The Hellmann–Feynman theorem at finite temperature
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 no direct access to the thermal average of the operators themselves.