The derivation of the Liouville equation from the Schrodinger equation and its implications

TL;DR

This paper presents a rigorous derivation of the classical Liouville and Boltzmann equations from the Schrödinger equation, bridging quantum and classical mechanics with a novel, mathematically sound approach.

Contribution

It introduces a new method to derive classical kinetic equations directly from quantum mechanics, maintaining compatibility with standard quantum transition rate calculations.

Findings

Derived the Liouville equation from the Schrödinger equation.

Showed how to obtain the non-collision part of the Boltzmann equation.

Established a formal, rigorous derivation of the Boltzmann equation from quantum principles.

Abstract

We present a new way of deriving classical mechanics from quantum mechanics. A key feature of the method is its compatibility with the standard approach used to derive transition rates between quantum states due to interactions. We apply the developed method to derive the main formulas of physical kinetics. We observe that, through the Liouville equation, we can deduce the non-collision part of the Boltzmann equation, and that, through the matrix of transition rates, we can deduce the collision integral. As a final result of the manuscript, we derive the Boltzmann equation from the Schr\"odinger equation as a single piece of formal mathematical manipulation, without any non-rigorous plausible reasoning used to glue together its different parts.