A particle of mass $m$ and charge $q$ moving in a uniform magnetic field $\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{A}=$ $(0,0, B)$ is described by the Hamiltonian

$$H=\frac{1}{2 m}(\mathbf{p}-q \mathbf{A})^{2}$$

where $\mathbf{p}$ is the canonical momentum, which obeys $\left[x_{i}, p_{j}\right]=i \hbar \delta_{i j}$. The mechanical momentum is defined as $\boldsymbol{\pi}=\mathbf{p}-q \mathbf{A}$. Show that

$$\left[\pi_{x}, \pi_{y}\right]=i q \hbar B$$

Define

$$a=\frac{1}{\sqrt{2 q \hbar B}}\left(\pi_{x}+i \pi_{y}\right) \quad \text { and } \quad a^{\dagger}=\frac{1}{\sqrt{2 q \hbar B}}\left(\pi_{x}-i \pi_{y}\right) \text {. }$$

Derive the commutation relation obeyed by $a$ and $a^{\dagger}$. Write the Hamiltonian in terms of $a$ and $a^{\dagger}$ and hence solve for the spectrum.

In symmetric gauge, states in the lowest Landau level with $k_{z}=0$ have wavefunctions

$$\psi(x, y)=(x+i y)^{M} e^{-q B r^{2} / 4 \hbar}$$

where $r^{2}=x^{2}+y^{2}$ and $M$ is a positive integer. By considering the profiles of these wavefunctions, estimate how many lowest Landau level states can fit in a disc of radius $R$.