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arXiv · 2006.12498

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

Pascal Szriftgiser·Edgardo S. Cheb-Terrab
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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.

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Pascal Szriftgiser, Edgardo S. Cheb-Terrab. 2020-06-22. Computer Algebra in Physics: The hidden SO(4) symmetry of the hydrogen atom. https://doi.org/10.1016/j.cpc.2021.108076

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