Spherical Symmetry and Spins - Group Theory Applied to Chemistry - page 177

This result identifies the X operator as a twofold rotation in a plane perpendicular to
the z -direction. Hence, a twofold rotation axis, C 2 , and its double group extension
ℵ C 2 belong to the same class only if the group contains additional binary elements
in a plane perpendicular to this axis. Otherwise, the first rule will apply. The excep-
tion referred to in this theorem is illustrated by the D 2 class structure as shown in

Table 7.4 .
So far, we have been concerned only with proper rotations because in the spinor
basis improper rotations are not defined. Since the electron spin is associated with
an internal “spinning” of the electron around its axis, the electron spin is assigned an
intrinsic positive parity. As we have seen before in Sect. 3.8 , every improper rotation
can be written as the product of a proper rotation and an inversion; therefore, when-
ever an improper rotation acts on a spinor, we simply take the matrix representation
for the proper factor in this improper rotation.

7.6 Kramers Degeneracy

In 1930, Kramers showed that in the presence of an arbitrary electrostatic field all
states with an odd number of spins must still have even degeneracy. 3

Theorem 18 The energy levels of a system that contains an odd number of spin- 2
particles are at least doubly degenerate in the absence of an external magnetic field .

This theorem reflects the influence of time-reversal symmetry. Already in
Chap. 2 , we showed that the time-reversal operator is an anti-linear operator. It
will turn any spatial wavefunction into its complex conjugate. Applying it twice
in succession will return the original wavefunction, and, hence, we may write for
spatial functions:

ϑ 2 vectors : =+ 1 (7.51)

However, the time-reversal properties of a spinor are different. Here we can use the
same argument as in Sect. 7.4 ,viz.inEq.( 7.38 ), by requiring that the time reversal
of the spinor components would lead to complex conjugation of their transformation
properties. This implies that time reversal must turn

| α into | β , and vice versa, but

with a sign difference:

ϑ | α =| β

(7.52)

ϑ | β =−| α

Hence, for spinors, applying time reversal twice leads to a sign change:

ϑ 2 spinors : =− 1 (7.53)

3 Adapted from [ 10 ].