Mathematical Methods for Physics and Engineering : A Comprehensive Guide

REPRESENTATION THEORY

Finally, for ozone, which is angular rather than linear, symmetry does not

place such tight constraints. A dipole-moment component parallel to the axis

BB′(figure 29.1(c)) is possible, since there is no symmetry operation that reverses

the component in that direction and at the same time carries the molecule into

an indistinguishable copy of itself. However, a dipole moment perpendicular to

BB′is not possible, since a rotation ofπaboutBB′would both reverse any

such component and carry the ozone molecule into itself – two contradictory

conclusions unless the component is zero.

In summary, symmetry requirements appear in the form that some or all

components of permanent electric dipoles in molecules are forbidden; they do

not show that the other components do exist, only that they may. The greater

the symmetry of the molecule, the tighter the restrictions on potentially non-zero

components of its dipole moment.

In section 23.11 other, more complicated, physical situations will be analysed

using results derived from representation theory. In anticipation of these results,

and since it may help the reader to understand where the developments in the

next nine sections are leading, we make here a broad, powerful, but rather formal,

statement as follows.

If a physical system is such that after the application of particular rotations or

reflections (or a combination of the two) the final system is indistinguishable from

the original system then its behaviour, and hence the functions that describe its

behaviour, must have the corresponding property of invariance when subjected to

the same rotations and reflections.

29.2 Choosing an appropriate formalism

As mentioned in the introduction to this chapter, the elements of a finite group

Gcan berepresentedby matrices; this is done in the following way. A suitable

column matrixu, known as abasis vector,§is chosen and is written in terms of

its componentsui,thebasis functions,asu=(u 1 u 2 ···un)T.Theuimay be of

a variety of natures, e.g. numbers, coordinates, functions or even a set of labels,

though for any one basis vector they will all be of the same kind.

Once chosen, the basis vector can be used to generate ann-dimensionalrep-

resentationof the group as follows. An elementXof the group is selected and

its effect on each basis functionuiis determined. If the action ofXonu 1 is to

produceu′ 1 , etc. then the set of equations

u′i=Xui (29.1)

§This usage of the termbasis vectoris not exactly the same as that introduced in subsection 8.1.1.