Strongly convergent method to solve one-dimensional quantum problems.

Abstract

An algorithm to solve the one-dimensional Schr\"odinger equation subject to Dirichlet boundary conditions is presented. The algorithm is based on a set of theorems that guarantee that when one solves the Schr\"odinger equation for a confined system and allows the boundaries to increase, the solutions converge strongly, in the norm of Hilbert space ${\mathit{L}}_{2}$(-\ensuremath{\infty},\ensuremath{\infty}), to the exact solutions of the unbounded problem. For the calculation of the solutions of the confined system we use a very efficient matrix method. By applying the algorithm to the harmonic oscillator and to the quartic and sextic potentials we show that with this method one can calculate the eigenvalues and eigenfunctions of a nonbounded one-dimensional problem with a high degree of accuracy and with very reasonable computational effort. We show that the eigenvalues corresponding to the sextic potential, V(x)=1/2${\mathit{x}}^{2}$+${\mathrm{\ensuremath{\alpha}}}_{2}$${\mathit{x}}^{4}$+${\mathrm{\ensuremath{\alpha}}}_{3}$${\mathit{x}}^{6}$, for different values of the parameter ${\mathrm{\ensuremath{\alpha}}}_{3}$ behave in a similar fashion as that described by Hioe et al. [Phys. Rep. 43C, 305 (1978)] for the quartic oscillator. \textcopyright{} 1996 The American Physical Society.