The Hellmann-Feynman theorem at finite temperature

TL;DR

This paper derives the Hellmann-Feynman theorem at finite temperature and demonstrates its usefulness in calculating expectation values in quantum systems through three illustrative examples.

Contribution

It provides a simple derivation of the finite-temperature Hellmann-Feynman theorem and shows its practical application in quantum mechanics models.

Findings

The theorem is valid for the harmonic oscillator, Ising, and Lipkin models.

It enables calculation of operator expectation values from free energy.

The approach simplifies thermal expectation calculations in quantum systems.

Abstract

We present a simple derivation of the Hellmann-Feynman theorem at finite temperature. We illustrate its validity by considering three relevant examples which can be used in quantum mechanics lectures: the one-dimensional harmonic oscillator, the one-dimensional Ising model and the Lipkin model. We show that the Hellmann-Feynman theorem allows one to calculate expectation values of operators that appear in the Hamiltonian. This is particularly useful when the total free-energy is available, but there is not direct access to the thermal average of the operators themselves.