Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1501.07697

Thomas Fermi approximation and large-NNN quantum mechanics

Sukla Pal, Jayanta K. Bhattacharjee

Published Jan 30, 2015• 0 citations
arXivHTMLPDF

Abstract

We note that the Thomas Fermi limit of Gross Pitaevskii equation and $N>>1$ limit of quantum mechanics, where $N$ is the dimensionality of space, are based on the same point of view. We combine these two to produce a modified Thomas Fermi approximation which gives a very good account of the energy of the condensate in harmonic trap.

DOI
10.48550/arxiv.1501.07697

Related papers

On the Thomas-Fermi approximation for Bose condensates in traps

Peter Schuck, X. Viñas

Thomas-Fermi theory for Bose condesates in inhomogeneous traps is revisited. The phase-space distribution function in the Thomas-Fermi limit is $f_0(\bold{R},\bold{p})$ $α$ $δ(μ- H_{cl})$ where $H_{cl}$ is the classical counterpart of the self-consistent Gross-Pitaevskii Hamiltonian. No assumption on the large N-limit is introduced and, e.g the kinetic energy is found to be in good agreement with the quantal results even for low and intermediate particle numbers N. The attractive case yields conclusive results as well.

arXiv:cond-mat/9810157•Oct 14, 1998•0 citations

Thomas-Fermi approximation for Bose-Einstein condensates in traps

Peter Schuck, X. Viñas

Thomas-Fermi theory for Bose-Einstein condensates in inhomogeneous traps is revisited. The phase-space distribution function of the condensate in the Thomas-Fermi limit $(\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Elzxh}}0)$ is ${f}_{0}(\mathit{R},\mathit{p})\ensuremath{\propto}\ensuremath{\delta}(\ensuremath{\mu}\ensuremath{-}{H}_{\mathrm{cl}})$ where ${H}_{\mathrm{cl}}$ is the classical counterpart of the self-consistent Gross-Pitaevskii Hamiltonian. Starting from this distribution function the Thomas-Fermi kinetic energy is calculated for any number of particles. Good agreement between the Gross-Pitaevskii and Thomas-Fermi kinetic energies is found even for low and intermediate particle numbers N. Application of this Thomas-Fermi theory to the attractive case and to the calculation of the frequencies of the monopole and quadrupole excitations in the sum rule approach yields conclusive results as well. The difference with the usual Thomas-Fermi approach to the Bose-Einstein condensates (large-N limit) is discussed in detail.

Physical Review A•Mar 8, 2000•32 citations

Ground state of a trapped Bose-Einstein condensate in two dimensions: Beyond the mean-field approximation

Jens O. Andersen, Hårek Haugerud

We consider the ground state of a trapped Bose-Einstein condensate in two dimensions. In the mean-field approximation the ground-state density profile satisfies the Gross-Pitaevskii equation. We compute the leading quantum corrections to the density profile to second order in an expansion around the Thomas-Fermi limit. By summing the ladder diagrams, we obtain a modified equation that includes gradient corrections.

arXiv:cond-mat/0012101•Physical Review A•Feb 20, 2002•16 citations

Simple model for interactions and corrections to the Gross-Pitaevskii equation

Hagar Veksler, Shmuel Fishman, Wolfgang Ketterle

One of the assumptions leading to the Gross-Pitaevskii equation (GPE) is that the interaction between atom pairs can be written effectively as a $\ensuremath{\delta}$ function so that the interaction range of the particles is assumed to vanish. A simple model that takes into account the extension of the interparticle potential is introduced. The correction to the GPE predictions for the energy of a condensate confined by a harmonic trap in the Thomas-Fermi regime is estimated. Although it is found to be small, we believe that in some situations it can be measured using its dependance on the frequency of the confining trap. Due to the simplicity of the model, it may have a wide range of applications.

arXiv:1404.2586•Physical Review A•Aug 13, 2014•36 citations

Self-similar approximations for a trapped Bose-Einstein condensate

Vyacheslav I. Yukalov, Elizaveta P. Yukalova, Vanderlei Salvador Bagnato

An approximate solution to the Gross-Pitaevskii equation for Bose-Einstein condensate in a spherical harmonic trap is suggested, which is valid in the whole interval of the coupling parameter, correctly interpolating between weak-coupling and strong-coupling limits. This solution is shown to be more accurate than the optimized Gaussian approximation as well as the Thomas-Fermi approximation. The derivation of the solution is based on the self-similar approximation theory. The possibility of obtaining interpolation formulas in the case of nonspherical traps is discussed.

arXiv:cond-mat/0211318•Physical Review A•Aug 27, 2002•13 citations

On the Thomas-Fermi approximation in the bulk of trapped Bose-Einstein condensed gases

Gyula Bene, P. Szépfalusy

Corrections to the Thomas-Fermi-type solution of the Gross-Pitaevskii equation are inevitable in order to get correctly the frequencies of the low lying modes out of the Bogolyubov equations. These corrections are important in the bulk, too, thus the failure of the Thomas-Fermi approximation is not confined to the surface. We discuss this effect quantitatively and consider similar phenomena of spin fluctuations in Bose-Einstein condensed gases in an optical trap.

arXiv:cond-mat/9910139•Oct 8, 1999•0 citations
We note that the Thomas Fermi limit of Gross Pitaevskii equation and N>>1N>>1N>>1 limit of quantum mechanics, where NNN is the dimensionality of space, are based on the same point of view. We combine these two to produce a modified Thomas Fermi approximation which gives a very good account of the energy of the condensate in harmonic trap.
Thomas-Fermi theory for Bose condesates in inhomogeneous traps is revisited. The phase-space distribution function in the Thomas-Fermi limit is f0(R,p)f_0(\bold{R},\bold{p})f0​(R,p) ααα δ(μ−Hcl)δ(μ- H_{cl})δ(μ−Hcl​) where HclH_{cl}Hcl​ is the classical counterpart of the self-consistent Gross-Pitaevskii Hamiltonian. No assumption on the large N-limit is introduced and, e.g the kinetic energy is found to be in good agreement with the quantal results even for low and intermediate particle numbers N. The attractive case yields conclusive results as well.
Thomas-Fermi theory for Bose-Einstein condensates in inhomogeneous traps is revisited. The phase-space distribution function of the condensate in the Thomas-Fermi limit $(\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Elzxh}}0)$ is ${f}_{0}(\mathit{R},\mathit{p})\ensuremath{\propto}\ensuremath{\delta}(\ensuremath{\mu}\ensuremath{-}{H}_{\mathrm{cl}})$ where Hcl{H}_{\mathrm{cl}}Hcl​ is the classical counterpart of the self-consistent Gross-Pitaevskii Hamiltonian. Starting from this distribution function the Thomas-Fermi kinetic energy is calculated for any number of particles. Good agreement between the Gross-Pitaevskii and Thomas-Fermi kinetic energies is found even for low and intermediate particle numbers N. Application of this Thomas-Fermi theory to the attractive case and to the calculation of the frequencies of the monopole and quadrupole excitations in the sum rule approach yields conclusive results as well. The difference with the usual Thomas-Fermi approach to the Bose-Einstein condensates (large-N limit) is discussed in detail.
One of the assumptions leading to the Gross-Pitaevskii equation (GPE) is that the interaction between atom pairs can be written effectively as a $\ensuremath{\delta}$ function so that the interaction range of the particles is assumed to vanish. A simple model that takes into account the extension of the interparticle potential is introduced. The correction to the GPE predictions for the energy of a condensate confined by a harmonic trap in the Thomas-Fermi regime is estimated. Although it is found to be small, we believe that in some situations it can be measured using its dependance on the frequency of the confining trap. Due to the simplicity of the model, it may have a wide range of applications.
An approximate solution to the Gross-Pitaevskii equation for Bose-Einstein condensate in a spherical harmonic trap is suggested, which is valid in the whole interval of the coupling parameter, correctly interpolating between weak-coupling and strong-coupling limits. This solution is shown to be more accurate than the optimized Gaussian approximation as well as the Thomas-Fermi approximation. The derivation of the solution is based on the self-similar approximation theory. The possibility of obtaining interpolation formulas in the case of nonspherical traps is discussed.