Local pinning of trial wave functions: An optimization method without integrals for finding approximate solutions of field equations

Abstract

We present an optimization method that requires no integrations for obtaining approximate solutions of field equations of the form F(Ψ(r{combining right arrow above}))=0. The expression F(Ψ(r{combining right arrow above})) would be identically zero everywhere if Ψ(r{combining right arrow above}) were an exact solution. The method consists of selecting an appropriate trial function Ψ0(r{combining right arrow above}), which depends on several parameters, and requiring that the values of these parameters be such that F(Ψ0(r{combining right arrow above})) equal zero in certain regions of space, especially around singularities. This requirement yields a locally pinned trial wave function that approximates the exact solution. We illustrate the method by applying it to a simple harmonic oscillator, a vortex in a superfluid, to the ground state of a Bose-Einstein condensate and to the ground state of the helium atom. © 2010 American Association of Physics Teachers.