Pseudospectral solution of the Schrödinger equation for the Rosen-Morse and Eckart potentials

Abstract

Pseudospectral methods based on non-classical quadratures are used to numerically compute the eigenvalues and eigenfunctions of the Schrodinger equation for the Rosen-Morse and Eckart potentials. The method uses a basis set of non-classical polynomials, $$\{ P_n(x) \}$$ , orthonormal with respect to a weight function, $$w(x)>0$$ , to construct an $$N \times N$$ matrix representative, $$\{ H_{nm} \}$$ , of the Hamiltonian, H. This matrix representative is transformed to an equivalent pseudospectral representative, $$\{H_{ij}\}$$ . The rate of convergence of the eigenvalues of $$\{ H_{ij} \}$$ to the eigenvalues of H, versus the grid size N, is reported for non-classical basis functions in comparison with the use of Legendre and Laguerre polynomials as well as a Fourier basis. The use of non-classical polynomials is shown to provide the fastest convergence for the eigenvalues. The pseudospectral method based on nonclassical quadratures proposed in this paper should find wide applicability to other problems in quantum and statistical mechanics. A review is provided of the use of a tensor product of one dimensional basis functions to describe two and three dimensional problems.