Homework 14

For this homework, you can submit up to 3 times.

Q14.1 \(\quad\) number of irreducible representations

Group \( C_{nv}=\{e,C_n,C_n^2,\cdots,C_n^{n-1},\sigma,C_n\sigma,\cdots, C_{n}^{n-1}\sigma\} \) denotes the symmetry group of a horizontal plane, which is invariant under rotation about the z-axis by \(2\pi/n\), and also invariant under reflection with respect to a perpendicular plane.

Please figure out the multiplication table of this discrete group, find the number of classes. How many irreducible representations does \(C_{10v}\) have ? Please think about the general case. Note, even or odd \(n\) results in different formulae.

Q14.2 \(\quad\) Character table of \(D_2\) (or \(C_{2v}\))

Group \( D_{2}=\{ e, C^{1/2}, \sigma, C^{1/2}\sigma \}\) is an Abelian group, so it has 4 different irreducible representations. The incomplete character table of \(D_2\) is listed below \[ \begin{bmatrix} & e & C^{1/2} & \sigma & C^{1/2}\sigma \\ A_1 & 1 & 1 & 1 & 1 \\ A_2 & 1 & 1 & -1 & -1 \\ B_1 & & & & \\ B_2 & & & & * \\ \end{bmatrix} \] Here \(A_1\) and \(A_2\) are both symmetric irreducible representations whose basis functions are symmetric under rotation \(C^{1/2}\), but the basis function of \(A_2\) is antisymmetric under reflection \(\sigma\). \(B_1\) and \(B_2\) are both antisymmetric irreducible representation. The difference is that the basis function of \(B_2\) is symmetric under reflection \(\sigma\). What is the number at position (*) ? Actually, there are 2 discrete groups of order 4, and both of them are Abelian. The other is \(C_4\). If you like, think about whether these two groups are isomorphic to each other or not.

Q14.3 \(\quad\) Invariant basis of \(C_{4v}\)

Each irreducible representation of a group is associated with an invariant subspace. As stated in Chaosbook, projection operator is the right tool to find the invariant subspaces of a group. For group \(C_{4v} = \{e, C^{1/4}, C^{2/4}, C^{3/4}, \sigma, \sigma C^{1/4}, \sigma C^{2/4}, \sigma C^{3/4} \} \), first figure out its character table, and then use projection operator to figure out all the invariant subspaces.

For the following candidates, which one is NOT an invariant subspace of \(C_{4v}\)? \begin{align*} \rho_0(\hat{x}) &=& \frac{1}{8}\left[ \rho(\hat{x}) ~+~ \rho(C^{2/4} \hat{x}) ~+~ \rho(C^{1/4} \hat{x}) ~+~ \rho(C^{3/4} \hat{x}) ~+~ \rho(C^{2/4} \sigma \hat{x}) ~+~ \rho(\sigma \hat{x}) ~+~ \rho(\sigma C^{1/4} \hat{x}) ~+~ \rho(\sigma C^{3/4} \hat{x}) \right] \\ \rho_1(\hat{x}) &=& \frac{1}{8}\left[ \rho(\hat{x}) ~+~ \rho(C^{2/4} \hat{x}) ~-~ \rho(C^{1/4} \hat{x}) ~-~ \rho(C^{3/4} \hat{x}) ~-~ \rho(\sigma C^{2/4} \hat{x}) ~-~ \rho(\sigma \hat{x}) ~+~ \rho(\sigma C^{1/4} \hat{x}) ~+~ \rho(\sigma C^{3/4} \hat{x}) \right] \\ \rho_2(\hat{x}) &=& \frac{1}{8}\left[ \rho(\hat{x}) ~+~ \rho(C^{2/4} \hat{x}) ~-~ \rho(C^{1/4} \hat{x}) ~-~ \rho(C^{3/4} \hat{x}) ~+~ \rho(C^{2/4} \sigma \hat{x}) ~+~ \rho(\sigma \hat{x}) ~-~ \rho(\sigma C^{1/4} \hat{x}) ~-~ \rho(\sigma C^{3/4} \hat{x}) \right] \\ \rho_3(\hat{x}) &=& \frac{1}{8}\left[ \rho(\hat{x}) ~+~ \rho(C^{2/4} \hat{x}) ~+~ \rho(C^{1/4} \hat{x}) ~+~ \rho(C^{3/4} \hat{x}) ~-~ \rho(C^{2/4} \sigma \hat{x}) ~-~ \rho(\sigma \hat{x}) ~-~ \rho(\sigma C^{1/4} \hat{x}) ~-~ \rho(\sigma C^{3/4} \hat{x}) \right] \\ \rho_4(\hat{x}) &=& \frac{1}{8}\left[ \rho(\hat{x}) ~-~ \rho(C^{2/4} \hat{x}) ~+~ \rho(C^{1/4} \hat{x}) ~+~ \rho(C^{3/4} \hat{x}) ~-~ \rho(C^{2/4} \sigma \hat{x}) ~+~ \rho(\sigma \hat{x}) ~-~ \rho(\sigma C^{1/4} \hat{x}) ~-~ \rho(\sigma C^{3/4} \hat{x}) \right] \end{align*}

Your e-mail

Please enter your e-mail (the same you are registered under in Piazza) to receive your grades: