Schrödinger and Dirac equations for the hydrogen atom, and Laguerre polynomials

Abstract

It is usually claimed that the Laguerre polynomials were popularized by Schrodinger when creating wave mechanics; however, we show that he did not immediately identify them in studying the hydrogen atom. In the case of relativistic Dirac equations for an electron in a Coulomb field, Dirac gave only approximations, Gordon and Darwin gave exact solutions, and Pidduck first explicitly and elegantly introduced the Laguerre polynomials, an approach neglected by most modern treatises and articles. That Laguerre polynomials were not very popular before their use in quantum mechanics, probably because they had been little used in classical mathematical physics, is confirmed by the fact that, as we show, they had been rediscovered independently several times during the nineteenth century, in published or unpublished studies of Abel, Murphy, Chebyshev, and Laguerre.