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The Journal of Chemical Physics

Deriving uniform semiclassical approximations for one-dimensional fermionic systems

Raphael F. Ribeiro, Kieron Burke

Published May 16, 2018• 13 citations
Publisher

Abstract

A complete derivation is provided of the uniform semiclassical approximations to the particle and kinetic energy densities of N noninteracting bounded fermions in one dimension. The employed methodology allows the inclusion of non-perturbative quantum effects, including tunneling and quantum oscillations, via an infinite resummation of the Poisson summation formula. We explore the analytic behavior, physical meaning, and the relationship between the semiclassical uniform approximations for the fermionic kinetic energy and particle densities.

DOI
10.1063/1.5025628

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A complete derivation is provided of the uniform semiclassical approximations to the particle and kinetic energy densities of N noninteracting bounded fermions in one dimension. The employed methodology allows the inclusion of non-perturbative quantum effects, including tunneling and quantum oscillations, via an infinite resummation of the Poisson summation formula. We explore the analytic behavior, physical meaning, and the relationship between the semiclassical uniform approximations for the fermionic kinetic energy and particle densities.
We investigate the particle and kinetic-energy densities for NNN non-interacting fermions confined in a local potential. Using Gutzwiller's semi-classical Green function, we describe the oscillating parts of the densities in terms of closed non-periodic classical orbits. We derive universal relations between the oscillating parts of the densities for potentials with spherical symmetry in arbitrary dimensions, and a ``local virial theorem'' valid also for arbitrary non-integrable potentials. We give simple analytical formulae for the density oscillations in a one-dimensional potential.
Using an analytic approximation to the quantum momentum previously derived to calculate single particle reduced density matrices in the semiclassical approximation, we have been able to construct equations of motion with which to perform trajectory calculations. Information calculated along these trajectories is used to calculate single particle reduced density matrices for one-dimensional systems. Charge densities for N fermions in a harmonic oscillator potential calculated using this quantum momentum approximation are in excellent agreement with the exact quantum mechanical calculations.
We investigate the particle and kinetic-energy densities for a system of NNN fermions confined in a potential $V(\bfr)$. In an earlier paper [J. Phys. A: Math. Gen. {\bf 36}, 1111 (2003)], some exact and asymptotic relations involving the particle density and the kinetic-energy density locally, i.e. at any given point $\bfr$, were derived for isotropic harmonic oscillators in arbitrary dimensions. In this paper we show that these {\it local virial theorems} (LVT) also hold exactly for linear potentials in arbitrary dimensions and for the one-dimensional box. We also investigate the validity of these LVTs when they are applied to arbitrary smooth potentials. We formulate generalized LVTs that are supported by a semiclassical theory which relates the density oscillations to the closed non-periodic orbits of the classical system. We test the validity of these generalized theorems numerically for various local potentials. Although they formally are only valid asymptotically for large particle numbers NNN, we show that they practically are surprisingly accurate also for moderate values of NNN.
We investigate the particle and kinetic-energy densities for a system of N fermions confined in a potential V (r). In an earlier paper (Brack and Murthy 2003 J. Phys. A: Math. Gen. 36 1111), some exact and asymptotic relations involving the particle density and the kinetic-energy density locally, i.e. at any given point r, were derived for isotropic harmonic oscillators in arbitrary dimensions. In this paper, we show that these local virial theorems (LVTs) also hold exactly for linear potentials in arbitrary dimensions and for the one-dimensional box. We also investigate the validity of these LVTs when they are applied to arbitrary smooth potentials. We formulate generalized LVTs that are suggested by a semiclassical theory which relates the density oscillations to the closed non-periodic orbits of the classical system. We test the validity of these generalized theorems numerically for various local potentials. Although formally they are only valid asymptotically for large particle numbers N, we show that practically they are surprisingly accurate also for moderate values of N.