Eigenvalues of the two-dimensional Schrodinger equation

A finite-difference method is used to compute the eigenvalues of the Schrodinger equation in two dimensions. A two-dimensional 'radial' equation may arise in considering the S state of the helium atom and its isoelectronic systems. The eigenvalue problem is reduced to a set of linear equations, i.e. to a matrix equation. The zeros of the determinant of the secular matrix are the eigenvalues. A difference equation of second order is used; the resulting matrix is banded and has a simple structure. A simple method that saves computation time and memory space has been devised to compute the eigenvalues of matrices of order 10 3 or 10 4 on relatively small computers. The accuracy is fourth order because the step size is extrapolated to zero. The lowest eigenvalue of the S state of helium was computed. Also, the application of this method to one-dimensional problems proved to be very efficient.