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

Published: 17 August 2021| Version 1 | DOI: 10.17632/knbckjrwfc.1


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.



Computational Physics, Quantum Mechanics, Computer Algebra System