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*}