Liouville-von Neumann approach and time-dependent Gaussian approximation

TL;DR

This paper demonstrates that the Liouville-von Neumann approach reproduces the time-dependent Gaussian approximation in quantum systems, revealing that many dynamic features depend mainly on the effective potential rather than detailed time evolution.

Contribution

It establishes a connection between the Liouville-von Neumann formalism and the Gaussian approximation, providing new insights into quantum dynamics and effective actions.

Findings

Reproduction of Gaussian approximation via Liouville-von Neumann approach

Wave-function squeezing depends on the effective potential

Many dynamical aspects are independent of detailed time evolution

Abstract

We show that Liouville-von Neumann approach to quantum mechanical systems, which demands the existence of invariant operators, reproduces the time-dependent variational Gaussian approximation. We find the effective action of the time-dependent systems and show that many aspects of the dynamics are independent of the details of time evolution, e.g., the squeezing of the wave-function is determined by the effective potential of the final stage of time-evolution.