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The Journal of Chemical Physics

One-Dimensional Fermion Density Formula

Herbert Payne

Published Apr 15, 1963• 3 citations
Publisher

Abstract

An expression for the sum of the position-probability densities for a one-dimensional system of fermions in the ground state is derived from wave mechanics. Its accuracy in application to three different potential-energy functions is shown. For the square well of infinite height it preserves the sinusoidal contributions which disappear in the corresponding Fermi—Thomas treatment. Where the potential energy exceeds the Fermi level, it is properly characterized.

DOI
10.1063/1.1733911

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An expression for the sum of the position-probability densities for a one-dimensional system of fermions in the ground state is derived from wave mechanics. Its accuracy in application to three different potential-energy functions is shown. For the square well of infinite height it preserves the sinusoidal contributions which disappear in the corresponding Fermi—Thomas treatment. Where the potential energy exceeds the Fermi level, it is properly characterized.
Using an integral expression which is based on the use of discretized path integrals, the one-particle density matrix is evaluated for individual fermions in a one- and three-dimensional harmonic well. In this regard, the present paper improves and extends work previously done by Handler. By deriving a closed form expression for the path integral with finite discretization, the density is expressed as a one-dimensional integral that can be evaluated numerically. In addition, the trace of this density matrix, which corresponds to the number of particles present as a function of the Fermi energy, is evaluated in closed form. For finite discretization, the function is continuous in the Fermi energy, but approaches the correct discontinuous limit as the discretization becomes large. Oscillations are observed at the discontinuities analogous to the Gibbs phenomenon of Fourier analysis. Calculations of the density show a marked improvement in the classically allowed region when the discretization is only one level beyond the simple Thomas–Fermi theory. The density also penetrates into the classically forbidden region as the discretization is increased, but the presence of negative oscillations requires a higher level of discretization for a proper description in this region.
Frequency sum rules are derived in extended quantum systems of non relativistic fermions from a minimal set of assumptions on dynamics in infinite volume, for ground and thermal states invariant under space translations or a lattice subgroup. For the jellium Coulomb model, they imply the one point result for the plasmon energy spectrum in the zero momentum limit. In general, the density waves energy spectrum is shown to converge, in the limit of large wavelenght, to a point measure at zero frequency, for any number of fermion fields and potentials with integrable second derivatives. For low momentum, ~ k^2 for potentials V with r^2 d_i d_j V integrable, ~ k^{a-d+2} for potentials decaying at infinity as 1/r^a, d-2 < a < d, d the space dimensions. For one component models with short range interactions, the fourth momentum of the frequency is expressed, at lowest order in k, purely in terms of the three point correlation function of the density.
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.
The density and correlations are studied for a system of fermions moving in an average harmonic oscillator potential in one and three dimensions when the number of fermions are large. The technique is based on the use of Mehler's formula for the product of two Hermite functions and con-. tour integrations. Wigner transform is then calculated using this technique. It is shown that it becomes a step function when number of fermions become large. By integrating out momenta, an expression for the density of fermions is derived. Two-particle correlation function is studied in detail. Dyson's type of correlations for a system of nucleons is studied. The connection between one dimensional system of fermions and eigenvalues of a random Hermitean Hamiltonian is shown. The Coulomb energy difference of isobars is studied using nucleon density and correlations.
The methods employed in the previous papers of this series for the ground-state energy of a normal Fermion system are extended to finite temperatures. The basic assumption is that the density matrix of the actual system in thermodynamic equilibrium can be derived by adiabatic transformation from the density matrix of system of noninteracting particles at the same temperature, volume, and chemical potential. However, the spectrum of the noninteracting system must be dependent on the intensive parameters. The self-consistency of the scheme is exhibited by demonstrating that the effective potential characterizing this system can be related to the self-energy operator of the actual system. Two equivalent versions of the theory are derived. In one of these the basic equations are precisely those of the Landau theory of Fermi liquids with the difference, however, that all quantities are given explicitly from first principles.