Monte Carlo Method for Evaluating the Quantum Real Time Propagator

A new exact representation of the quantum propagator is derived in terms of semiclassical initial value representations. The resulting expression may be expanded in a series, of which the leading order term is the semiclassical one. Motion of a Gaussian wave packet on a symmetric double well potential is used to demonstrate numerical convergence of the series and the ability to compute each element in the series using Monte Carlo methods.

Figure 1

The real (a) and imaginary (b) parts of the correlation function of a Gaussian wave packet moving on a symmetric quartic double well potential. The solid line is the numerically exact quantum result; the solid line with points, the dot-dashed line, and the solid line with circles are the respective approximations $c_0(t)$, $c_1(t)$, and $c_2(t)$. For further details, see the text.

Figure 2

The convergence of the semiclassical IVR representation of the propagator. The function $Q_i(t)$ measures the average deviation of the $i$th approximant from the numerically exact answer. Note the increasing accuracy as one goes from the zeroth order approximation to the second order approximation. The solid line is $Q_0(t)$, the dash-dotted line is $Q_1(t)$, and the dotted solid line is $Q_2(t)$.

Figure 3

Quantum effects in the density autocorrelation function of a Gaussian wave packet evolving on a symmetric quartic double well potential. The solid line is the numerically exact quantum function, the dash-dotted line is the mixed quantum classical approximation, and the solid line with stars is the result obtained from $|c_2(t){|}^2$.