Kinetic energy functional for Fermi vapors in spherical harmonic confinement

Minguzzi, Anna; March, N. H.; Tosi, M. P.

doi:10.1016/s0375-9601(01)00125-6

Cited by 18 publications

(28 citation statements)

References 9 publications

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“…Howard, March, and Nieto [9], following previous work on two and three dimensions by Minguzzi, March, and Tosi [10], gave the generalization of Eq. …”

Section: Generalization Of Relativistic Difference Equation To D-dimementioning

confidence: 95%

Fermion particle density equations in relation to relativistic density functional theory

Howard¹,

March²

2004

Int J of Quantum Chemistry

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ABSTRACT:Relativistic density functional theory goes back at least to the work of Vallarta and Rosen. This represents the generalization of the Thomas-Fermi method to embody the kinetic energy of an electron of momentum p and rest mass m 0 as given by special relativity theory in the chemical potential Euler equation. Here we set up a relativistic density functional theory by heuristic arguments that start from well-established exact nonrelativistic differential equations for the Fermion density n(r) in model systems. We focus first on the differential equation of Lawes and March for one-dimensional harmonic confinement. Into their result we introduce the finiteness of the velocity of light c by replacing the Lawes-March differential equation for the Fermion density n(x) by a difference equation for the relativistic Fermion density in which the Compton wavelength h/m 0 c, with m 0 the Fermion rest mass, plays an important role. As we take the nonrelativistic limit c 3 ϱ, we regain the exact nonrelativistic result. Contact is then established with the Vallarta-Rosen treatment in the limit when a large number N of Fermions are harmonically confined. The remaining models investigated are two-electron in character. Relativistic difference equations to determine the density are again presented, for two such models.

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“…Howard, March, and Nieto [9], following previous work on two and three dimensions by Minguzzi, March, and Tosi [10], gave the generalization of Eq. …”

Section: Generalization Of Relativistic Difference Equation To D-dimementioning

confidence: 95%

Fermion particle density equations in relation to relativistic density functional theory

Howard¹,

March²

2004

Int J of Quantum Chemistry

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“…Differentiating both sides of (C.1), we can rewrite it as a third-order differential equation (3ODE) for ρ(r): This equation had been previously derived for IHOs with D = 1 in [45] and with D = 2 in [6]. Its form for D = 3 was surmised and numerically tested in [7], and general solutions for ρ(r) in the three-dimensional case were discussed in [10]. For D = 1 dimensional systems, we can expect the IDE (C.1) to be approximately valid, since the SLVT (81) is exact and the generalized LVT (78) numerically found to be well fulfilled everywhere.…”

Section: Appendix C (Integro-) Differential Equations For the Densitmentioning

confidence: 99%

“…Recent experimental success confining fermion gases in magnetic traps [2] has led to a renewed interest in theoretical studies of confined degenerate fermion systems at zero [3,4,5,6,7,8,9,10,11,12] and finite temperatures [13,14]. Quite some effort has been devoted in these articles to establish local virial theorems for various types of confining potentials.…”

Section: Introductionmentioning

confidence: 99%

Exact and asymptotic local virial theorems for finite fermionic systems

Brack

Koch

Murthy

et al. 2010

J. Phys. A: Math. Theor.

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Abstract.We investigate the particle and kinetic-energy densities for a system of N fermions confined in a potential V (r). In an earlier paper [J. Phys. A: Math. Gen. 36, 1111(2003], 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 (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 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 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 .

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“…1 It is of interest here, for this harmonic potential, to note that the ground-state density qðrÞ satisfies, as well as the Slater sum, a third-order ordinary differential equation for this case, which applies to any arbitrary number of closed shells [16,17]. Essentially the density qðrÞ is related to Sðr; bÞ by an inverse Laplace transform on b: The equation given by Minguzzi et al [16] reads…”

mentioning

confidence: 99%

Propagator and Slater sum in one‐body potential theory

March

Howard

2003

Physica Status Solidi (b)

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Dedicated to Professor Jozef T. Devreese on the occasion of his 65th birthday PACS 71.15.Mb For a one-body potential VðrÞ generating eigenfunctions w i ðrÞ and corresponding eigenvalues E i , the Feynman propagator Kðr; r 0 ; tÞ is simply related to the canonical density matrix Cðr; r 0 ; bÞ by b ! it: The diagonal element Sðr; r; bÞ of C is the so-called Slater sum of statistical mechanics. Differential equations for the Slater sum are first briefly reviewed, a quite general equation being available for a one-dimensional potential VðxÞ: This equation can be solved for a sech 2 potential, and some physical properties of interest such as the local density of states are derived by way of illustration. Then, the Coulomb potential ÀZe 2 =r is next considered, and it is shown that what is essentially the inverse Laplace transform of Sðr; bÞ=b can be calculated for an arbitrary number of closed shells. Blinder has earlier determined the Feynman propagator in terms of Whittaker functions and contact is here established with his work. The currently topical case of Fermion vapours which are harmonically confined is then treated, for both two and three dimensions. Finally, in an Appendix, a perturbation series for the Slater sum is briefly summarized, to all orders in the one-body potential VðrÞ: The corresponding kinetic energy is thereby accessible.

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Kinetic energy functional for Fermi vapors in spherical harmonic confinement

Cited by 18 publications

References 9 publications

Fermion particle density equations in relation to relativistic density functional theory

Fermion particle density equations in relation to relativistic density functional theory

Exact and asymptotic local virial theorems for finite fermionic systems

Propagator and Slater sum in one‐body potential theory

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