A novel wave function factorization simplifying the matrix representation of the Schrödinger equation

Abstract

In quantum scattering theory, coordinate systems with nontrivial Jacobians may arise and cause difficulty in the reduction of close coupled equations to the desired, simple, obviously Hermitian form {−I(d2/dx2)+W(x)}f=0 with W=W°. We consider x to be the translational coordinate, orthogonal to the surface coordinates, and the Schrödinger equation is represented in a basis of surface functions. We introduce a novel wave function factorization which permits reduction to the above form if the basis is (locally) independent of x for arbitrary Jacobian and general weight function for the surface functions. This factorization is compared with the more common factorization where all coordinates are treated equally. Applications to wave function matching and the calculation of surface integrals are mentioned. Several three-dimensional, orthogonal coordinate systems provide examples simplified by the novel factorization.