Abstract
A hydrogen atom in a constant magnetic field is a problem whose Schrödinger equation has been considered to be analytically unsolvable. However, it has been found for the case of two-dimensional space some specific values of the magnetic field for which the problem has analytical solutions. In this paper, we give a full theory for these analytical solutions. First, we construct an equation to calculate of exact-solution-points, the values of magnetic field which allow exact solutions for the Schrödinger equation. The explicit forms of the wave functions corresponding to these analytical solutions are also given. Second, we analyse the exact-solution-points and discover some important rules. (i) There exist (n − 1)2 different values of the magnetic field so that the Schrödinger equation of a two-dimensional hydrogen atom in such field has exact analytical solutions with the principal quantum number n. (ii) For more details, among these (n − 1)2 exact analytical solutions, there are nr=n−|m|−1 solutions with the same magnetic quantum number m. Both properties are also proved mathematically in this paper. For illustration, some concrete values are shown for the cases of quantum number n up to 8.
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