(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 $H$ and number operator $N$ in terms of $a$ and $a^{\dagger}$. 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) The operator $K_{r}$ is defined to be

$$K_{r}=\frac{\left(a^{\dagger}\right)^{r} a^{r}}{r !}$$

for $r=0,1,2, \ldots$ Show that $K_{r}$ commutes with $N$. Show that if $r \leqslant n$, then

$$K_{r}|n\rangle=\frac{n !}{(n-r) ! r !}|n\rangle,$$

and $K_{r}|n\rangle=0$ otherwise. By considering the action of $K_{r}$ on the state $|n\rangle$, deduce that

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