(a) For a particle of charge $q$ moving in an electromagnetic field with vector potential $\boldsymbol{A}$ and scalar potential $\phi$, write down the classical Hamiltonian and the equations of motion.

(b) Consider the vector and scalar potentials

$$\boldsymbol{A}=\frac{B}{2}(-y, x, 0), \quad \phi=0$$

(i) Solve the equations of motion. Define and compute the cyclotron frequency $\omega_{B}$.

(ii) Write down the quantum Hamiltonian of the system in terms of the angular momentum operator

$$L_{z}=x p_{y}-y p_{x}$$

Show that the states

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

for any function $f$, are energy eigenstates and compute their energy. Define Landau levels and discuss this result in relation to them.

(iii) Show that for $f(w)=w^{M}$, the wavefunctions in ( $\dagger$ ) are eigenstates of angular momentum and compute the corresponding eigenvalue. These wavefunctions peak in a ring around the origin. Estimate its radius. Using these two facts or otherwise, estimate the degeneracy of Landau levels.