Stochastic Solutions to the Schroedinger Equation for Fermions

Candidate: Arnow,David Moss

Abstract

An exact stochastic method has been developed for generating the antisymmetric eigensolution of lowest index and its associated eigenvalue for the Schrodinger wave equation in 3N dimensions. The method is called the Green's function Monte Carlo method for fermions (FGFMC) because it is based on a Monte Carlo solution to the integral form of the Schrodinger equation (using Green's function) and because it is the fermion class of particles in physics which require antisymmetric solutions. The solution consists of two sets of 3N-dimensional points, {R(,j)('+)} and {R(,j)('-)}, distributed by density functions (psi)('+) and (psi)('-), whose difference, (psi)('+)-(psi)('-), is proportional to the eigensolution, (psi)(,F). These sets may be used to estimate integrals of the form (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) where R = (x(,1),...,x(,3N)) and where f(R) and g(R) are antisymmetric functions. By setting g(R) to (psi)(,T)(R) and f(R) to H(psi)(,T)(R), where (psi)(,T) is an antisymmetric trial wave function satisfying the boundary conditions, E(,F) is obtained. The method is exact because the only sources of error are variance and bias, both of which can be estimated and reduced, either by employing larger sample sizes, or by reconstructing the sampling procedure in ways that make greater use of our understanding of the problem (importance sampling). There are no physical or mathematical approximations other than the statistical one. The crux of the method is a sampling procedure which constructs the two sets of points in linear time (as a function of accuracy). Earlier methods were exponential in cost. The FGFMC method is successfully applied to a one dimensional problem and a nine dimensional problem, the results of which are presented here. These results demonstrate that this method can be successfully applied to small physical problems on medium-scale computing machines. The key to this success was the transformation of the problem from exponential to linear cost as a function of accuracy. The strong dependence on dimensionality, however, currently results in an exponential cost as a function of problem size, and this, until overcome, imposes a servere barrier to calculations on large systems.