Chapter 25, Representation theory Video Solutions, Mathematical Methods for Physics and Engineering: A Comprehensive Guide

Problem 1

A group $\mathcal{G}$ has four elements $I, X, Y$ and $Z$, which satisfy $X^{2}=Y^{2}=Z^{2}=$ $X Y Z=I$. Show that $\mathcal{G}$ is Abelian and hence deduce the form of its character table.
Show that the matrices
$$
\begin{aligned}
&\mathrm{D}(I)=\left(\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right), & \mathrm{D}(X)=\left(\begin{array}{cc}
-1 & 0 \\
0 & -1
\end{array}\right) \\
&\mathrm{D}(\boldsymbol{Y})=\left(\begin{array}{cc}
-1 & -p \\
0 & 1
\end{array}\right), & \mathrm{D}(Z)=\left(\begin{array}{cc}
1 & p \\
0 & -1
\end{array}\right)
\end{aligned}
$$
where $p$ is a real number, form a representation $\mathrm{D}$ of $\mathcal{G} .$ Find its characters and decompose it into irreps.

Nick Johnson

Numerade Educator

05:51

Problem 2

Using a square whose corners lie at coordinates $(\pm 1, \pm 1)$, form a natural representation of the dihedral group $\mathcal{D}_{4-}$ Find the characters of the representation, and, using the information (and class order) in table $25.4$ (p. 944$)$, express the representation in terms of irreps.

Now form a representation in terms of eight $2 \times 2$ orthogonal matrices, by considering the effect of each of the elements of $\mathcal{D}_{4}$ on a general vector $(x, y)$. Confirm that this representation is one of the irreps found using the natural representation.

William Mead

Numerade Educator

05:10

Problem 3

The quaternion group $Q$ (see exercise $24.20$ ) has eight elements $\{\pm 1, \pm i, \pm j, \pm k\}$ obeying the relations
$$
i^{2}=j^{2}=k^{2}=-1, \quad i j=k=-j i
$$
Determine the conjugacy classes of $\mathcal{Q}$ and deduce the dimensions of its irreps. Show that $Q$ is homomorphic to the four-element group $\mathcal{V}$, which is generated by two distinct elements $a$ and $b$ with $a^{2}=b^{2}=(a b)^{2}=I .$ Find the one-dimensional irreps of $\mathcal{V}$ and use these to help determine the full character table for $Q$.

Ely Crowder

Numerade Educator

05:10

Problem 4

(a) By considering the possible forms of its cycle notation, determine the number of elements in each conjugacy class of the permutation group $S_{4}$ and show that $S_{4}$ has five irreps. Give the logical reasoning that shows they must consist of two three-dimensional, one two-dimensional, and two one-dimensional irreps.
(b) By considering the odd and even permutations in the group $S_{4}$ establish the characters for one of the one-dimensional irreps.
(c) Form a natural matrix representation of $4 \times 4$ matrices based on a set of objects $\{a, b, c, d\}$, which may or may not be equal to each other, and, by selecting one example from each conjugacy class, show that this natural representation has characters $4,2,1,0,0 .$ The one-dimensional vector subspace spanned by sets of the form $\{a, a, a, a\}$ is invariant under the permutation group and hence transforms according to the invariant irrep $\mathrm{A}_{1}$. The remaining three-dimensional subspace is irreducible; use this and the characters deduced above to establish the characters for one of the three-dimensional irreps, $T_{1}$
(d) Complete the character table using orthogonality properties, and check the summation rule for each irrep. You should obtain table $25.8 .$

Ely Crowder

Numerade Educator

00:27

Problem 5

In exercise $24.10$, the group of pure rotations taking a cube into itself was found to have 24 elements. The group is isomorphic to the permutation group $S_{4}$, considered in the previous question, and hence has the same character table, once corresponding classes have been established. By counting the number of elements in each class make the correspondences below (the final two cannot be decided purely by counting, and should be taken as given).
$\begin{aligned}&\begin{array}{c}\text { Permutation } \\ \text { class type }\end{array} & \begin{array}{c}\text { Symbol Action } \\ \text { (physics) }\end{array} & \begin{array}{l}\text { Syme } \\ (1) & I & \text { none } \\ (123) & 3 & \text { rotations about a body diagonal } \\ (12)(34) & 2_{z} & \text { rotation of } \pi \text { about the normal to a face } \\ (1234) & 4_{z} & \text { rotations of } \pm \pi / 2 \text { about the normal to a face } \\ \text { (12) } & 2_{d} \text { rotation of } \pi \text { about an axis through the } \\ \text { centres of opposite edges }\end{array} \\& & & \end{aligned}$
Reformulate the character table $25.8$ in terms of the elements of the rotation symmetry group ( 432 or $O$ ) of a cube and use it when answering exercises $25.7$ and $25.8$

Sheryl Ezze

Numerade Educator

05:51

Problem 6

Consider a regular hexagon orientated so that two of its vertices lie on the $x$-axis. Find matrix representations of a rotation $R$ through $\pi / 6$ and a reflection $m_{y}$ in the $y$-axis by determining their effects on vectors lying in the $x y$-plane. Show that a reflection $m_{x}$ in the $x$-axis can be written as $m_{x}=m_{y} R^{3}$ and that the $(12)$ elements of the symmetry group of the hexagon are given by $R^{n}$ or $R^{n} m_{y} .$

Using the representations of $R$ and $m_{y}$ as generators, find a two-dimensional representation of the symmetry group, $C_{6}$, of the regular hexagon. Is it a faithful representation?

William Mead

Numerade Educator

02:07

Problem 7

In a certain crystalline compound, a thorium atom lies at the centre of a regular octahedron of six sulphur atoms at positions $(\pm a, 0,0),(0, \pm a, 0),(0,0, \pm a)$. These can be considered as being positioned at the centres of the faces of a cube of side $2 a$. The sulphur atoms produce at the site of the thorium atom an electric field that has the same symmetry group as a cube ( 432 or $O$ ).

The five degenerate $d$-electron orbitals of the thorium atom can be expressed, relative to any arbitrary polar axis, as
$$
\left(3 \cos ^{2} \theta-1\right) f(r), \quad e^{\pm i \phi} \sin \theta \cos \theta f(r), \quad e^{\pm \Delta \phi} \sin ^{2} \theta f(r)
$$
A rotation about that polar axis by an angle $\phi^{\prime}$ effectively changes $\phi$ to $\phi-\phi^{\prime}$. Use this to show that the character of the rotation in a representation based on the orbital wavefunctions is given by
$$
1+2 \cos \phi^{\prime}+2 \cos 2 \phi^{\prime}
$$ and hence that the characters of the representation, in the order of the symbols given in exercise $25.5$, is $5,-1,1,-1,1 .$ Deduce that the five-fold degenerate level is split into two levels, a doublet and a triplet.

Zachary Warner

Numerade Educator

00:57

Problem 8

Sulphur hexafluoride is a molecule with the same structure as the crystalline compound in exercise $25.7$, except that a sulphur atom is now the central atom. The following are the forms of some of the electronic orbitals of the sulphur atom, together with the irreps according to which they transform under the symmetry group 432 (or $O$ ).
$$
\begin{array}{ll}
\Psi_{s}=f(r) & \mathrm{A}_{1} \\
\Psi_{p_{1}}=z f(r) & \mathrm{T}_{1} \\
\Psi_{d_{1}}=\left(3 z^{2}-r^{2}\right) f(r) & \mathrm{E} \\
\Psi_{d_{2}}=\left(x^{2}-y^{2}\right) f(r) & \mathrm{E} \\
\Psi_{d_{3}}=x y f(r), & \mathrm{T}_{2}
\end{array}
$$
The function $x$ transforms according to the irrep $T_{1}$. Use the above data to determine whether dipole matrix elements of the form $J=\int \phi_{1} x \phi_{2} d \tau$ can be non-zero for the following pairs of orbitals $\phi_{1}, \phi_{2}$ in a sulphur hexafluoride molecule: (a) $\Psi_{d 1}, \Psi_{s} ;(\mathrm{b}) \Psi_{d 1}, \Psi_{n 1} ;(\mathrm{c}) \Psi_{d 2}, \Psi_{d 1} ;(\mathrm{d}) \Psi_{s}, \Psi_{d \hat{3}} ;(\mathrm{e}) \Psi_{n \mathrm{l}}, \Psi_{s=}$

Raghvendra Singh

Numerade Educator

02:24

Problem 9

The hydrogen atoms in a methane molecule $\mathrm{CH}_{4}$ form a perfect tetrahedron with the carbon atom at its centre. The molecule is most conveniently described mathematically by placing the hydrogen atoms at the points $(1,1,1),(1,-1,-1)$, $(-1,1,-1)$ and $(-1,-1,1) .$ The symmetry group to which it belongs, the tetrahedral group $\left(\overline{4} 3 m\right.$ or $T_{d}$ ) has classes typified by $I, 3,2_{z}, m_{d}$ and $\overline{4}_{z}$, where the first three are as in exercise $25.5, m_{d}$ is a reflection in the mirror plane $x-y=0$ and $\overline{4}_{2}$ is a rotation of $\pi / 2$ about the $z$-axis followed by an inversion in the origin. $A$ reflection in a mirror plane can be considered as a rotation of $\pi$ about an axis perpendicular to the plane, followed by an inversion in the origin.

The character table for the group $43 m$ is very similar to that for the group 432, and has the form shown in table $25.9 .$ By following the steps given below, determine how many different internal vibration frequencies the $\mathrm{CH}_{4}$ molecule has.
(a) Consider a representation based on the 12 coordinates $x_{i}, y_{i}, z_{l}$ for $i=$ $1,2,3,4 .$ For those hydrogen atoms that transform into themselves, a rotation through an angle $\theta$ about an axis parallel to one of the coordinate axes gives rise in the natural representation to the diagonal elements 1 for the corresponding coordinate and $2 \cos \theta$ for the two orthogonal coordinates. If the rotation is followed by an inversion then these entries are multiplied by -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 is
$$
\mathrm{A}_{1} \oplus \mathrm{E} \oplus \mathrm{T}_{1} \oplus 2 \mathrm{~T}_{2}
$$
(b) The irreps of the bodily translational and rotational motions are included in this expression and need to be identified and removed. Show that when this is done it can be concluded that there are three different internal vibration frequencies in the $\mathrm{CH}_{4}$ molecule. State their degeneracies and check that they are consistent with the expected number of normal coordinates needed to describe the internal motions of the molecule.

Manik Pulyani

Numerade Educator

00:27

Problem 10

(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 cycle notation.
(b) Assume that all permutations with the same cycle structure belong to the same conjugacy class. Show that this leads to a contradiction and hence demonstrates that even if two permutations have the same cycle structure they do not necessarily belong to the same class.
(c) By evaluating the products $p_{1}=(123)(4) \cdot(12)(34) \cdot(132)(4)$ and $p_{2}=$ $(132)(4) \cdot(12)(34) \cdot(123)(4)$ 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 $(1 \alpha)(\beta \gamma) \bullet(123)(4) \bullet(1 \alpha)(\beta \gamma)$, where $\alpha, \beta, \gamma$ are 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 of the form $(124)$ (3) have order 3, find the character table for $A_{4}$

Sheryl Ezze

Numerade Educator

06:41

Problem 11

Use the results of exercise $24.23$ to find the character table for the dihedral group $\mathcal{D}_{5}$, the symmetry group of a regular pentagon.

Ely Crowder

Numerade Educator

02:24

Problem 12

Demonstrate that equation (25.24) does indeed generate a set of vectors transforming according to an irrep $\lambda$, by sketching and superposing drawings of an equilateral triangle of springs and masses, based on that shown in figure $25.7 .$
Figure $25.7$ The three normal vibration modes of the equilateral array. Mode ( $a)$ is known as the "breathing mode'. Modes $(b)$ and $(c)$ transform according to irrep $\mathrm{E}$ and have equal vibrational frequencies.
(a) Make an initial sketch showing an arbitrary small mass displacement from, say, vertex $C$. Draw the results of operating on the initial sketch with each of the symmetry elements of the group $3 m\left(C_{3 v}\right)$.
(b) Superimpose the results, weighting them according to the characters of irrep $\mathrm{A}_{1}$ (table $25.1$ in section $25.6$ ) and verify that the resultant is a symmetrical arrangement in which all three masses move symmetrically towards (or away from) the centroid of the triangle. The mode is illustrated in figure $25.7(a)$.
958 (c) Start again, now considering a displacement $\delta$ of $C$ parallel to the $x$-axis. Form a similar superposition of sketches weighted according to the characters of irrep $\mathrm{E}$ (note that the reflections are not needed). The resultant contains some bodily displacement of the triangle, since this also transforms according to E. Show that the displacement of the centre of mass is $\bar{x}=\delta, \bar{y}=0$ Subtract this out and verify that the remainder is of the form shown in figure $25.7(c)$
(d) Using an initial displacement parallel to the $y$-axis, and an analogous procedure, generate the remaining normal mode, degenerate with that in $(c)$ and shown in figure $25.7(b)$

Manik Pulyani

Numerade Educator

01:16

Problem 13

Further investigation of the crystalline compound considered in exercise $25.7$ shows that the octahedron is not quite perfect but is elongated along the $(1,1,1)$ direction with the sulphur atoms at positions $\pm(a+\delta, \delta, \delta), \pm(\delta, a+\delta, \delta), \pm(\delta, \delta, a+$ $\delta$ ), where $\delta \ll a$. This structure is invariant under the (crystallographic) symmetry group 32 with three two-fold axes along directions typified by $(1,-1,0)$. The latter axes, which are perpendicular to the $(1,1,1)$ direction, are axes of twofold symmetry for the perfect octahedron. The group 32 is really the threedimensional version of the group $3 \mathrm{~m}$ and has the same character table as table $25.1$ (section 25.6). Use this to show that, when the distortion of the octahedron is included, the doublet found in exercise $25.7$ is unsplit but the triplet breaks up into a singlet and a doublet.

Doruk Isik

Numerade Educator