Mathematical Methods for Physics and Engineering - Matematica.NET

29.11 PHYSICAL APPLICATIONS OF GROUP THEORYHowever, we have so far allowed x i , y i to be completely general, and we must now identifyand remove those irreps that do not correspond to vibrations. These will be the irrepscorresponding to bodily translations of the triangle and to its rotation without relativemotion of the three masses.Bodily translations are linear motions of the centre of mass, which has coordinatesx =(x 1 + x 2 + x 3 )/3 and y =(y 1 + y 2 + y 3 )/3).Table 29.1 shows that such a coordinate pair (x, y) transforms according to the twodimensionalirrep E; this accounts for one of the two such irreps found in the naturalrepresentation.It can be shown that, as stated in table 29.1, planar bodily rotations of the triangle –rotations about the z-axis, denoted by R z – transform as irrep A 2 . Thus, when the linearmotions of the centre of mass, and pure rotation about it, are removed from our reducedrepresentation, we are left with E⊕A 1 . So, E and A 1 must be the irreps corresponding to theinternal vibrations of the triangle – one doubly degenerate mode and one non-degeneratemode.The physical interpretation of this is that two of the normal modes of the system havethe same frequency and one normal mode has a different frequency (barring accidentalcoincidences for other reasons). It may be noted that in quantum mechanics the energyquantum of a normal mode is proportional to its frequency. ◭In general, group theory does not tell us what the frequencies are, since it isentirely concerned with the symmetry of the system and not with the values ofmasses and spring constants. However, using this type of reasoning, the resultsfrom representation theory can be used to predict the degeneracies of atomicenergy levels and, given a perturbation whose Hamiltonian (energy operator) hassome degree of symmetry, the extent to which the perturbation will resolve thedegeneracy. Some of these ideas are explored a little further in the next sectionand in the exercises.29.11.4 Breaking of degeneraciesIf a physical system has a high degree of symmetry, invariant under a group G ofreflections and rotations, say, then, as implied above, it will normally be the casethat some of its eigenvalues (of energy, frequency, angular momentum etc.) aredegenerate. However, if a perturbation that is invariant only under the operationsof the elements of a smaller symmetry group (a subgroup of G) is added, some ofthe original degeneracies may be broken. The results derived from representationtheory can be used to decide the extent of the degeneracy-breaking.The normal procedure is to use an N-dimensional basis vector, consisting of theN degenerate eigenfunctions, to generate an N-dimensional representation of thesymmetry group of the perturbation. This representation is then decomposed intoirreps. In general, eigenfunctions that transform according to different irreps nolonger share the same frequency of vibration. We illustrate this with the followingexample.1111