Mathematical Methods for Physics and Engineering - Matematica.NET

REPRESENTATION THEORYthe corresponding coordinate and 2 cos θ for the two orthogonal coordinates.If the rotation is followed by an inversion then these entries are multipliedby −1. Atoms not transforming into themselves give a zero diagonal contribution.Show that the characters of the natural representation are 12, 0, 0,0, 2 and hence that its expression in terms of irreps isA 1 ⊕ E ⊕ T 1 ⊕ 2T 2 .(b) The irreps of the bodily translational and rotational motions are included inthis expression and need to be identified and removed. Show that when thisis done it can be concluded that there are three different internal vibrationfrequencies in the CH 4 molecule. State their degeneracies and check thatthey are consistent with the expected number of normal coordinates neededto describe the internal motions of the molecule.29.10 Investigate the properties of an alternating group and construct its charactertable as follows.(a) The set of even permutations of four objects (a proper subgroup of S 4 )is known as the alternating group A 4 . List its twelve members using cyclenotation.(b) Assume that all permutations with the same cycle structure belong to thesame conjugacy class. Show that this leads to a contradiction, and hencedemonstrates that, even if two permutations have the same cycle structure,they do not necessarily belong to the same class.(c) By evaluating the productsp 1 = (123)(4) • (12)(34) • (132)(4) and p 2 = (132)(4) • (12)(34) • (123)(4)deduce that the three elements of A 4 with structure of the form (12)(34)belong to the same class.(d) By evaluating products of the form (1α)(βγ) • (123)(4) • (1α)(βγ), where α, β, γare various combinations of 2, 3, 4, show that the class to which (123)(4)belongs contains at least four members. Show the same for (124)(3).(e) By combining results (b), (c) and (d) deduce that A 4 has exactly four classes,and determine the dimensions of its irreps.(f) Using the orthogonality properties of characters and noting that elements ofthe form (124)(3) have order 3, find the character table for A 4 .29.11 Use the results of exercise 28.23 to find the character table for the dihedral groupD 5 , the symmetry group of a regular pentagon.29.12 Demonstrate that equation (29.24) does, indeed, generate a set of vectors transformingaccording to an irrep λ, by sketching and superposing drawings of anequilateral triangle of springs and masses, based on that shown in figure 29.5.CCCA B AB 30 ◦ AB30 ◦ (a) (b) (c)Figure 29.7 The three normal vibration modes of the equilateral array. Mode(a) is known as the ‘breathing mode’. Modes (b) and (c) transform accordingto irrep E and have equal vibrational frequencies.1116