The radial wavefunction of a relativistic binary of two fermions bound by the Coulomb force

Abstract

The exact radial eigenfunctions of a relativistic binary atom bound by the static Coulomb force are calculated. We consider the two-fermion Dirac equation for two distinguishable fermions (like, e.g., in positronium) including the static Coulomb potential but no radiative corrections. As shown in a previous paper and its addendum this problem can be solved exactly [1]. Here we provide the exact solution in terms of generalized hypergeometric functions for the radial eigenfunctions of the hamiltonian, which are determined through a matrix recursion relation.Also, an equivalent set of coupled first-order, or uncoupled second-order, differential equations that may be solved numerically is provided.

An effective Dirac equation for a binary of two fermions

“…The eigenfunctions of the spin-orbit-coupling operators are four-component spinors forming a basis in the spin product state spanned by the spin singlet-and triplet-like eigenfunctions of the total angular momentum operator [5]. The four-component radial wavefunctions in the particle-antiparticle state can be expressed in terms of generalized hypergeometric functions which are determined through a matrix recursion relation [6].…”