Short-Time Expansion of Amplitudes for Time-Dependent Potentials
In this Demonstration we show that short-time quantum-mechanical transition amplitudes can be very accurately calculated if their expansion in the time of propagation is known to high orders. Here we consider imaginary-time amplitudes for the one-dimensional forced harmonic oscillator, with the time-dependent potential . For a quantum system in a time-dependent potential, the probability of a transition from an initial state to a final state in time is equal to , where is the transition amplitude. The evolution operator has to take into account explicit time-dependence of the potential. In a recently developed effective action approach, the amplitude is expressed in terms of an effective potential, and a set of recursive relations allows the systematic analytic derivation of the terms in the expansion of the effective potential in time t. The effective action thus obtained is characterized by a chosen level corresponding to the maximal order in its expansion. If level effective action is used, errors in the calculation of the transition amplitudes are proportional to .
Mathematica programs developed for symbolic derivation of higher-order effective actions, as well as the C programs developed and used in numerical simulations in the above papers, can be found at http://www.scl.rs/speedup.