On the Wigner-Kirkwood Expansion of the Free Energy and the Evaluation of the Quantum Correction

Abstract

The Wigner-Kirkwood expansion of the quantum correction to the classical free energy is generally said to be in powers of ℏ2 and only its first few terms are presented. In this work, we use the Bloch differential equation to obtain a general description of all terms in a dimensionless form. The first corrective term turns out to be proportional to the product of λ2/a2, where λ is the thermal de Broglie wavelength and a3 is the volume per particle, by an effective coupling constant. This dimensionless parameter can be used to assess the magnitude of the quantum correction. Using the one-component plasma as an illustration we highlight the importance of the magnitude of the potential on the quantum correction. The results presented are not formally new; the emphasis is placed on a simple and didactic presentation.