10. Appendix

⎧ ⎫
2apple
⎪⎨ cos (y z) ⎪⎬
a
⎪⎩
2apple
i sin (y z)
⎪⎭
a
1
2
|011〉 |011〉
|011〉 |011〉
Partial solution to Problem 2.8 587
(2.8.2)
⎧ ⎫
2apple
⎪⎨ cos (y z) ⎪⎬
a
⎪⎩
2apple
i sin (y z)
⎪⎭
a
1
|011〉 |011〉
(2.8.1)
2 |011〉 |011〉
We shall denote these six symmetrized wave functions as:
2apple
æ1 ∼ cos (y z)
a
2apple
æ2 ∼ sin (y z)
a
2apple
2apple
æ3 ∼ sin (y z) cos (y z)
a
a
(2.8.4)
2apple
2apple
æ4 ∼ sin (y z) cos (y z)
a
a
2apple
2apple
æ5 ∼ cos x sin x
a
a
2apple
2apple
æ6 ∼ cos x sin x
a
a
To determine their irreducible representations we shall test their symmetry
by applying the 2-fold rotation C2 4 (y′ ) to all these 6 wave functions. First we
have to find out the effect of this operation on the point (x, y, z) inO. When
refered to O ′ this point has the coordinate: x ′ x (a/8); y ′ y (a/8) and
z ′ z (a/8). We may also say that the operation T transforms (x, y, z) to
(x ′ , y ′ , z ′ ). The 2-fold rotation then transforms (x ′ , y ′ , z ′ ) into (x ′ , y ′ , z ′ ).
Finaly T1 transforms this point into x ′ a
8 , y′ a
8 , z′ a
a
8 x 4 ,
y, z a
2
4 . Thus the net effect of the operation C4 (y ′ ) on the crystal can be
represented by the change:
(x, y, z) → x a a
, y, z . (2.8.5)
4 4
The effect of this operation on the six wavefunctions æ1 to æ6 can now be
shown to be:
æ1 ∼ cos(2apple/a)(y z) → cos(2apple/a)(y z a/4)
sin(2apple/a)(y z) æ2
æ2 ∼ sin(2apple/a)(y z) → sin(2apple/a)(y z a/4)
cos(2apple/a)(y z) æ1
(2.8.6a)
(2.8.6b)