The states of the hydrogen atom are denoted by $|n l m\rangle$ with $l<n,-l \leq m \leq l$ and associated energy eigenvalue $E_{n}$, where

$$E_{n}=-\frac{e^{2}}{8 \pi \epsilon_{0} a_{0} n^{2}} .$$

A hydrogen atom is placed in a weak electric field with interaction Hamiltonian

$$H_{1}=-e \mathcal{E} z$$

a) Derive the necessary perturbation theory to show that to $O\left(\mathcal{E}^{2}\right)$ the change in the energy associated with the state $|100\rangle$ is given by

$$\Delta E_{1}=e^{2} \mathcal{E}^{2} \sum_{n=2}^{\infty} \sum_{l=0}^{n-1} \sum_{m=-l}^{l} \frac{|\langle 100|z| n l m\rangle|^{2}}{E_{1}-E_{n}}$$

The wavefunction of the ground state $|100\rangle$ is

$$\psi_{n=1}(\mathbf{r})=\frac{1}{\left(\pi a_{0}^{3}\right)^{1 / 2}} e^{-r / a_{0}}$$

By replacing $E_{n}, \forall n>1$, in the denominator of $(*)$ by $E_{2}$ show that

$$\left|\Delta E_{1}\right|<\frac{32 \pi}{3} \epsilon_{0} \mathcal{E}^{2} a_{0}^{3}$$

b) Find a matrix whose eigenvalues are the perturbed energies to $O(\mathcal{E})$ for the states $|200\rangle$ and $|210\rangle$. Hence, determine these perturbed energies to $O(\mathcal{E})$ in terms of the matrix elements of $z$ between these states.

[Hint:

$$\begin{aligned} \langle n l m|z| n l m\rangle &=0 & & \forall n, l, m \\ \left\langle n l m|z| n l^{\prime} m^{\prime}\right\rangle &=0 & & \forall n, l, l^{\prime}, m, m^{\prime}, \quad m \neq m^{\prime} \end{aligned}$$