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Physical Review

Nature of the Quantum Corrections to the Statistical Model

Herbert Payne

Published Dec 15, 1963• 16 citations
Publisher

Abstract

The scheme of correcting the Fermi-Thomas particle density formula by a power series in $\ensuremath{\hbar}$, procedures for which have been proposed by a number of authors, is examined through its application to a one-dimensional linear potential, which yields an analytical expression for the exact wave mechanical density for comparison. It is concluded that this is an asymptotic series, valid only where the particle density is large. Furthermore, terms of an oscillatory nature, which may very well transcend the so-called quantum corrections, are missing. A reason for this is offered.

DOI
10.1103/physrev.132.2544

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math-phCross-listedarXiv:2610.30887•Oct 23, 2026
The scheme of correcting the Fermi-Thomas particle density formula by a power series in \ensuremathℏ\ensuremath{\hbar}\ensuremathℏ, procedures for which have been proposed by a number of authors, is examined through its application to a one-dimensional linear potential, which yields an analytical expression for the exact wave mechanical density for comparison. It is concluded that this is an asymptotic series, valid only where the particle density is large. Furthermore, terms of an oscillatory nature, which may very well transcend the so-called quantum corrections, are missing. A reason for this is offered.
Quantum corrections are obtained for the single-particle density matrix in a semiclassical ensemble where the distribution is unrestricted. A form for the density matrix containing explicitly a function of the wave-mechanical Hamiltonian operator is devised; a formalism is then developed to decompose this operator function into effectively classical and nonclassical parts. The classical part corresponds to the semiclassical density matrix and the quantum corrections are obtained from the nonclassical terms.The quantum-corrected density matrix for a spherically symmetrical and a one-dimensional system have been evaluated. The densities for a linear harmonic oscillator and a Coulomb potential for a Fermi-Dirac distribution have also been examined in some detail.
Any proposed approximation to the ground-state kinetic energy of a system of noninteracting fermions in terms of the particle density \ensuremath{\rho}(r) can be used to estimate the potential field v(r) that will give rise to a given \ensuremath{\rho}(r), or the \ensuremath{\rho}(r) that will result from placing a given number of particles in a given v(r). Comparison with exact quantum-mechanical calculations thus gives two possible types of tests for any proposed kinetic-energy functional. This paper reports such tests for a recently proposed nonlocal functional applicable to one-dimensional systems, comparing its predictions for several simple problems not only with the correct answers but also with the predictions of Thomas-Fermi and Thomas-Fermi-Weizs\"acker functionals, to which the new functional proves considerably superior. The comparisons yield useful insights on the virtues and defects of the new functional, and on the directions in which improvements should be sought. Particular attention is devoted to Friedel oscillations and shell structure. It is shown that functionals of the Thomas-Fermi-Weizs\"acker type can never predict multiple maxima in \ensuremath{\rho} if v has only a single minimum and no maxima; the new functional does not have this defect, though it yields density oscillations rather weaker than the exact quantum-mechanical calculations. A natural general inference from the present tests is that any satisfactory kinetic-energy functional must (as ours does) replace the entire Thomas-Fermi term by a nonlocal expression.
A systematic method is presented for deriving the Thomas-Fermi equation for an atom and the quantum corrections from the many-body description. The novel feature of the method is that it does not require any a priori assumptions about the assignment of electrons to fully occupied single-particle states or about the distribution of electrons in phase space, but shows instead that the distribution which is usually assumed, or derived from the assumption of fully occupied single particle states, is a direct consequence of specifying that the many particle system is in its ground state. The procedure used in the derivation is the expansion of the mixed position-momentum representation of the Green's function in a series of powers of $\ensuremath{\hbar}$. The lowest order term is found to correspond with the Thomas-Fermi density. The form of the higher order terms, which are to be considered as corrections to zeroth order term, depends on the approximations made in the many-body equations for obtaining the Green's function. This paper deals only with the Hartree-Fock approximation, but the methods presented here allow generalization to other approximations which can include correlation effects.
Knowledge of a series expansion of the equation of state provides a deep insight into the physical nature of a quantum system. Starting from a generic ``perturbative'' equation of state of a homogeneous ultracold gas we make predictions for the properties of the gas in the presence of harmonic confinement. The local density approximation is used to obtain the chemical potential, total and release energies, Thomas-Fermi size, and density profile of a trapped system in three-, two-, and one-dimensional geometries. The frequencies of the lowest breathing modes are calculated using scaling and sum-rule approaches and could be used in an experiment as a high-precision tool for obtaining the expansion terms of the equation of state. The derived formalism is applied to dilute Bose and Fermi gases in different dimensions and to integrable one-dimensional models. The physical meaning of the expansion terms in a number of systems is discussed.
We first prove a sharp Lieb-Thirring inequality with semi-classical constant and gradient correction in one dimension. In the periodic setting, the corresponding inequality also includes a parity dependent shell-correction term, which appears to be new. Moreover, we also characterize the optimizers of these inequalities. In particular, the optimizing density profiles form a two-dimensional manifold, which we call here the Poincaré density manifold. We then apply these results to study the Thomas--Fermi--von Weizsäcker (TFW) approximation of the kinetic energy in Density Functional Theory. More precisely, we first obtain an explicit and non-asymptotic bound on the global and local error made by the TFW approximation. These bounds are controlled by the distance of the normalized density profile from the Poincaré density manifold times the number of particles. In particular, these bounds show that the TFW approximation is exact on the Poincaré manifold. We then show that, in the periodic setting, the distance between the Kohn-Sham potential and the inverse TFW potential in the natural dual Sobolev space can also be controlled by explicit density dependent constants times the distance to the Poincaré manifold. As a corollary, we prove that for a fixed density profile, the Kohn-Sham potential converges towards the inverse Thomas--Fermi potential in the high-density limit N→∞N \rightarrow \inftyN→∞ with asymptotic rate of order N−2N^{-2}N−2. This rate is shown to be optimal.