(i) Write the Hamiltonian for the harmonic oscillator,

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

in terms of creation and annihilation operators, defined by

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

Obtain an expression for $\left[a^{\dagger}, a\right]$ by using the usual commutation relation between $p$ and $x$. Deduce the quantized energy levels for this system.

(ii) Define the number operator, $N$, in terms of creation and annihilation operators, $a^{\dagger}$ and $a$. The normalized eigenvector of $N$ with eigenvalue $n$ is $|n\rangle$. Show that $n \geq 0$.

Determine $a|n\rangle$ and $a^{\dagger}|n\rangle$ in the basis defined by $\{|n\rangle\}$.

Show that

$$a^{\dagger m} a^{m}|n\rangle=\left\{\begin{aligned} \frac{n !}{(n-m) !}|n\rangle, & m \leq n \\ 0, & m>n \end{aligned}\right.$$

Verify the relation

$$|0\rangle\langle 0|=\sum_{m=0} \frac{1}{m !}(-1)^{m} a^{\dagger m} a^{m}$$

by considering the action of both sides of the equation on an arbitrary basis vector.