Time Evolution In Macroscopic Systems. I: Equations of Motion

TL;DR

This paper re-examines the time evolution of macroscopic systems by extending the equation of motion for the density matrix to account for both classical and quantum probabilities, highlighting the need for a more comprehensive approach.

Contribution

It introduces an extended equation of motion for the density matrix that incorporates classical probability changes alongside quantum dynamics.

Findings

Standard treatments often neglect classical probability changes.

The extended equation naturally includes classical probabilities within the density matrix framework.

A model demonstrates the necessity of this extension for accurate time evolution analysis.

Abstract

Time evolution of macroscopic systems is re-examined primarily through further analysis and extension of the equation of motion for the density matrix $\rho(t)$. Because $\rho$ contains both classical and quantum-mechanical probabilities it is necessary to account for changes in both in the presence of external influences, yet standard treatments tend to neglect the former. A model of time-dependent classical probabilities is presented to illustrate the required type of extension to the conventional time-evolution equation, and it is shown that such an extension is already contained in the definition of the density matrix.