (ψ1 (r) , . . . , ψn (r)). The basic example is the spin-1/2
particle discussed in Appendix C.1.1-A. In the general case, the Hilbert space is a
tensor product, H = Horbital — Hspin , where the orbital (spatial) and spin degrees of
freedom are represented by Horbital and Hspin respectively. Thus the spatial and spin
degrees of freedom are kinematically independent.
Minimal coupling

Since L acts only on the spatial arguments of the wave functions, i.e. on Horbital ,
it can be expressed in the form L = L — Ispin . The spin angular momentum,
S = Iorbital — S acts only on the internal degrees of freedom, and satis¬es the standard
commutation relations
[Si , Sj ] = i ijk Sk . (C.100)
Since L and S act on di¬erent parts of the product space H they must commute:

[Li , Sj ] = [Li — Ispin , Iorbital — Sj ] = [Li , Iorbital ] — [Ispin , Sj ] = 0 , (C.101)

and the total angular momentum J = L + S satis¬es

[Ji , Jj ] = i ijk Jk . (C.102)

This shows that J is the generator of both spatial and spin rotations. In particular,
vector operators will satisfy
[Ji , Vj ] = i ijk Vk . (C.103)
The decomposition of the total angular momentum into the sum of orbital and spin
parts is only possible when L and S commute, i.e. when the spatial and spin degrees
of freedom are kinematically independent.

C.6 Minimal coupling
In minimal coupling, the standard momentum operator ’i ∇ is replaced by

’i ∇ ’ ’i ∇ ’ qA , (C.104)

where A is the vector potential for an external, classical ¬eld. This notion is usually
presented as the simplest way to guarantee the gauge invariance of the quantum theory
for a charge interacting with an external electromagnetic ¬eld; but there is a simpler
explanation, which only involves classical electrodynamics and the correspondence
principle (Cohen-Tannoudji et al., 1977b, Appendix III.3).
The classical Lagrangian for a point particle with charge q interacting with the
classical ¬eld determined by the scalar potential ¦ and the vector potential A is
r ’ q¦ + q r · A .
™ ™
L= (C.105)
The canonical momentum p conjugate to r is de¬ned by
p= = m™ + qA ,
r (C.106)

so that the kinetic momentum m™ is

m™ = p ’ qA .
r (C.107)

The Hamiltonian is de¬ned as a function of r and p by

H (r, p) = p · r ’ L ,
™ (C.108)
Quantum theory

where eqn (C.107) is used to eliminate r in favor of r and p. This leads to

1 2
(p ’ qA) + q¦ .
H= (C.109)
The transition to quantum theory is now made by the correspondence-principle
replacement, p ’ p = ’i ∇. For transverse ¬elds (∇ · A = 0), the quantum Hamil-
tonian is
1 2
(p ’ qA (r)) + q¦
q 2 A (r)2
p2 q
’ A (r) · p +
= + q¦ . (C.110)
2m m 2m
For many applications the external ¬eld is weak, so the A (r) -term can be neglected
and the Hamiltonian becomes
p2 q
+ q¦ ’ A (r) · p .
H= (C.111)
2m m
In accord with the classical terminology,

p = ’i ∇ (C.112)

is called the canonical momentum operator, and

pkin = p ’ qA (r) (C.113)

is called the kinetic momentum operator. The velocity operator is v = dr/dt,
and the Heisenberg equation of motion for r (i dr/dt = [r, H]) yields

mv = pkin = p ’ qA (r) . (C.114)

Thus the kinetic momentum operator pkin approaches mvclass in the classical limit,
but the canonical momentum operator p is the generator of spatial translations.

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