Computer algebra in physics: The hidden SO(4) symmetry of the hydrogen atom

Abstract

Pauli first noticed the hidden SO(4) symmetry for the hydrogen atom in the early stages of quantum mechanics [1]. Starting from that symmetry, one can recover the spectrum of a spinless hydrogen atom and the degeneracy of its states without explicitly solving Schrödinger's equation [2], [3]. In this paper, we derive that SO(4) symmetry and spectrum using a computer algebra system (CAS). While this problem is well known [4], [5], its solution involves several steps of manipulating expressions with tensorial quantum operators, including 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 and to showcase the CAS technique. 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 two alternative patterns of computer algebra steps that can be used for systematically tackling more complicated symbolic problems of this kind. Program summaryProgram Title: Maple specific worksheetCPC Library link to program files:https://doi.org/10.17632/knbckjrwfc.1Developer's repository link:https://www.maplesoft.com/applications/view.aspx?SID=154764Licensing provisions: GPLv3Programming language: Maple (Maplesoft)Nature of problem: Provide a general framework to handle Quantum Mechanics' non-commutative algebra systematically. Non-commutative tensor calculus, subject to algebra rules, is tedious, time-consuming and error-prone. This Maple worksheet provides a general framework to implement, test and address a wide range of non-commutative problems commonly found in advanced quantum mechanics. It can be customized and adapted to potentially more complicated algebra problems.Solution method: Make use of Maple's Physics package. First, configure the problem parameters: space dimension, tensor objects and non-commutative quantities. Then provide the most basic commutation rules, such as position, linear and angular momentum commutators. From there, using the Simplify command and a short set of other key commands, demonstrate further commutation rules until the complete target algebra is reconstructed. One should notice that the Physics package commands employed here take into account custom algebra rules, the sum rule for repeated indices, and use tensor-simplification algorithms.Additional comments including restrictions and unusual features: This article mirrors a Maple worksheet. As such, it is not an usual code intended to make a computation for a given set of parameters. It is rather a general template or guideline, useful to extend the present problem with new computations or, more generally, widely open to tackle new quantum mechanics problem involving tensor and commutator heavy calculus.