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Physical Review A

Hartree solutions for the self-Yukawian boson sphere

Manuel C. Membrado, Amalio F. Pacheco, Jaime Sañudo A

Published Apr 1, 1989• 99 citations
Publisher

Abstract

The ground state of an N-boson assembly interacting through two-body attractive Yukawa forces is analyzed in the Hartree approximation. The solution, which is not universal, can be adequately characterized by a unique parameter \ensuremath{\mu}\ifmmode \tilde{}\else \~{}\fi{} proportional to \ensuremath{\mu}${N}^{\mathrm{\ensuremath{-}}1}$, where ${\ensuremath{\mu}}^{\mathrm{\ensuremath{-}}1}$ is the range of the force. It is found that for \ensuremath{\mu}\ifmmode \tilde{}\else \~{}\fi{} larger than a critical value, there is no bound solution for the system. Our solutions are shown to rigorously fulfill the virial theorem.

DOI
10.1103/physreva.39.4207

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The ground state of an N-boson assembly interacting through two-body attractive Yukawa forces is analyzed in the Hartree approximation. The solution, which is not universal, can be adequately characterized by a unique parameter \ensuremath{\mu}\ifmmode \tilde{}\else ~{}\fi{} proportional to \ensuremath{\mu}N\ensuremath−1{N}^{\mathrm{\ensuremath{-}}1}N\ensuremath−1, where \ensuremathμ\ensuremath−1{\ensuremath{\mu}}^{\mathrm{\ensuremath{-}}1}\ensuremathμ\ensuremath−1 is the range of the force. It is found that for \ensuremath{\mu}\ifmmode \tilde{}\else ~{}\fi{} larger than a critical value, there is no bound solution for the system. Our solutions are shown to rigorously fulfill the virial theorem.
We consider a system of N nonrelativistic bosons in two dimensions, interacting weakly via a short-range attractive potential. We show that for N large, but below some critical value, the properties of the N-boson bound state are universal. In particular, the ratio of the binding energies of (N+1)- and N-boson systems, B(N+1)/B(N), approaches a finite limit, approximately 8.567, at large N. We also confirm previous results that the three-body system has exactly two bound states. We find for the ground state B(0)(3)=16.522 688(1)B2 and for the excited state B(1)(3)=1.270 409 1(1)B2.
The exact solution for the ground state of the quantum one-dimensional NNN-boson problem with attractive $\ensuremath{\delta}$-function two-body potentials is compared with the (exact) self-consistent solution of the corresponding variational Hartree problem.
Systems of N(>>1) fermions (or bosons) interacting via either repulsive long-ranged forces (e.g., Coulomb) or short-ranged forces (e.g., repulsive barrier plus attractive Yukawa tail) are clearly of interest in problems from nuclear physics to astrophysics. By comparing expectation values for determinants (or permanents) of plane-wave orbitals in the lowest kinetic-energy state, and Wannier-like orbitals localized in the points of a perfect lattice, the conditions on the coupling and particle density for the existence of energetically-preferred, spatially-periodic, nontrivial (i.e.. nonplane-wave) solutions to the Hartree - Fock equations were deduced. That an electron gas in a crystalline confirmation, the Wigner solid,'' has lower energy than the fluid (plane-waves) configuration at low densities (or all densities, if the charge is increased indefinitely) can be shown to imply the existence of lower-energy, spatially-periodic HF solutions. Analogously, for the short-ranged force case cited before one can show very simply that a determinant (or permanent) of nonover-lapping Wannier orbitals gives lower energy for high enough repulsive barrier in an intermediate range of densities (or all densities if the barrier height is increased indefinitely to the hard core limit) in three dimensions. Lower-energy non-trivial solutions therefore emerge under these conditions, giving a finite energy per particle evenmore » in the hard core limit. These results differ qualitatively from the one-dimensional case first studied by Overhauser and agree with those conjectured for 3-dimensions by several subsequent authors. The inclusion of attractive forces can, moreover, provide a condensed (negative energy) spatiallyperiodic state lower than the trivial state. One can furthermore show that there will exist a whole family of aperiodic singledeterminantal states of higher (but finite) energy than the perfect hep lattice state but lower still than the trivial state. (auth)« less
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/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.
We study a system composed of N identical bosons interacting in three dimensions via attractive Yukawa pair potentials V(r)=-${\mathit{e}}^{\mathrm{\ensuremath{-}}\ensuremath{\lambda}\mathit{r}}$/r. Four approaches to the problem are related: the ``equivalent two-body method,'' translation-invariant Gaussian trial functions, collective field theory, and Hartree trial functions. Upper and lower energy bounds are given and are compared to some recent results which were obtained independently by an optimized Hartree method [M. Membrado, F. Pacheco, and J. Sa\~nudo, Phys. Rev. A 39, 4207 (1989)]. In the pure gravitational limit \ensuremath{\lambda}\ensuremath{\rightarrow}0 the N-body energy is determined for all N and all values of the coupling with an error of less than 7.2%.
The problem of N bosons interacting with each other via repulsive delta-function interactions and with an external, attractive, delta-function potential is solved, within the Hartree approximation, exactly. If the interparticle interactions are above a certain value, there is no bound state. Thus the bound state does not just expand to compensate for the increase in the repulsive Hartree potential. Moreover as the interaction strength is increased to that value, the ground-state wavefunction develops a pole at the position of the attractive potential.