I'm interested to understand the algebraic method to analyse the Landau level problem.
To set up the problem (I will be using natural units), I consider the Hamiltonian: $$ \hat{H}=(\hat{\mathbf{p}}-\hat{\mathbf{A}})^2 $$ In the algebraic method, I can find two sets of bosonic operators which commute with the Hamiltonian. One set is obtained by $\hat{b}=\frac{1}{\sqrt{2}}(\hat{\pi}_x-i\hat{\pi}_y)$ and $\hat{b}^\dagger=\frac{1}{\sqrt{2}}(\hat{\pi}_x+i\hat{\pi}_y)$ where $\hat{\pi}=\hat{\mathbf{p}}-\hat{\mathbf{A}}$ is the mechanical momentum. The Hamiltonian in terms of these operators is $$\hat{H}=2(\hat{b}^\dagger\hat{b}+\frac{1}{2})$$ which is just the harmonic oscillator.
The other set of bosonic operators are given by $\hat{d}=\frac{1}{\sqrt{2}}(\hat{X}+i\hat{Y})$ and $\hat{d}^\dagger=\frac{1}{\sqrt{2}}(\hat{X}-i\hat{Y})$ where $\hat{X}=\hat{x}-\hat{\pi}_y$ and $\hat{Y}=\hat{y}+\hat{\pi}_x$ are the guiding centers.
Both of these two sets of bosonic algebras obey Heisenberg algebras, and I believe the eigenfunctions can be written as: $|n,m>$ where $n$ and $m$ are eigenvalues of the $\hat{b}^\dagger\hat{b}$ and $\hat{d}^\dagger\hat{d}$ operators respectively, and so $n,m=0, 1,...$. Explicitly, this is given by:$$|n,m> \propto (\hat{b}^\dagger)^n(\hat{d}^\dagger)^m|0,0>.$$
I have also heard that coherent states can be used to represent the same eigenstates (see Stack Exchange answer here). Which of the bosonic operators should I use to create coherent states? Why are coherent states used to represent the Landau level eigenstates?
