Note to the solution of the Schrödinger equation for finite systems

Patrick L. Nash, Liao Yuan Chen

Valentino A. Simpao

This brief note presents the exact analytical solution of an example class of dynamical systems, realized by differential equations. These problems occur in quantum mechanics and their generalizations in theoretical physics, but they particu- larly find application to chemical systems. Addressed is the exact analytical solution of the radial Schrodinger equation for the nonrelativistic N-particle system with pair- wise 1

Richa Rani

Robert L. W. Chen

Luigi E. Picasso

Francesco R. Ruggeri

In a previous note (1), it was noted one could write a solution of the bound state Schrodinger equation as W(x)=Wo(x)Dn(x) where W(x) is the wavefunction and Wo(x), the ground state wavefunction. Inserting into the Schrodinger equation one may obtain the Schrodinger equation for Wo(x) and an equation coupling Wo(x) and Dn(x). In this note, we suggest such an approach may be applied to the one dimensional bound Dirac equations. This allows one to obtain decoupled equations for u1(x) and v1(x) where u(x)=uo(x)u1(x) and v(x)=v1(x)vo(x) are the full solutions. Thus, if one is able to obtain a solution for uo(x) and vo(x) by inspection, the decoupled equations allow one to find the general solution. We apply this approach to the case of a scalar potential g|x|. The Dirac problem for this potential has been solved in the literature (2) with decoupling occurring through the taking of spatial derivatives. Here we wish to suggest a general method.