Variational principle for confined quantum systems.

Abstract

A new variational principle for the energy eigenvalues of a confined quantum system is presented. Whereas the exact wave function \ensuremath{\Psi} must vanish on the bounding surface of the region, the trial function \ensuremath{\psi} in this principle need not obey any specific boundary condition. Calculationally, the method is similar to the conventional variational method except that kinetic energy turns out to be a weighted average of -F\ensuremath{\psi}${\mathrm{\ensuremath{\nabla}}}^{2}$\ensuremath{\psi} d\ensuremath{\tau} and F(\ensuremath{\nabla}\ensuremath{\psi})\ensuremath{\cdot}(\ensuremath{\nabla}\ensuremath{\psi})d\ensuremath{\tau} in the ratio 2:-1. Although the principle is not a definite (minimum) one, good results are obtained in several examples.