Metaplectic-c Quantized Energy Levels of the Hydrogen Atom

TL;DR

This paper introduces a novel geometric approach using metaplectic-c prequantization to compute hydrogen atom energy levels, aligning with quantum predictions without symplectic reduction.

Contribution

It presents a new method for quantizing hydrogen energy levels via metaplectic-c bundles and regular level sets, simplifying the traditional symplectic reduction process.

Findings

Results match standard quantum mechanical energy levels.

Method avoids symplectic reduction by using regular level sets.

Utilizes Ligon-Schaaf regularization to transform the problem.

Abstract

This paper calculates the quantized energy levels of the hydrogen atom, using a metaplectic-c prequantization bundle and a definition of a quantized energy level that was introduced by the author in a previous paper. The calculation makes use of a computational technique also demonstrated in that paper. The result is consistent with the standard quantum mechanical prediction. Unlike other treatments of the hydrogen atom, this approach does not require the construction of the symplectic reduction, but takes place over a regular level set of the energy function. The Ligon-Schaaf regularization map is used to transform the problem into one of determining the quantized energy levels of a free particle on the tangent bundle for S3.