Time-Dependent Gaussian Solution for the Kostin Equation around Classical Trajectories

TL;DR

This paper derives time-dependent Gaussian solutions for the Kostin equation in dissipative quantum mechanics, showing classical-like motion of the wave packet's center and solving the free particle case analytically.

Contribution

It introduces a perturbation theory for the free Kostin equation and analyzes the wave packet's width and Wigner function in dissipative settings.

Findings

Center of mass follows classical dynamics with damping.

Wave packet width satisfies a non-conservative Pinney equation.

Analytic solutions for free Kostin equation are obtained.

Abstract

The structure of time-dependent Gaussian solutions for the Kostin equation in dissipative quantum mechanics is analyzed. Expanding the generic external potential near the center of mass of the wave packet, one conclude that: the center of mass follows the dynamics of a classical particle under the external potential and a damping proportional to the velocity; the width of the wave packet satisfy a non-conservative Pinney equation. An appropriate perturbation theory is developed for the free particle case, solving the long standing problem of finding analytic expressions for square integrable solutions of the free Kostin equation. The associated Wigner function is also studied.