Nonperturbative approach for a time-dependent quantum mechanical system

We present a variational method which uses a quartic exponential function as a trial wave function to describe time-dependent quantum mechanical systems. We introduce a new physical variable y which is appropriate to describe the shape of a wave packet, and calculate the effective action as a function of both the dispersion root<(q) over cap (2)> and y. The effective potential successfully describes the transition of the system from the false vacuum to the true vacuum. The present method well describes the time evolution of the wave function of the system for a short period for the quantum roll problem and describes the long-time evolution up to 75% accuracy. These are shown in comparison with direct numerical computations of the wave function. We briefly discuss the large N behavior of the present approximation.