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

摘要

We present an optimization method that requires no integrations for obtaining approximate solutions of field equations of the form F(Ψ()) = 0. The expression F(Ψ()) would be identically zero everywhere if Ψ() were an exact solution. The method consists of selecting an appropriate trial function Ψ0(), which depends on several parameters, and requiring that the values of these parameters be such that F(Ψ0()) 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.