Mathematical Methods for Physics and Engineering - Matematica.NET

REPRESENTATION THEORYFinally, for ozone, which is angular rather than linear, symmetry does notplace such tight constraints. A dipole-moment component parallel to the axisBB ′ (figure 29.1(c)) is possible, since there is no symmetry operation that reversesthe component in that direction and at the same time carries the molecule intoan indistinguishable copy of itself. However, a dipole moment perpendicular toBB ′ is not possible, since a rotation of π about BB ′ would both reverse anysuch component and carry the ozone molecule into itself – two contradictoryconclusions unless the component is zero.In summary, symmetry requirements appear in the form that some or allcomponents of permanent electric dipoles in molecules are forbidden; they donot show that the other components do exist, only that they may. The greaterthe symmetry of the molecule, the tighter the restrictions on potentially non-zerocomponents of its dipole moment.In section 23.11 other, more complicated, physical situations will be analysedusing results derived from representation theory. In anticipation of these results,and since it may help the reader to understand where the developments in thenext 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 orreflections (or a combination of the two) the final system is indistinguishable fromthe original system then its behaviour, and hence the functions that describe itsbehaviour, must have the corresponding property of invariance when subjected tothe same rotations and reflections.29.2 Choosing an appropriate formalismAs mentioned in the introduction to this chapter, the elements of a finite groupG can be represented by matrices; this is done in the following way. A suitablecolumn matrix u, known as a basis vector, § is chosen and is written in terms ofits components u i ,thebasis functions, asu =(u 1 u 2 ··· u n ) T .Theu i may be ofa 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 an n-dimensional representationof the group as follows. An element X of the group is selected andits effect on each basis function u i is determined. If the action of X on u 1 is toproduce u ′ 1 , etc. then the set of equationsu ′ i = Xu i (29.1)§ This usage of the term basis vector is not exactly the same as that introduced in subsection 8.1.1.1078