Relativistic remnants in the reduction of the Bethe-Salpeter equation to the Schrödinger equation

Abstract

Following Salpeter, the Bethe-Salpeter equation for the bound system of two oppositely charged particles is reduced to a Schrödinger equation for each of the following cases: (a) both particles are spin 1/2 particles, (b) one particle is a spinor while the other is spinless, and (c) both particles are spinless. It is shown that ife is the magnitude of charge carried by each of the particles whose masses are set equal to the electron and proton masses then, strictly speaking, only in case (a) do we obtain the familiar Schrödinger equation for the hydrogen atom. The latter equation is recovered in the other two cases only if relativistic remnants—terms of the order of 10⁻⁵ and smaller—are neglected in comparison with unity. Attention is drawn to a situation where such remnants may not be negligibly small, viz. the problem of confinement of quarks.