Fast Converging Path Integrals for Time-Dependent Potentials I: Recursive Calculation of Short-Time Expansion of the Propagator

In this and subsequent paper arXiv:1011.5185 we develop a recursive approach for calculating the short-time expansion of the propagator for a general quantum system in a time-dependent potential to orders that have not yet been accessible before. To this end the propagator is expressed in terms of a discretized effective potential, for which we derive and analytically solve a set of efficient recursion relations. Such a discretized effective potential can be used to substantially speed up numerical Monte Carlo simulations for path integrals, or to set up various analytic approximation techniques to study properties of quantum systems in time-dependent potentials. The analytically derived results are numerically verified by treating several simple models.

💡 Research Summary

In this work the authors present a systematic recursive scheme for constructing high‑order short‑time expansions of the quantum propagator in the presence of an explicitly time‑dependent potential V(q,t). Starting from the standard path‑integral representation, they rewrite the propagator for a single time slice Δτ as

K(q_f,t_f; q_i,t_i)= (2πħΔτ)^{‑1/2} exp