Numerical Study of the Multiple Gaussian Representation of Time Dependent Wave Functions of a Morse Oscillator

摘要

The authors have studied the accuracy and the numerical stability of several methods for solving time dependent quantum problems in which a localized system is disturbed for a finite time. The methods assume that the wave function can be represented throughout the interaction time as a sum of Gaussian wave packets; the time evolution of the wave function is given by the evolution of the parameters in each packet. One of the methods propagates the packets independently and the other couples them by using a variational principle. Applications were made to calculate the time evolution of the ground state of a Morse oscillator and of a linear combination of the first four Morse states, and to calculate the vibrational excitation by coupling to a strong laser field. We have found that the coupled packets method provides accurate, numerically stable solutions for these problems, while the decoupled packet method fails for some of them. We have investigated the dependence of the results on the method used to fit the initial wave function to the sum of Gaussians and found that the non-linear, least square fitting is superior to other methods. A particularly important observation is that the results obtained by the coupled Gaussians method are improved as the number of packets is increased but the propagation equations become unstable if too many packets are used to fit the wave function. Methods for monitoring and avoiding such instablilites are discussed in the paper.