Energy Levels of Hydrogenlike Atoms in a Magnetic Field

Abstract

A new expansion of the wave function in Laguerre polynomials for hydrogenlike atoms in a magnetic field has been developed. This expansion reduces the Schr\"odinger equation to an infinite set of algebraic equations for which a fast, accurate, and unambiguous calculation of the bound states of a hydrogen atom in a magnetic field is possible. Four quantum numbers ($K, C, M, N$) are used to label the energy levels of the system in cylindrical coordinates. When the magnetic field is zero, the new labeling of the states is equivalent to the standard ($n, l, m$) quantum numbers obtained for the Coulomb potential in spherical coordinates. Numerical results are presented for the first 14 energy levels with principal quantum number $n=1, 2, 3$. Several energy-level crossings occur in the region $0<\ensuremath{\gamma}<1$, where $\ensuremath{\gamma}=\frac{\ensuremath{\hbar}{\ensuremath{\omega}}_{c}}{2\mathcal{R}}$ and $\mathcal{R}$ is the effective Rydberg energy.