Let

$$\hat{x}=\left(\frac{\hbar}{2 m \omega}\right)^{1 / 2}\left(a+a^{\dagger}\right), \quad \hat{p}=\left(\frac{\hbar m \omega}{2}\right)^{1 / 2} i\left(a^{\dagger}-a\right)$$

be the position and momentum operators for a one-dimensional harmonic oscillator of mass $m$ and frequency $\omega$. Write down the commutation relations obeyed by $a$ and $a^{\dagger}$ and give an expression for the oscillator Hamiltonian $H(\hat{x}, \hat{p})$ in terms of them. Prove that the only energies allowed are $E_{n}=\hbar \omega\left(n+\frac{1}{2}\right)$ with $n=0,1,2, \ldots$ and give, without proof, a formula for a general normalised eigenstate $|n\rangle$ in terms of $|0\rangle$.

A three-dimensional oscillator with charge is subjected to a weak electric field so that its total Hamiltonian is

$$H_{1}+H_{2}+H_{3}+\lambda m \omega^{2}\left(\hat{x}_{1} \hat{x}_{2}+\hat{x}_{2} \hat{x}_{3}+\hat{x}_{3} \hat{x}_{1}\right)$$

where $H_{i}=H\left(\hat{x}_{i}, \hat{p}_{i}\right)$ for $i=1,2,3$ and $\lambda$ is a small, dimensionless parameter. Express the general eigenstate for the Hamiltonian with $\lambda=0$ in terms of one-dimensional oscillator states, and give the corresponding energy eigenvalue. Use perturbation theory to compute the changes in energies of states in the lowest two levels when $\lambda \neq 0$, working to the leading order at which non-vanishing corrections occur.