Subatomic Physics

8.2. Symmetry Breaking by a Magnetic Field 223
of F and find the eigenfunctions and eigenvalues. This procedure is not necessary
because F is an old friend. Equation (5.3) shows that
F = − Lz
. (8.6)
�
Not unexpectedly, F is proportional to the z component of the orbital angular
momentum. Invariance of a system under rotation around the z axis leads to conservation
of F and thus also of Lz.
Two generalizations are physically reasonable, and we give them without proof:
(1) If the system has a total angular moment J (spin plus orbital), then Lz is
replaced by Jz. (2)U for a rotation by an angle δ around the arbitrary direction ˆn
(where ˆn is a unit vector) is
�
−iδ ˆn · J
Un(δ) =exp
�
�
. (8.7)
If the system is invariant under rotation about ˆn, the Hamiltonian will commute
with Un and consequently also with ˆn · J:
[H, Un] =0−→ [H, ˆn · J] =0. (8.8)
The component of the angular momentum along ˆn is conserved. If ˆn can be taken
to be any direction, all components of J are conserved, and J is a constant of the
motion.
With Eq. (8.7) it is straightforward to find the commutation relations for the
components of J:
[Jx,Jy] = i�Jz, (5.6)
cyclic.
The steps in the derivation are outlined in Problem 8.1 The commutation relations
(Eq. (5.6)) are a consequence of the unitary transformation (Eq. (8.7)), which in
turn is a consequence of the invariance of H under rotation.
8.2 Symmetry Breaking by a Magnetic Field
AparticlewithspinJ and magnetic moment µ can be described by a Hamiltonian
H = H0 + Hmag, (8.9)
where Hmag is given in Eq. (5.20). Usually, H0 is isotropic, and the system described
by H0 is invariant under rotations about any direction. This fact is expressed by
[H0, J] =0. (8.10)
The energy of the particle is independent of its orientation in space. If a magnetic
field is switched on, the symmetry is broken, and Eq. (8.10) no longer holds:
[H, J] =[H0 + Hmag, J] �= 0. (8.11)