Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2602.13006

Effective classical potential for quantum statistical averages

Vijay Ganesh Sadhasivam, Stuart C. Althorpe, Venkat Kapil

quant-phphysics.chem-phSubmitted Feb 13, 2026• Announced Feb 16, 2026• 0 citations
arXivHTMLPDF

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 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.

Related papers

Effective classical potential for quantum statistical averages

Anonymous, Stuart C. Althorpe, Venkat Kapil

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.

Physical review. E•Jul 10, 2026•0 citations

Calculation of equilibrium averages with Feynman-Hibbs effective classical potentials and similar variational approximations

Gregory A. Voth

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.

Physical Review A•Oct 1, 1991•36 citations

Improving the variational path integral approach to the quantum double-well potential

Jing-Dong Bao, Hongyu Wang

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.

Journal of Physics A Mathematical and General•Aug 7, 2002•3 citations

Quantum Statistical Mechanics in Classical Phase Space. Test Results for Quantum Harmonic Oscillators

Phil Attard

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.

arXiv:1811.02032•Nov 5, 2018•0 citations

A Modification of the Quasi-Classical Approximation of Expectation Values for Harmonic Oscillator Stationary States

David K. Sahm

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.

Chinese Journal of Physics•Dec 1, 1999•0 citations

Approximate Functional Integral Methods in Statistical Mechanics. I. Moment Expansions

Armand Siegel, Terence Burke

In this paper, four distinct ideas are combined, which under a wide range of circumstances can give very rapidly converging series expansions for functional integrals. (1) Expansion of the functional being integrated in functional Taylor series. In the familiar case arising in quantum statistical mechanics, that of the Wiener integral of exp[−∫V(x(t))dt],V(x) being the perturbing potential, this is equivalent to expanding the characteristic functional of the probability functional of V(x(t)) in central moments of V(x(t)). The lowest-order term of the series is the approximation obtained by Feynman and Hibbs through a variational method. (2) Transfer of the harmonic term of the potential, when the functional integral is the quantum-statistical density matrix (Green's function of the Bloch equation), to the weighting function. This transforms the functional integral from a Wiener to an Uhlenbeck-Ornstein integral. The formal expressions for the terms of the expansion are somewhat more complicated, but they can be worked out, and the result is a great improvement in the speed of convergence of the series with decreasing temperature and/or decreasing relative magnitude of the anharmonic part of the potential. (3) ``Reservation of variables'' in the integration. This amounts to breaking the averaging process down into an average over subsets of the distributed random function (conditional average), followed by an averaging of these averages. Any step of this kind (it may be repeated within the subsets, etc.) gives an improvement of accuracy. (4) When the quantity being evaluated through functional integration is the partition function, the device introduced by Feynman and Hibbs, of interchanging the functional integration with the integration of the Green's function over the equated initial and final configuration-space points, may be combined with the above techniques. This eliminates one integration in the terms of the expansion and seems to improve accuracy at the same time. The general series obtained is correlated with more conventional operator techniques of quantum-mechanical perturbation theory, in order to answer the perennial question, does the path-integral method bring with it anything that could not be derived by other methods? It is in some sense a Feynman-Dyson expansion of the Green's function, but one that is further modified mathematically in a way characteristic only of the path-integral point of view, and which, moreover, improves its accuracy. It thus appears unlikely that the result is merely one of standard type disguised as a functional integral result. Sample numerical calculations are given to assay the accuracy of the methods, which are shown to compare very favorably with the traditional approximation of finite subdivision of the time interval.

Journal of Mathematical Physics•Nov 1, 1972•13 citations
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.
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.
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.
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.
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.
In this paper, four distinct ideas are combined, which under a wide range of circumstances can give very rapidly converging series expansions for functional integrals. (1) Expansion of the functional being integrated in functional Taylor series. In the familiar case arising in quantum statistical mechanics, that of the Wiener integral of exp[−∫V(x(t))dt],V(x) being the perturbing potential, this is equivalent to expanding the characteristic functional of the probability functional of V(x(t)) in central moments of V(x(t)). The lowest-order term of the series is the approximation obtained by Feynman and Hibbs through a variational method. (2) Transfer of the harmonic term of the potential, when the functional integral is the quantum-statistical density matrix (Green's function of the Bloch equation), to the weighting function. This transforms the functional integral from a Wiener to an Uhlenbeck-Ornstein integral. The formal expressions for the terms of the expansion are somewhat more complicated, but they can be worked out, and the result is a great improvement in the speed of convergence of the series with decreasing temperature and/or decreasing relative magnitude of the anharmonic part of the potential. (3) ``Reservation of variables'' in the integration. This amounts to breaking the averaging process down into an average over subsets of the distributed random function (conditional average), followed by an averaging of these averages. Any step of this kind (it may be repeated within the subsets, etc.) gives an improvement of accuracy. (4) When the quantity being evaluated through functional integration is the partition function, the device introduced by Feynman and Hibbs, of interchanging the functional integration with the integration of the Green's function over the equated initial and final configuration-space points, may be combined with the above techniques. This eliminates one integration in the terms of the expansion and seems to improve accuracy at the same time. The general series obtained is correlated with more conventional operator techniques of quantum-mechanical perturbation theory, in order to answer the perennial question, does the path-integral method bring with it anything that could not be derived by other methods? It is in some sense a Feynman-Dyson expansion of the Green's function, but one that is further modified mathematically in a way characteristic only of the path-integral point of view, and which, moreover, improves its accuracy. It thus appears unlikely that the result is merely one of standard type disguised as a functional integral result. Sample numerical calculations are given to assay the accuracy of the methods, which are shown to compare very favorably with the traditional approximation of finite subdivision of the time interval.