Less trouble with orbits: The semi-classical hydrogen problem in parabolic and spherical coordinates

TL;DR

This paper addresses discrepancies in the semi-classical hydrogen problem by proposing amended Sommerfeld orbits, enabling consistent orbit comparisons across parabolic and spherical coordinates.

Contribution

It introduces amended Sommerfeld orbits to resolve historical differences in orbit eccentricity calculations between coordinate systems.

Findings

Amended orbits provide consistent eccentricity values.

The approach clarifies previous discrepancies in semi-classical hydrogen models.

Improves accuracy of orbit quantization in different coordinate systems.

Abstract

Historically, the eccentricity of Sommerfeld orbits from quantization conditions in either parabolic or spherical coordinates was found to differ in almost all cases. To do the orbit comparison correctly, one must use amended instead of traditional Sommerfeld orbits.