2023/12/15 by Michel Caffarel, Caffarel, Michel
Physics and Astronomy · #Atomic Physics (physics.atom-ph) #Chemical Physics (physics.chem-ph) #FOS: Physical sciences #Quantum Electrodynamics and Casimir Effect #Quantum Physics (quant-ph) #Quantum, superfluid, helium dynamics #Spectroscopy and Quantum Chemical Studies #Statistical Mechanics (cond-mat.stat-mech)
paper · pdf · doi:10.48550/arxiv.2312.09859
openalex publication_date 2023/12/15 · openalex created_date 2023/12/19 · openalex updated_date 2026/08/01
In this work an approximate analytic expression for the quantum partition function of the quartic oscillator described by the potential V(x) = (1)/(2) ω2 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→ 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, E0(g) and E1(g) are also presented.