Solutions to Problems
1.1 The diagram for the product ˆ
C
z
2 ˆ
ı is the same as in Fig. 1.1, except for the
intermediate point P 2 , which should be denoted by a circle instead of a cross,
since it is now below the gray disc. However, the end point P 3 remains the same,
irrespective of the order of the operators. This implies that their commutator
vanishes.
1.2 Represent the rotation of the coordinates by the rotational matrix D as given by
x 2
y 2
= D
x 1
y 1
=
cos α − sin α
sin α cos α
x 1
y 1
x 2 y 2
=
x 1 y 1
D
T =
x 1 y 1
cos α sin α
− sin α cos α
Express the sum x 2
2 + y 2
2 as the scalar product of the coordinate row with the
coordinate column and verify that this scalar product remains invariant under
the matrix transformation.
1.3 In general, the radius does not change if D is orthogonal, i.e., if
D
T × D = I
1.4 Apply the general rule that a displacement of the function corresponds to an opposite coordinate displacement. As a result of the transformation, the function
acquires an additional phase factor:
T a e
ikx = e
ik(x−a) = e
−ika e
ikx
1.5 The action of a rotation about the z-axis can be expressed by a differential
operator as
ˆ
O(α) = cos α
x
∂
∂x
+ y
∂
∂y
+ sin α
y
∂
∂x
− x
∂
∂y
The unit element corresponds to α = 0, and hence,
ˆ
E = ˆ
O(0) =
x
∂
∂x
+ y
∂
∂y
A.J. Ceulemans, Group Theory Applied to Chemistry, Theoretical Chemistry and
Computational Modelling, DOI 10.1007/978-94-007-6863-5,
© Springer Science+Business Media Dordrecht 2013
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