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Physical review. E

Effective classical potential for quantum statistical averages

Anonymous, Stuart C. Althorpe, Venkat Kapil

Published Jul 10, 2026• 0 citations
Publisher

Abstract

We present an effective potential that allows quantum thermal expectation values of a position-dependent observable to be estimated as a classical ensemble average of the corresponding function. We follow the approach of Feynman and Hibbs but perform the mean-field treatment of quantum fluctuations about the path starting point rather than the path centroid. Furthermore, rather than performing a full variational optimization of the potential, we explore approximate functional forms that yield numerical robustness. The resulting closed-form potential is exact in the classical and harmonic limits; benchmarks against exact position distributions for one-dimensional quartic, Morse, and double-well potentials show good agreement for potentials with harmonic support.

DOI
10.1103/zr37-gtc5

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We present an effective potential that allows quantum thermal expectation values of a position-dependent observable to be estimated as a classical ensemble average of the corresponding function. We follow the approach of Feynman and Hibbs but perform the mean-field treatment of quantum fluctuations about the path starting point rather than the path centroid. Furthermore, rather than performing a full variational optimization of the potential, we explore approximate functional forms that yield numerical robustness. The resulting closed-form potential is exact in the classical and harmonic limits; benchmarks against exact position distributions for one-dimensional quartic, Morse, and double-well potentials show good agreement for potentials with harmonic support.
We present an effective potential that allows quantum thermal expectation values of a position-dependent observable to be estimated as a classical ensemble average of the corresponding function. We follow the approach of Feynman and Hibbs, but perform the mean-field treatment of quantum fluctuations about the path starting point rather than the path centroid. Furthermore, rather than performing a full variational optimization of the potential, we explore approximate functional forms that yield a numerical robustness. The resulting closed-form potential is exact in the classical and harmonic limits; benchmarks against exact position distributions for one-dimensional quartic, Morse, and double-well potentials, show good agreement for potentials with harmonic support.
A number of years ago, Feynman and Hibbs derived an insightful expression for a quasiclassical effective potential in order to estimate the free energy for a quantum-mechanical system that obeys Boltzmann statistics. In the present paper, it is demonstrated how these ideas can be extended to calculate an approximate equilibrium average of any general coordinate-dependent operator. The central feature of this analysis is the replacement of the operator by an effective Gaussian-averaged quasiclassical function.
An improved variational path integral approach is developed and applied to the quantum double-well potential, in which part of the quartic term of the potential is included in the trial action. The expression of the effective classical potential (ECP) under a non-Gaussian expectation is obtained. Here the frequency and fourth-order derivative of the potential are treated as two variational parameters, determined by the minimization of the ECP at each point. We calculate the ECP, the free energy and the level splitting of a symmetrical double-well potential. It is shown that the present results are better than those of the Feynman–Kleinert Gaussian variational method.
I introduce, for one degree-of-freedom harmonic oscillator stationary states, a modi-fication of the quasi-classical approximation for expectation values of observables which are polynomial functions of the position and momentum. After time averaging a suit-able function determined by the observable, instead of evaluating the average at the energy eigenvalue, if one then averages over the energy with a certain gamma distribu-tion, one gets more accurate expectation values for large quantum numbers. I consider Monte Carlo evaluations of the averages with the random phase points chosen accord-ing to the respective distributions. Although the Monte Carlo error is greater for the modification than for the quasi-classical approximation when the same number of phase points is used, it is less than that for a Monte Carlo evaluation of a certain exact ex-pression in which the same time averaged function is averaged over the energy with Wigners quasiprobability distribution. PACS. 03.65.Sq- Semiclassical theories and applications. PACS. 02.7O.Lq- Monte Carlo and statistical methods. I.
The von Neumann trace form of quantum statistical mechanics is transformed to an integral over classical phase space. Formally exact expressions for the resultant position-momentum commutation function are given. A loop expansion for wave function symmetrization is also given. The method is tested for quantum harmonic oscillators. For both the boson and fermion cases, the grand potential and the average energy obtained by numerical quadrature over classical phase space are shown to agree with the known analytic results. A mean field approximation is given which is suitable for condensed matter, and which allows the quantum statistical mechanics of interacting particles to be obtained in classical phase space.
The classical Wigner model is one way to approximate the quantum dynamics of atomic nuclei. Here, a new method is presented for sampling the initial quantum mechanical distribution that is required in the classical Wigner model. The new method is tested for the position, position-squared, momentum, and momentum-squared autocorrelation functions for a one-dimensional quartic oscillator and double well potential as well as a quartic oscillator coupled to harmonic baths of different sizes. Two versions of the new method are tested and shown to possibly be useful. Both versions always converge toward the classical Wigner limit. For the one-dimensional cases, some results that are essentially converged to the classical Wigner limit are acquired and others are not far off. For the multi-dimensional systems, the convergence is slower, but approximating the sampling of the harmonic bath with classical mechanics was found to greatly improve the numerical performance. For the double well, the new method is noticeably better than the Feynman-Kleinert linearized path integral method at reproducing the exact classical Wigner results, but they are equally good at reproducing exact quantum mechanics. The new method is suggested as being interesting for future tests on other correlation functions and systems.