Electron problem on sphere and cylinder Under perpendicular magnetic field conditions

An electron in a magnetic field is a classical problem, some interesting effects of which appear in the transition to quantum mechanics. We consider the motion of an electron in a plane in a Landau and symmetric model and in a magnetic field created by a Dirac magnetic monopole. We present two different cases: spherical and cylindrical geometries. When considering motion on a spherical surface, we find our own wave functions not by solving the Schrödinger equation, but with the help of algebraic methods. First we find the wave function corresponding to the lowest Landau level, and then the general result. When considering the cylindrical geometry, we will solve the problem in two different ways: the first - in the cylindrical coordinate plane, and the second - in the Landau form by imposing a periodic boundary condition for the problem solved, and we will see that the energies of the Landau levels and the eigenwave functions are the same in both cases.