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In this work an approximate analytic expression for the quantum partitionfunction of the quartic oscillator described by the potential $V(x) =\frac{1}{2} \omega^2 x^2 + g x^4$ is presented. Using a path integralformalism, the exact partition function is approximated by the partitionfunction of a harmonic oscillator with an effective frequency depending both onthe temperature and coupling constant $g$. By invoking a Principle of MinimalSensitivity (PMS) of the path integral to the effective frequency, we derive amathematically well-defined analytic formula for the partition function. Quiteremarkably, the formula reproduces qualitatively and quantitatively the keyfeatures of the exact partition function. The free energy is accurate to a fewpercent over the entire range of temperatures and coupling strengths $g$. Boththe harmonic ($g\rightarrow 0$) and classical (high-temperature) limits areexactly recovered. The divergence of the power series of the ground-stateenergy at weak coupling, characterized by a factorial growth of theperturbational energies, is reproduced as well as the functional form of thestrong-coupling expansion along with accurate coefficients. Explicit accurateexpressions for the ground- and first-excited state energies, $E_0(g)$ and$E_1(g)$ are also presented.

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