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. We consider imaginarytime amplitudes of the onedimensional harmonic oscillator , anharmonic oscillator , and doublewell potential . For a general quantum system described by the Hamiltonian , the probability for a transition from an initial state to a final state in time is equal to , where is the transition amplitude. In a recently developed effective action approach, the amplitude is expressed in terms of the effective potential and a set of recursive relations allows systematic analytic derivation of terms in an 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.
