A numerical method for the accurate evaluation of a propagator in a grid space is presented. The algorithm is stable and accurate up to a microscopically significant long time ( = 100 ps for proton tunneling processes). The procedure is achieved by first initializing the small time propagator to be consistent with the grid space, and then a numerical matrix multiplication scheme is adopted to yield the propagator of a future time. The method is applied to study the motion of a proton in a bistable potential. The time invariance of the quantum thermal equilibrium density matrix and the energy eigenvalues of the Hamiltonian are used to judge the goodness of the resulting propagator.
Iwt)>=K(t)ln * (1) K(t) is a unitary operator and, for a time-independent Hamiltonian, can be explicitly written as
The representation of K(t) in a complete or overcomplete basis set { (vi) 1 is often referred to as a path integral, because the matrix elements { (pi) K( t) 1 pi)} can be formally written in terms of functional integrals [ 11. For a continuous real space ( 1 x) }, the representation of K( t) is the Feynman path integral [3]. Using coherent states as basis, which is an over-