Computer Algebra in Physics: The hidden SO(4) symmetry of the hydrogen atom

TL;DR

This paper demonstrates how a computer algebra system can derive the hidden SO(4) symmetry and spectrum of the hydrogen atom, showcasing CAS's potential in complex quantum tensor algebra computations.

Contribution

It introduces a systematic CAS-based method to derive the hydrogen atom's SO(4) symmetry and spectrum, highlighting its utility for complex quantum algebra problems.

Findings

CAS effectively derives the SO(4) symmetry and spectrum.

The method simplifies tensorial quantum operator manipulations.

Pattern recognition for systematic symbolic problem-solving in quantum algebra.

Abstract

Pauli first noticed the hidden SO(4) symmetry for the Hydrogen atom in the early stages of quantum mechanics [1]. Departing from that symmetry, one can recover the spectrum of a spinless hydrogen atom and the degeneracy of its states without explicitly solving Schr\"odinger's equation [2]. In this paper, we derive that SO(4) symmetry and spectrum using a computer algebra system (CAS). While this problem is well known [3, 4], its solution involves several steps of manipulating expressions with tensorial quantum operators, simplifying them by taking into account a combination of commutator rules and Einstein's sum rule for repeated indices. Therefore, it is an excellent model to test the current status of CAS concerning this kind of quantum-and-tensor-algebra computations. Generally speaking, when capable, CAS can significantly help with manipulations that, like non-commutative tensor calculus subject to algebra rules, are tedious, time-consuming and error-prone. The presentation also shows a pattern of computer algebra operations that can be useful for systematically tackling more complicated symbolic problems of this kind.