An approach to construct wave packets with complete classical-quantum correspondence in non-relativistic quantum mechanics

TL;DR

This paper presents a method to construct wave packets in non-relativistic quantum mechanics that exhibit complete correspondence with classical trajectories, utilizing initial conditions and de-Broglie Bohm interpretation for validation.

Contribution

It introduces a novel approach to select initial wave functions that ensure wave packets follow classical trajectories in quantum systems.

Findings

Wave packets closely follow classical trajectories.

Bohmian trajectories agree with classical paths.

Method is independent of oscillator specifics.

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 a complete agreement with their classical counterparts.