Leading quantum correction to the classical free energy

The quantum free energy of a system governed by a standard Hamiltonian is larger than its classical counterpart. The lowest-order correction, first calculated by Wigner, is proportional to ℏ2 and involves the sum of the mean squared forces. We present an elementary derivation of this result by drawing upon the Zassenhaus formula, an operator-generalization for the main functional relation of the exponential map. Our approach highlights the central role of non-commutativity between kinetic and potential energy and is more direct than Wigner's original calculation, or even streamlined variations thereof found in modern textbooks. We illustrate the quality of the correction for the simple harmonic oscillator (analytically) and the purely quartic oscillator (numerically) in the limit of high temperature. We also demonstrate that the Wigner correction fails in situations with sufficiently rapidly changing potentials, for instance, the particle in a box.