In this Demonstration we show that shorttime quantummechanical transition amplitudes can be very accurately calculated if their expansion in the time of propagation is known to high orders. Here we consider imaginarytime amplitudes for the onedimensional forced harmonic oscillator, with the timedependent potential . For a quantum system in a timedependent 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 timedependence 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 .
The effective action approach was originally developed in a series of papers: The extension of the approach to manybody systems is presented in the following papers: This method has been successfully applied to numerical studies of properties of various quantum systems: Recently, this approach has been extended to systems in timedependent potentials: Mathematica programs developed for symbolic derivation of higherorder 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.
