The 'accidental' degeneracy of the hydrogen atom is no accident

摘要

The Schrodinger equation does not account for the 2n(2) degeneracy of the hydrogen atom, which it dismisses as an "accidental" degeneracy. The factor of "2" in the 2n(2) degeneracy is well-accounted-for in the relativistic formulation by the two spin states of the electron. The n(2) 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 n(2) 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 n(2) degeneracy of the hydrogen atom explained, but its energy spectrum and eigenfunctions are obtained without even using the Schrodinger equation, employing only the fundamental principles of quantum mechanics rather than the Schrodinger equation.