Quantum harmonic oscillators and thermalization

TL;DR

This paper investigates the thermalization process of a quantum harmonic oscillator by extending the ELR invariant method, deriving a thermodynamics-like first law, and analyzing energy changes during thermalization.

Contribution

It introduces a generalized ELR invariant approach to describe quantum harmonic oscillator thermalization and formulates a thermodynamics-like law for the process.

Findings

Thermalization modeled via ELR frequency change.

Derived energy as a function of entropy and its rate.

Established a thermodynamics analogy for quantum thermalization.

Abstract

We study a quantum harmonic oscillator undergoing thermalization. To describe the thermalization process, we generalize the Ermakov-Lewis-Riesenfeld (ELR) invariant method for the oscillator. After imposing appropriate conditions on the thermalization process, we introduce an ansatz equation that describes the time evolution effectively. We write down the first law for thermalization in the same form as that for ordinary thermodynamics. Here, the thermalization effect appears through a change of the ELR frequency. Finally, we obtain the oscillator's energy undergoing thermalization as a function of entropy and its time derivative.