(i) Hermitian operators $\hat{x}, \hat{p}$, satisfy $[\hat{x}, \hat{p}]=i \hbar$. The eigenvectors $|p\rangle$, satisfy $\hat{p}|p\rangle=p|p\rangle$ and $\left\langle p^{\prime} \mid p\right\rangle=\delta\left(p^{\prime}-p\right)$. By differentiating with respect to $b$ verify that

$$e^{-i b \hat{x} / \hbar} \hat{p} e^{i b \hat{x} / \hbar}=\hat{p}+b$$

and hence show that

$$e^{i b \hat{x} / \hbar}|p\rangle=|p+b\rangle$$

Show that

$$\langle p|\hat{x}| \psi\rangle=i \hbar \frac{\partial}{\partial p}\langle p \mid \psi\rangle$$

and

$$\langle p|\hat{p}| \psi\rangle=p\langle p \mid \psi\rangle .$$

(ii) A quantum system has Hamiltonian $H=H_{0}+H_{1}$, where $H_{1}$ is a small perturbation. The eigenvalues of $H_{0}$ are $\epsilon_{n}$. Give (without derivation) the formulae for the first order and second order perturbations in the energy level of a non-degenerate state. Suppose that the $r$ th energy level of $H_{0}$ has $j$ degenerate states. Explain how to determine the eigenvalues of $H$ corresponding to these states to first order in $H_{1}$.

In a particular quantum system an orthonormal basis of states is given by $\left|n_{1}, n_{2}\right\rangle$, where $n_{i}$ are integers. The Hamiltonian is given by

$$H=\sum_{n_{1}, n_{2}}\left(n_{1}^{2}+n_{2}^{2}\right)\left|n_{1}, n_{2}\right\rangle\left\langle n_{1}, n_{2}\left|+\sum_{n_{1}, n_{2}, n_{1}^{\prime}, n_{2}^{\prime}} \lambda_{\left|n_{1}-n_{1}^{\prime}\right|,\left|n_{2}-n_{2}^{\prime}\right|}\right| n_{1}, n_{2}\right\rangle\left\langle n_{1}^{\prime}, n_{2}^{\prime}\right|,$$

where $\lambda_{r, s}=\lambda_{s, r}, \lambda_{0,0}=0$ and $\lambda_{r, s}=0$ unless $r$ and $s$ are both even.

Obtain an expression for the ground state energy to second order in the perturbation, $\lambda_{r, s}$. Find the energy eigenvalues of the first excited state to first order in the perturbation. Determine a matrix (which depends on two independent parameters) whose eigenvalues give the first order energy shift of the second excited state.