Computer algebra in physics: The hidden SO(4) symmetry of the hydrogen atom
Pauli first noticed the hidden SO(4) symmetry for the hydrogen atom in the early stages of quantum mechanics . 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 , . In this paper, we derive that SO(4) symmetry and spectrum using a computer algebra system (CAS). While this problem is well known , , 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.