Methods of Solution for the Magnetized Coulomb Problem

Measuring energies in units of the Rydberg energy E ∞, lengths in units of the Bohr radius a 0, and the magnetic field strength in units of B 0, the Hamiltonian of an electron in a static Coulomb potential and in a uniform magnetic field then reads for spin-down states $$^{H = - \Delta - \frac{2} {r} + 2\beta {l_z} + {\beta ^2}{\varrho ^2} - 2\beta }$$ (3.1) where the magnetic field is assumed to point in the z-direction and ϱ 2 = x 2+y 2. The energies of the corresponding spin-up states are obtained by simply adding 4β. The eigenstates of (3.1) can be classified according to z-parity π and the z-component l z of orbital angular momentum, which are exact symmetries of H, but in general no further separation of the two-dimensional problem is possible.