An Approach to Construct Wave Packets with Complete Classical-Quantum Correspondence in Non-relativistic Quantum Mechanics

Abstract

We introduce a method to construct wave packets with complete classical and quantum correspondence in one-dimensional non-relativistic quantum mechanics. First, we consider two similar oscillators with equal total energy. In classical domain, we can easily solve this model and obtain the trajectories in the space of variables. This picture in the quantum level is equivalent with a hyperbolic partial differential equation which gives us a freedom for choosing the initial wave function and its initial slope. By taking advantage of this freedom, we propose a method to choose an appropriate initial condition which is independent from the form of the oscillators. We then construct the wave packets for some cases and show that these wave packets closely follow the whole classical trajectories and peak on them. Moreover, we use de-Broglie Bohm interpretation of quantum mechanics to quantify this correspondence and show that the resulting Bohmian trajectories are also in complete agreement with their classical counterparts.

Counterpropagating Wavepacket Solutions of the Time-Dependent Schroedinger Equation for a Decaying Potential Field

“…In non-relativistic quantum theory attempts have been made to find wavepacket solutions for such potential fields as the Morse and the generalized Morse potentials [1,2], harmonic and pseudoharmonic potentials [3], the Woods-Saxon potentials [4,5], and in the presence of a Coulomb field [6]. In these cases a complete correspondence with the classical theory can be established in general [7], although for time-dependent potentials this correspondence is usually more complicated and in most cases not possible in the sense of Ehrenfest theorem [8]. A persistent feature of wavepackets, both in case of time independent as well as time dependent potentials, is their spatial spreading with time.…”