The Hartree solution for the ground state of N >>1 bosons placed within a spherical impenetrable shell is calculated. The bosons are assumed to interact through Coulomb repulsive forces and therefore occupy a thin volume adjacent to the shell wall. The resulting density profile is compared with its parallel Thomas-Fermi solution for fermions.
The Hartree solution for the ground state of N >>1 bosons placed within a spherical impenetrable shell is calculated. The bosons are assumed to interact through Coulomb repulsive forces and therefore occupy a thin volume adjacent to the shell wall. The resulting density profile is compared with its parallel Thomas-Fermi solution for fermions.
We consider the T=0 ground state of mixtures of bosons and fermions. Applying the Thomas-Fermi approximation we show that, depending on the strength of boson-boson and boson-fermion interactions, the Fermi gas may constitute a ``shell'' around or a ``core'' inside the Bose condensate, or even both.
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 ground-state properties of a single-component one-dimensional Coulomb gas are investigated. We use Bose-Fermi mapping for the ground-state wave function which permits solution of the Fermi sign problem in the following respects: (i) the nodal surface is known, permitting exact calculations; and (ii) evaluation of determinants is avoided, reducing the numerical complexity to that of a bosonic system and, thus, allowing simulation of a large number of fermions. Due to the mapping, the energy and local properties in one-dimensional Coulomb systems are exactly the same for Bose-Einstein and Fermi-Dirac statistics. The exact ground-state energy is calculated in homogeneous and trapped geometries using the diffusion Monte Carlo method. We show that in the low-density Wigner crystal limit an elementary low-lying excitation is a plasmon, which is to be contrasted with the high-density ideal Fermi gas/Tonks-Girardeau limit, where low-lying excitations are phonons. Exact density profiles are compared to the ones calculated within the local density approximation, which predicts a change from a semicircular to an inverted parabolic shape of the density profile as the value of the charge is increased.
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>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.
Many-particle systems pose commonly known computational challenges in quantum theory. The obstacles arise from the difficulty in finding sets of eigenvalues and eigenvectors of the underlying Hamiltonian while enforcing fermion or boson statistics, not to mention the prohibitive increase in the computational cost with the system’s size. The first obvious step in this direction is to elaborate the theory for Fermi or Bose gases without inter-particle interactions. The traditional approach to the work is with the ideal gases confined in a cubic container with impenetrable walls (in arbitrary dimensions). This approach allows one to find the particle’s spectra and compute all thermodynamic quantities of the confined gas. In the present work, we consider the gas confined in a spherical container (in other words, an infinitely deep spherical potential well in D dimensions), solving the corresponding Schrödinger equation using zero boundary conditions. We address the case of a finite number of particles N, either bosons or fermions, in the spherical potential box, as well as the thermodynamic limit, corresponding to N → ∞. Owing to Weyl’s relations, in the latter limit, the results do not depend on the shape of the box and thus approach the commonly known ones valid in the infinite space. Owing to the underlying SO(D) symmetry, we are dealing with particles carrying a well-defined angular momentum that, together with sorted energy eigenvalues, imparts a shell structure to the system.
The validity of the essential ingredients of the microscopic formulation of the interacting boson model: (i) truncation of the shell-model or fermion space and (ii) the procedures for mapping the truncated fermion space on to the boson space, is tested over various (spherical, deformed and transitional) regions both in the simplistic single j-shell, identical boson systems and the more realistic multi j-shell, proton-neutron systems depending on the feasibility of such calculations. In the situations where the truncation and mapping procedures fail to reproduce the exact shell-model results, it is demonstrated that renormalization and higher order mapping procedures cure the associated problems, respectively and provide excellent agreements. In addition, a Hamiltonian of the interacting boson-fermion model which is relevant for a study of odd mass nuclei is derived for a single j-shell case in the shell-model framework, which yields results identical to those in the shell-model for the monopole pairing interaction.