What are the commutation relations between the position operator $\hat{x}$ and momentum operator $\hat{p}$ ? Show that this is consistent with $\hat{x}, \hat{p}$ being hermitian.

The annihilation operator for a harmonic oscillator is

$$a=\sqrt{\frac{1}{2 \hbar}}(\hat{x}+i \hat{p})$$

in units where the mass and frequency of the oscillator are 1 . Derive the relation $\left[a, a^{\dagger}\right]=1$. Write down an expression for the Hamiltonian

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

in terms of the operator $N=a^{\dagger} a$.

Assume there exists a unique ground state $|0\rangle$ of $H$ such that $a|0\rangle=0$. Explain how the space of eigenstates $|n\rangle$, is formed, and deduce the energy eigenvalues for these states. Show that

$$a|n\rangle=A|n-1\rangle, \quad a^{\dagger}|n\rangle=B|n+1\rangle$$

finding $A$ and $B$ in terms of $n$.

Calculate the energy eigenvalues of the Hamiltonian for two harmonic oscillators

$$H=H_{1}+H_{2}, \quad H_{i}=\frac{1}{2} \hat{p}_{i}^{2}+\frac{1}{2} \hat{x}_{i}^{2}, \quad i=1,2 .$$

What is the degeneracy of the $n^{\text {th }}$energy level? Suppose that the two oscillators are then coupled by adding the extra term

$$\Delta H=\lambda \hat{x}_{1} \hat{x}_{2}$$

to $H$, where $\lambda \ll 1$. Calculate the energies for the states of the unperturbed system with the three lowest energy eigenvalues to first order in $\lambda$ using perturbation theory.

[You may assume standard perturbation theory results.]