The right angle: precise numerical orthogonality in eigenstates

Solutions of the Schrodinger equation that pertain to different energies are orthogonal by virtue of quantum dynamics. However, when we obtain such solutions numerically using library differential equation solvers, and when the inner product is defined by numerical quadrature, the result is not sufficiently orthogonal for certain purposes. This paper shows how to construct stable finite-difference schemes that preserve accurate numerical orthogonality of the solutions.