The two-dimensional hydrogen atom revisited

TL;DR

This paper revisits the two-dimensional hydrogen atom, revealing degeneracies due to a Runge-Lenz vector analogue, reformulating the problem in momentum space, and deriving new integral relations involving special functions.

Contribution

It introduces a novel momentum space formulation of the 2D hydrogen atom and derives previously untabulated integral relations involving special functions.

Findings

Degeneracy in energy eigenvalues due to Runge-Lenz vector analogue

Reformulation in momentum space using spherical harmonics

New integral relations involving special functions

Abstract

The bound state energy eigenvalues for the two-dimensional Kepler problem are found to be degenerate. This "accidental" degeneracy is due to the existence of a two-dimensional analogue of the quantum-mechanical Runge-Lenz vector. Reformulating the problem in momentum space leads to an integral form of the Schroedinger equation. This equation is solved by projecting the two-dimensional momentum space onto the surface of a three-dimensional sphere. The eigenfunctions are then expanded in terms of spherical harmonics, and this leads to an integral relation in terms of special functions which has not previously been tabulated. The dynamical symmetry of the problem is also considered, and it is shown that the two components of the Runge-Lenz vector in real space correspond to the generators of infinitesimal rotations about the respective coordinate axes in momentum space.