Towards many-dimensional real-time quantum theory for heavy-particle dynamics. I. Semiclassics in the Lagrange picture of classical phase flow

Abstract

We study a quantum theory in terms of action decomposed function (ADF), a class of quantum wave function, towards many-dimensional applications to quantum dynamics of heavy particles as in chemical reactions. The equation of motion for the complex-valued amplitude of ADF represents a coupling between the internal diffusive motion of a wave packet and dynamics of its group velocity in a hierarchical manner ascending from classical to purely quantum mechanics via semiclassical dynamics. We attempt to solve this equation of motion dividing it into two stages: a semiclassical level and beyond. In this paper, as the first stage, we develop a semiclassical approximation in the Lagrange picture of classical phase flow. In the Euler picture (as in the standard WKB picture), continuous integration of the stability matrix along the paths is required. By adopting the Lagrange picture, on the other hand, we represent the semiclassical amplitude in terms of what we call deviation determinant, which can be evaluated readily in many-dimensional systems. Numerical tests show that ADF reproduces quantum wave packets at each space-time point along classical paths very well. However, the ADF in this stage is not free of the semiclassical singularity. In other words, the wave functions diverge at turning points or caustics, depending on the initial conditions chosen. This divergence is known to take place at points where classical paths smoothly distributed in phase space have ``focuses'' in configuration space (or momentum space) and reflects an intrinsic relationship between quantum and classical mechanics. Therefore, it is by studying the mechanism of removing the singularity that the essential feature of quantum mechanics will be clarified. This aspect will be discussed in a companion paper [K. Takatsuka and S. Takahashi, Phys. Rev. A 89, 012109 (2014)] as the second stage of many-body quantum theory.