Eigenvalues of the two-dimensional Schrodinger equation

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Abstract
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 103 or 104 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.