Mathematical Methods for Physics and Engineering : A Comprehensive Guide

29.11 PHYSICAL APPLICATIONS OF GROUP THEORY

However, we have so far allowedxi,yito be completely general, and we must now identify
and remove those irreps that do not correspond to vibrations. These will be the irreps
corresponding to bodily translations of the triangle and to its rotation without relative
motion of the three masses.
Bodily translations are linear motions of the centre of mass, which has coordinates

x=(x 1 +x 2 +x 3 )/3andy=(y 1 +y 2 +y 3 )/3).

Table 29.1 shows that such a coordinate pair (x, y) transforms according to the two-
dimensional irrep E; this accounts for one of the two such irreps found in the natural
representation.
It can be shown that, as stated in table 29.1, planar bodily rotations of the triangle –
rotations about thez-axis, denoted byRz– transform as irrep A 2. Thus, when the linear
motions of the centre of mass, and pure rotation about it, are removed from our reduced
representation, we are left with E⊕A 1. So, E and A 1 must be the irreps corresponding to the
internal vibrations of the triangle – one doubly degenerate mode and one non-degenerate
mode.
The physical interpretation of this is that two of the normal modes of the system have
the same frequency and one normal mode has a different frequency (barring accidental
coincidences for other reasons). It may be noted that in quantum mechanics the energy
quantum of a normal mode is proportional to its frequency.

In general, group theory does not tell us what the frequencies are, since it is

entirely concerned with the symmetry of the system and not with the values of

masses and spring constants. However, using this type of reasoning, the results

from representation theory can be used to predict the degeneracies of atomic

energy levels and, given a perturbation whose Hamiltonian (energy operator) has

some degree of symmetry, the extent to which the perturbation will resolve the

degeneracy. Some of these ideas are explored a little further in the next section

and in the exercises.

29.11.4 Breaking of degeneracies

If a physical system has a high degree of symmetry, invariant under a groupGof

reflections and rotations, say, then, as implied above, it will normally be the case

that some of its eigenvalues (of energy, frequency, angular momentum etc.) are

degenerate. However, if a perturbation that is invariant only under the operations

of the elements of a smaller symmetry group (a subgroup ofG) is added, some of

the original degeneracies may be broken. The results derived from representation

theory can be used to decide the extent of the degeneracy-breaking.

The normal procedure is to use anN-dimensional basis vector, consisting of the

Ndegenerate eigenfunctions, to generate anN-dimensional representation of the

symmetry group of the perturbation. This representation is then decomposed into

irreps. In general, eigenfunctions that transform according to different irreps no

longer share the same frequency of vibration. We illustrate this with the following

example.