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. We consider imaginary-time amplitudes of the one-dimensional harmonic oscillator , anharmonic oscillator , and double-well 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 .
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.