Reduction of the Breit Coulomb equation to an equivalent Schrödinger equation, and investigation of the behavior of the wave function near the origin

Abstract

The Breit equation for two equal-mass spin-1/2 particles interacting through an attractive Coulomb potential is separated into its angular and radial parts, obtaining coupled sets of first-order differential equations for the radial wave functions. The radial equations for the /sup 1/J/sub J/, /sup 3/J/sub J/, and /sup 3/P/sub 0/ states are further reduced to a single, one-dimensional Schroedinger equation with a relatively simple effective potential. No approximations, other than the initial one of an instantaneous Coulomb interaction, are made in deriving this equation; it accounts for all relativistic effects, as well as for mixing between different components of the wave function. Approximate solutions are derived for this Schroedinger equation, which gives the correct O(..cap alpha../sup 4/) term for the 1 /sup 1/S/sub 0/ energy and for the n/sup 1/J/sub J/ energies, for J>0. The radial equations for the /sup 3/(J +- 1)/sub J/ states are reduced to two second-order coupled equations. At small r, the Breit Coulomb wave functions behave as r/sup ..nu..//sup -1/, where ..nu.. is either ..sqrt..J(J+1)+1-..cap alpha../sup 2//4 or ..sqrt..J(J+1)-..cap alpha../sup 2//4 . The /sup 1/S/sub 0/ and /sup 3/P/sub 0/ wave functions therefore diverge at the origin as r/sup //sup ..sqrt..//sup 1-//sup ..cap alpha..//sup <2//4 -1$.more » This divergence of the J = 0 states, however, does not occur when the spin-spin interaction, -(..cap alpha../r)..cap alpha..x..cap alpha.., is added to the Coulomb potential.« less