The position and momentum operators of the harmonic oscillator can be written as

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

where $m$ is the mass, $\omega$ is the frequency and the Hamiltonian is

$$H=\frac{1}{2 m} \hat{p}^{2}+\frac{1}{2} m \omega^{2} \hat{x}^{2}$$

Assuming that

$$[\hat{x}, \hat{p}]=i \hbar$$

derive the commutation relations for $a$ and $a^{\dagger}$. Construct the Hamiltonian in terms of $a$ and $a^{\dagger}$. Assuming that there is a unique ground state, explain how all other energy eigenstates can be constructed from it. Determine the energy of each of these eigenstates.

Consider the modified Hamiltonian

$$H^{\prime}=H+\lambda \hbar \omega\left(a^{2}+a^{\dagger 2}\right)$$

where $\lambda$ is a dimensionless parameter. Use perturbation theory to calculate the modified energy levels to second order in $\lambda$, quoting any standard formulae that you require. Show that the modified Hamiltonian can be written as

$$H^{\prime}=\frac{1}{2 m}(1-2 \lambda) \hat{p}^{2}+\frac{1}{2} m \omega^{2}(1+2 \lambda) \hat{x}^{2} .$$

Assuming $|\lambda|<\frac{1}{2}$, calculate the modified energies exactly. Show that the results are compatible with those obtained from perturbation theory.