The "accidental" degeneracy of the hydrogen atom is no accident
Canadian Journal of Physics
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The Schrödinger equation does not account for the 2n2 degeneracy of the hydrogen atom, which it dismisses as an "accidental" degeneracy. The factor of "2" in the 2n2 degeneracy is well-accounted-for in the relativistic formulation by the two spin states of the electron. The n2 degeneracy is nevertheless not quite an "accident"; it is due to the SO(4), rather than SO(3), symmetry of the hydrogen atom. This result is well known, but is inadequately commented upon in most courses in quantum mechanics and atomic physics, leaving the student wondering about the origins of the n2 degeneracy of the hydrogen atom. A pedagogical analysis of this interesting aspect, which highlights the fundamental principles of quantum mechanics, is presented in this article. While doing so, not only is the n2 degeneracy of the hydrogen atom explained, but its energy spectrum and eigenfunctions are obtained without even using the Schrödinger equation, employing only the fundamental principles of quantum mechanics rather than the Schrödinger equation.