10. Appendix

Partial solution to Problem 2.12 593
sion operator with eignenvalues: 1. Hence, the double group character of the
improper rotations such as S4 can be obtained by first calculating the character
for the corresponding proper rotation. In the book it was pointed out in 2.6.2
that for some symmetry operations Ci both Ci and ÊCi belong to the same
class. It can be shown [see H. Jones: The Theory of Brillouin Zones and Electronic
States in Crystals, p. 251] that this is not true if the trace of the spin operator
corresponding to Ci (i.e. Tr[O(Ci)]) is non-zero. In case Tr[O(Ci)] 0
then Ci and ÊCi will belong to the same class if Ci contains a 2-fold rotation
about an axis perpendicular to the axis n.
Some references on the effect of symmetry operations on spin wave functions:
E. Merzbacher: Quantum Mechanics. (John Wiley & Sons, Inc. New York, 1961).
M. Tinkham: Group Theory and Quantum Mechanics. (McGraw-Hill, New York, 1964).
V. Heine: Group Theory in Quantum Mechanics. (Dover, New York, 1993).
Solution to Problem 2.12
(a) The operator p has three components px, py and pz which transform under
the symmetry operations of Td like the components of a vector x, y, and
z. Similarly the °4 conduction and valence bands of the zincblende-type semiconductor
are three-dimensional whose basis functions transform under the
symmetry operations of Td like the components of a vector x, y, and z. We
shall denote the valence band basis functions as |X〉, |Y〉 and |Z〉 respectively.
The corresponding conduction band basis functions will be denoted by °4c(x),
°4c(y) and °4c(z), respectively. Under a C2 or 180◦ rotation around say the
x-axis the coordinate transformation can be represented as: xyz ⇒ xyz. Thus,
under the C2 operation about the x-axis, any terms of the form xxy, xxz and
yyy etc., which contain only an odd number of y or z will change sign and
therefore vanish. The same will be true for terms of the form xyy, xxx etc
which contain an odd number of x. Similarly all matrix elements of the form
〈X|px|°4c(x)〉, 〈Y|px|°4c(x)〉 etc., can be shown to vanish under appropriate C2
rotations.
(b) In the case of matrix elements of the form 〈X|py|°4c(z)〉 which do not
vanish since they do not transform like the matrix elements considered in (a),
we can show that they transform into each other under the three-fold rotations
C3. For example, a 120◦ rotation about the [111] axis will result in the
following coordinate transformation: xyz ⇒ yzx and its effect on the matrix
element 〈X|py|°4c(z)〉 would be to transform it into 〈Y|pz|°4c(x)〉. Since the
crystal is invariant under this C3 rotation we conclude that: 〈X|py|°4c(z)〉
〈Y|pz|°4c(x)〉. The other cyclic permutations of 〈X|py|°4c(z)〉 can also be
shown to be equal by using C3 rotations about the other equivalent [111] axes.
(c) This remaining problem is left as an exercise.