Developing the Pair Function Using de Broglie Waves to Describe the Particles

In this centennial anniversary year of the Schrödinger equation, we present this manuscript in which we revisit the pair function describing the motions of a pair of particles in a Coulomb potential. Interest in the pair function is rekindled by recent reports that the multi-particle Hamiltonian can be written as a sum of pair Hamiltonians and coupling interactions between pairs. The coupling interactions represent correlated motions between pairs. As is shown recently, the canonical results – obtained by direct solution of the eigenequation – yield local momenta that are complex-valued. Following the suggestion of Schrödinger, we use de Broglie waves to determine eigensolutions to the Schrödinger equation that uniformly have real-valued local momenta. The angular functions obtained here resemble the canonical spherical harmonics: the Legendre polynomial is the real part of the de Broglie wave while the equatorial function is identical to the canonical result. The radial expression for bound states has similar behaviour: the canonical function (which is the associated Laguerre function) is the real part of the corresponding de Broglie wave.