On computer simulation of Feynman-Kac path-integrals

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:66

Article; Proceedings Paper

Korzeniowski, A

关键词: diffusion random walks;    harmonic oscillator;    hydrogen atom;   

DOI  :  10.1016/0377-0427(95)00170-0

来源: Elsevier

【 摘 要 】

Consider a path-integral E(x)exp{integral(0)(t) V(X(s))ds} f(X(t)) which is the solution to a diffusion version of the generalized Schrodinger's equation partial derivative u/partial derivative t = Hu, u(0, x) = f(x). Here H = A + V, where A is an infinitesimal generator of a strongly continuous Markov semigroup corresponding to the diffusion process {X(s), 0 less than or equal to s less than or equal to t, X(0) = x}. To see a connection to quantum mechanics, take A = 1/2 Delta and replace V by - V. Then one obtains (H) over bar = - H = -H = -1/2 Delta + V, which is a quantum mechanical Hamiltonian corresponding to a particle of mass 1 (in atomic units) subject to interaction with potential V. Path-integrals play a role in obtaining physical quantities such as ground state energies. This paper will be concerned with explanations of two approaches in the actual computer evaluations of path-integrals through simulations of the diffusion processes. The results will be presented by comparing, in concrete examples, the computational advantages or disadvantages depending on whether the diffusion process X(t) is ergodic or not.

【 授权许可】

Free