We briefly review a recently developed semiclassical theory1 for quantum oscillations in the spatial (particle and kinetic energy) densities of finite fermion systems and present some examples of its results. We then discuss the inclusion of correlations (finite temperatures, pairing correlations) in the semiclassical theory.
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.
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 a system of N 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 N, we show that they practically are surprisingly accurate also for moderate values of N.
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.