(i) The creation and annihilation operators for a harmonic oscillator of angular frequency $\omega$ satisfy the commutation relation $\left[a, a^{\dagger}\right]=1$. Write down an expression for the Hamiltonian $H$ in terms of $a$ and $a^{\dagger}$.

There exists a unique ground state $|0\rangle$ of $H$ such that $a|0\rangle=0$. Explain how the space of eigenstates $|n\rangle, n=0,1,2, \ldots$ of $H$ is formed, and deduce the eigenenergies for these states. Show that

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

(ii) Write down the number operator $N$ of the harmonic oscillator in terms of $a$ and $a^{\dagger}$. Show that

$$N|n\rangle=n|n\rangle$$

The operator $K_{r}$ is defined to be

$$K_{r}=\frac{a^{\dagger r} a^{r}}{r !}, \quad r=0,1,2, \ldots$$

Show that $K_{r}$ commutes with $N$. Show also that

$$K_{r}|n\rangle= \begin{cases}\frac{n !}{(n-r) ! r !}|n\rangle & r \leq n \\ 0 & r>n\end{cases}$$

By considering the action of $K_{r}$ on the state $|n\rangle$ show that

$$\sum_{r=0}^{\infty}(-1)^{r} K_{r}=|0\rangle\langle 0|$$