Energy spectra of the hydrogen atom and the harmonic oscillator in two dimensions from Bogomolny's semiclassical transfer operator.

Abstract

The semiclassical quantization scheme formulated by Bogomolny [E.B. Bogomolny, Nonlinearity 5, 805 (1992); Chaos 2, 5 (1992)], employing a suitably chosen Poincar\'e surface of section, has been used to calculate the energy eigenvalues of the hydrogen atom in one and two dimensions and the anisotropic harmonic oscillator in two dimensions. For the two-dimensional systems it was found to be advantageous to decompose Bogomolny's transfer operator into two "half-mapping" operators. This approach, developed by Haggerty [M.R. Haggerty, Ph.D. thesis, Massachusetts Institute of Technology, 1994 (unpublished); Phys. Rev. E 52, 389 (1995)], leads to an analytical solution for the energy eigenvalues of the hydrogen atom. However, the energies are found to depend on the quantum number $n$ as ${(n\ensuremath{-}\frac{1}{4})}^{\ensuremath{-}2}$, unlike the exact quantum energies, which go as ${(n\ensuremath{-}\frac{1}{2})}^{\ensuremath{-}2}$. An attempt to explain this one-quarter shift on the basis of the Langer-modified WKB approximation is only partly successful. For the two-dimensional harmonic oscillator, numerical calculations yield results close to the exact quantum energies.