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Path integral for the quartic oscillator: An accurate analytic formula for the partition function

Published 15 Dec 2023 in quant-ph, cond-mat.stat-mech, physics.atom-ph, and physics.chem-ph | (2312.09859v4)

Abstract: In this work an approximate analytic expression for the quantum partition function of the quartic oscillator described by the potential $V(x) = \frac{1}{2} \omega2 x2 + g x4$ is presented. Using a path integral formalism, the exact partition function is approximated by the partition function of a harmonic oscillator with an effective frequency depending both on the temperature and coupling constant $g$. By invoking a Principle of Minimal Sensitivity (PMS) of the path integral to the effective frequency, we derive a mathematically well-defined analytic formula for the partition function. Quite remarkably, the formula reproduces qualitatively and quantitatively the key features of the exact partition function. The free energy is accurate to a few percent over the entire range of temperatures and coupling strengths $g$. Both the harmonic ($g\rightarrow 0$) and classical (high-temperature) limits are exactly recovered. The divergence of the power series of the ground-state energy at weak coupling, characterized by a factorial growth of the perturbational energies, is reproduced as well as the functional form of the strong-coupling expansion along with accurate coefficients. Explicit accurate expressions for the ground- and first-excited state energies, $E_0(g)$ and $E_1(g)$ are also presented.

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