(Solved): A particle confined to a plane moving in a uniform magnetic field perpendicular to that plane can b ...

A particle confined to a plane moving in a uniform magnetic field perpendicular to that plane can be described by the Hamiltonian $H=\frac{1}{2m}{\left(-i\hslash \frac{\partial }{\partial x}+\frac{e}{2c}By\right)}^2+\frac{1}{2m}{\left(-i\hslash \frac{\partial }{\partial y}-\frac{e}{2c}Bx\right)}^2\text{. }$ This corresponds to choosing the vector potential in the form $\mathbf{A}=\frac{1}{2}\mathbf{B}\times \mathbf{r}$, where $\mathbf{B}$ points along the $z$-axis. Also the charge is defined as $e>0$. Find the energy levels s of the corresponding Schrödinger equation $H\psi =E\psi$. Find also the wave functions corresponding the lowest Landau level states (no need to find the wave functions of higher energy states).