On computer simulation of Feynman-Kac path-integrals (Q1919383)

Using the Feynman-Kac formula \(u(t, x)= E_x \exp\{- \int^t_0 V(X(s)) ds\}\) for the solution of a diffusion version of the generalized Schrödinger equation \(\partial u/\partial t= Hu\), \(u(0, x)= f(x)\), quantum mechanical systems are studied. The generalization involves the introduction of a new potential as a perturbation of \(V\), \(U(x)= V(x)- {1\over 2} {\Delta\varphi(x)\over \varphi(x)}\). Various choices of \(\varphi\) correspond to important sampling techniques. As examples the one-dimensional harmonic oscillator and hydrogen atom are analyzed by the Feynman-Kac formula and important (space and time averages) technique. By combining the important sampling techniques with parallel computing quantum mechanical particle systems may be treated.