Improved Feynman’s path integral method with a large time step: Formalism and applications

Abstract

We describe an efficient path integral scheme for calculating the propagator of an arbitrary quantum system, as well as that of a stochastic system in special cases where the Fokker–Planck equation obeys strict detailed balance. The basic idea is to split the respective Hamiltonian into two exactly solvable parts and then to employ a symmetric decomposition of the time evolution operator, which is exact up to a high order in the time step. The resulting single step propagator allows rather large time steps in a path integral and leads to convergence with fewer time slices. Because it involves no system-specific reference system, the algorithm is amenable to all known numerical schemes available for evaluating quantum path integrals. In this way one obtains a highly accurate method, which is simultaneously fast, stable, and computationally simple. Numerical applications to the real time quantum dynamics in a double well and to the stochastic dynamics of a bistable system coupled to a harmonic mode show our method to be superior over the approach developed by the Makri group in their quasiadiabatic propagator representation, to say nothing about the propagation scheme based on the standard Trotter splitting.