In two-dimensional space xy (with B in the z-direction) we have a proton. With a Landau gauge choice A=xBe y^, write (in closed analytical form) the common eigenfunctions of Hamiltonian H and X0, and then (separately) the common
eigenfunctions of H and Y0. (X0 and Y0 are of course guiding center operators. X0=c+c*Pi y/qB, Y0=y-cPi x/qB, whereas Pi x, Pi y canonical momentum px-qAx, py-qAy).
Suppose our system is inside a rectangle (Lx,Ly) of macroscopic dimensions. Then, as a function of the well-known Landau wave functions Psi n,ky of the above system, give an analytical
expression for the local probability current J(x,y) (it should be in the form J =SOMETHING*|Psi |2 and you should clearly show its direction [which of course can change at different points in space – it obviously depends on what the exact form of SOMETHING is]); at the end find how much the global is (always in 2D).