Integral equation for numerical solution of stationary quantum-mechanical problems

Abstract

The work objective is to describe the numerical solution method for the stationary Schrödinger equation based on the application of the integral equation identical to the Schrödinger equation. The structure of this integral equation is close to the structure of the Fredholm equation of second kind and allows obtaining the problem numerical solution. The method under study allows finding the energy eigenvalues and eigensolutions to the quantum-mechanical problems of various dimensions. The test results of the solving problems method for one-dimensional and two-dimensional quantum oscillators are obtained. The found numerical values of eigenenergy and eigenfunctions of the oscillator are compared to the known analytical solutions, and then the error of result is evaluated. The highest accuracy of the solution is obtained for the first energy levels. The numerical solution error increases with the number of the energy eigenvalue. For the subsequent energy level, the error increases almost by an order of magnitude. The solution error for the fourth energy level is less than 2% if the integration domain contains 500 elements. If the energy level is degenerate, it is possible to obtain all eigenfunctions corresponding to the given level. For this purpose, various auxiliary functions the symmetry of which is coherent with the eigenfunction symmetry are used.