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Physical Review C

Three-dimensional, spherically symmetric, saturating model of anN-boson condensate

A. C. Merchant, M. P. Isidro Filho

Published Oct 1, 1988• 2 citations
Publisher

Abstract

The variational Hartree method is applied to a system of N bosons interacting via Skyrme-type attractive and repulsive forces. When the system condenses into its ground state, the wave function for a single boson, \ensuremath{\varphi}, may be described by means of the nonlinear, time-independent Schr\"odinger equation, ${\mathrm{\ensuremath{\nabla}}}^{2}$\ensuremath{\varphi}=${\mathrm{\ensuremath{\epsilon}}}_{1}$\ensuremath{\varphi}-A\ensuremath{\Vert}\ensuremath{\varphi}${\mathrm{\ensuremath{\Vert}}}^{2}$\ensuremath{\varphi}+B\ensuremath{\Vert}\ensuremath{\varphi}${\mathrm{\ensuremath{\Vert}}}^{4}$\ensuremath{\varphi}. In one dimension, this equation has a single, analytic, bound state solution and exhibits the property of saturation. In the physically more interesting case of three dimensions with spherical symmetry, no analytic solutions are known, so that a numerical solution must be resorted to, and this shows that saturation is again obtained. However, an analytic approximation to the three-dimensional wave function, which is very accurate for values of ${\ensuremath{\epsilon}}_{1}$B/${A}^{2}$>0.1, is deduced and studied. Approximate analytic expressions for the Hartree potential energy, kinetic energy, and mean square radius of the system are thereby derived, and applications to finite nuclei and infinite nuclear matter are considered.

DOI
10.1103/physrevc.38.1911

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The variational Hartree method is applied to a system of N bosons interacting via Skyrme-type attractive and repulsive forces. When the system condenses into its ground state, the wave function for a single boson, \ensuremath{\varphi}, may be described by means of the nonlinear, time-independent Schr"odinger equation, \ensuremath∇2{\mathrm{\ensuremath{\nabla}}}^{2}\ensuremath∇2\ensuremath{\varphi}=\ensuremathϵ1{\mathrm{\ensuremath{\epsilon}}}_{1}\ensuremathϵ1​\ensuremath{\varphi}-A\ensuremath{\Vert}\ensuremath{\varphi}\ensuremath∥2{\mathrm{\ensuremath{\Vert}}}^{2}\ensuremath∥2\ensuremath{\varphi}+B\ensuremath{\Vert}\ensuremath{\varphi}\ensuremath∥4{\mathrm{\ensuremath{\Vert}}}^{4}\ensuremath∥4\ensuremath{\varphi}. In one dimension, this equation has a single, analytic, bound state solution and exhibits the property of saturation. In the physically more interesting case of three dimensions with spherical symmetry, no analytic solutions are known, so that a numerical solution must be resorted to, and this shows that saturation is again obtained. However, an analytic approximation to the three-dimensional wave function, which is very accurate for values of \ensuremathϵ1{\ensuremath{\epsilon}}_{1}\ensuremathϵ1​B/A2{A}^{2}A2>0.1, is deduced and studied. Approximate analytic expressions for the Hartree potential energy, kinetic energy, and mean square radius of the system are thereby derived, and applications to finite nuclei and infinite nuclear matter are considered.
Application of the variational Hartree method to a system of N bosons interacting via Skyrme-type forces allows us to describe the single-boson wave function for the ground-state condensate in terms of the nonlinear time-independent Schr\"odinger equation, ${\mathit{\ensuremath{\nabla}}\phantom{\rule{0ex}{0ex}}}^{2}$\ensuremath{\varphi}=${\mathrm{\ensuremath{\varepsilon}}}_{1}$\ensuremath{\varphi}-A\ensuremath{\Vert}\ensuremath{\varphi}${\mathrm{\ensuremath{\Vert}}}^{2}$\ensuremath{\varphi}+B\ensuremath{\Vert}\ensuremath{\varphi}${\mathrm{\ensuremath{\Vert}}}^{4}$\ensuremath{\varphi}. In one dimension, not only is \ensuremath{\varphi} found by analytic integration of this equation, but the single-boson separation energy, the Hartree potential energy, the kinetic energy, and the mean-square radius are also all evaluated in closed form. The resulting expressions show that the model exhibits the property of saturation.
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Stationary solutions for the cubic nonlinear Schrödinger equation modeling Bose-Einstein condensates (BECs) confined in three spatial dimensions by general forms of a potential are studied through a perturbation method and also numerically. Note that we study both repulsive and attractive BECs under similar frameworks in order to deduce the effects of the potentials in each case. After outlining the general framework, solutions for a collection of specific confining potentials of physical relevance to experiments on BECs are provided in order to demonstrate the approach. We make several observations regarding the influence of the particular potentials on the behavior of the BECs in these cases, comparing and contrasting the qualitative behavior of the attractive and repulsive BECs for potentials of various strengths and forms. Finally, we consider the nonperturbative where the potential or the amplitude of the solutions is large, obtaining various qualitative results. When the kinetic energy term is small (relative to the nonlinearity and the confining potential), we recover the expected Thomas-Fermi approximation for the stationary solutions. Naturally, this also occurs in the large mass limit. Through all of these results, we are able to understand the qualitative behavior of spherical three-dimensional BECs in weak, intermediate, or strong confining potentials.
We investigate the behavior of a homogeneous one-dimensional Bose gas composed of NNN identical bosons experiencing attractive two-body and repulsive three-body interactions. Utilizing intermediate Hartree theory, we rigorously derive the homogeneous cubic-quintic nonlinear Schrödinger functional as the mean-field limit of the many-body Hamiltonian. Particular attention is given to the impact of the repulsive three-body term on ground state energy estimates and mass concentration properties
The wave functions of boson and fermion gases are known even when the particles have harmonic interactions. Here we generalize these results by solving exactly the N–body Schrödinger equation for potentials V that can be any function of the sum of the squares of the distances of the particles from one another in three dimensions. For the harmonic case that function is linear in r2. Explicit N–body solutions are given when U(r) = –2Mh–2V(r) = 𝛇r–1 – 𝛇2r–2. Here M is the sum of the masses and r2 = 1/2 M–2ΣmImJ(xI – xJ)2. For general U(r) the solution is given in terms of the one– or two–body problem with potential U(r) in three dimensions. The degeneracies of the levels are derived for distinguishable particles, for bosons of spin zero and for spin 1/2 fermions. The latter involve significant combinatorial analysis which may have application to the shell model of atomic nuclei. For large N the fermionic ground state gives the binding energy of a degenerate white dwarf star treated as a giant atom with an N–body wave function. The N–body forces involved in these extraordinary N–body problems are not the usual sums of two–body interactions, but nor are forces between quarks or molecules. Bose–Einstein condensation of particles in three dimensions interacting via these strange potentials can be treated by this method.
Abstract We demonstrate the possibility of creating a self-bound stable three-dimensional matter-wave spherical boson–fermion quantum ball in the presence of an attractive boson–fermion interaction and a small repulsive three-boson interaction. The two-boson interaction could be attractive or repulsive whereas the fermions are taken to be in a fully-paired super-fluid state in the Bardeen–Cooper–Schreifer (quasi-noninteracting weak-coupling) limit. We also include the Lee–Huang–Yang (LHY) correction to a repulsive bosonic interaction term. The repulsive three-boson interaction and the LHY correction can stop a global collapse while acting jointly or separately. The present study is based on a mean-field model, where the bosons are subject to a Gross–Pitaevskii (GP) Lagrangian functional and the fully-paired fermions are described by a Galilean-invariant density functional Lagrangian. The boson–fermion interaction is taken to be the mean-field Hartree interaction, quite similar to the interaction term in the GP equation. The study is illustrated by a variational and a numerical solution of the mean-field model for the boson–fermion 7 Li– 6 Li system.