(i) A Hamiltonian $H_{0}$ has energy eigenvalues $E_{r}$ and corresponding non-degenerate eigenstates $|r\rangle$. Show that under a small change in the Hamiltonian $\delta H$,

$$\delta|r\rangle=\sum_{s \neq r} \frac{\langle s|\delta H| r\rangle}{E_{r}-E_{s}}|s\rangle,$$

and derive the related formula for the change in the energy eigenvalue $E_{r}$ to first and second order in $\delta H$.

(ii) The Hamiltonian for a particle moving in one dimension is $H=H_{0}+\lambda H^{\prime}$, where $H_{0}=p^{2} / 2 m+V(x), H^{\prime}=p / m$ and $\lambda$ is small. Show that

$$\frac{i}{\hbar}\left[H_{0}, x\right]=H^{\prime}$$

and hence that

$$\delta E_{r}=-\lambda^{2} \frac{i}{\hbar}\left\langle r\left|H^{\prime} x\right| r\right\rangle=\lambda^{2} \frac{i}{\hbar}\left\langle r\left|x H^{\prime}\right| r\right\rangle$$

to second order in $\lambda$.

Deduce that $\delta E_{r}$ is independent of the particular state $|r\rangle$ and explain why this change in energy is exact to all orders in $\lambda$.