Elliptical orbits in the phase-space quantization

TL;DR

This paper revisits the phase-space quantization of hydrogen-like atoms, focusing on elliptical orbits and comparing three methods to derive energy levels, providing detailed calculations suitable for educational purposes.

Contribution

It presents a comprehensive analysis of phase-space quantization methods for elliptical orbits, clarifying a common textbook gap and enhancing understanding of old quantum theory.

Findings

Derivation of energy levels using three different methods

Clarification of elliptical orbit implications in phase-space quantization

Accessible detailed calculations for educational use

Abstract

The energy levels of hydrogen-like atoms are obtained from the phase-space quantization, one of the pillars of the old quantum theory, by three different methods - (i) direct integration, (ii) Sommerfeld's original method, and (iii) complex integration. The difficulties come from the imposition of elliptical orbits to the electron, resulting in a variable radial component of the linear momentum. Details of the calculation, which constitute a recurrent gap in textbooks that deal with phase-space quantization, are shown in depth in an accessible fashion for students of introductory quantum mechanics courses.