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arXiv:0802.3312

Semi-classical Theory of Spatial Density Oscillations in Finite Fermion Systems

Jérôme Roccia, M. Brack

Published Feb 22, 2008• 0 citations
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Abstract

We investigate the particle and kinetic-energy densities for $N$ 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.

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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.
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.
For the system of noninteracting fermions in a one-body potential , the local virial theorems (LVT) are relations, at a given point in space, between this potential, kinetic energy and particle densities. It was recently shown (Brack et al 2010 J. Phys. A: Math. Theor. 43 255204) that for d-dimensional linear and also for isotropic harmonic oscillator potentials these LVTs are exactly satisfied. We present alternative and simple proofs of these theorems, by consideration of the canonical or Bloch density matrix and its relation to the kinetic energy density. The explicit analytical forms of the Bloch density matrix are used for the above-mentioned potentials to achieve the proofs. For the case of linear potential, we obtain a more general result for the so-called semilocal virial theorem, and for the harmonic oscillator potential case we derive a new relationship between the diagonal part of the canonical bloch density and the kinetic energy density.
We investigate the particle and kinetic energy densities of harmonically trapped fermion gases at zero temperature in arbitrary dimensions. We derive differential equations connecting these densities, which so far have been proved only in one or two dimensions, and give other interesting relations involving several densities or the particle density alone. We show that in the asymptotic limit of large particle numbers, the densities go over into the semi-classical Thomas-Fermi (TF) densities. Hereby the Fermi energy to be used in the TF densities is uniquely identified. We derive an analytical expansion for the remaining oscillating parts and obtain very simple closed forms for the leading-order oscillating densities. Finally, we show that the simple TF functional relation tau(TF)[rho] between kinetic and particle densities is also fulfilled for the asymptotic quantum densities tau(r) and rho(r) including their leading-order oscillating terms.
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