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arXiv:cond-mat/0004153 · Journal of Physics B Atomic Molecular and Optical Physics

Separable structure of many-body ground-state wavefunction

Yeong E Kim, Alexander L Zubarev

Published Sep 14, 2000• 3 citations
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Abstract

We have investigated a general structure of the ground-state wavefunction for the Schrödinger equation for N identical interacting particles (bosons or fermions) confined in a harmonic anisotropic trap in the limit of large N . It is shown that the ground-state wavefunction can be written in a separable form. As an example of its applications, this form is used to obtain the ground-state wavefunction describing collective dynamics for N trapped bosons interacting via contact forces.

DOI
10.1088/0953-4075/33/19/305

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We have investigated a general structure of the ground-state wavefunction for the Schrödinger equation for N identical interacting particles (bosons or fermions) confined in a harmonic anisotropic trap in the limit of large N . It is shown that the ground-state wavefunction can be written in a separable form. As an example of its applications, this form is used to obtain the ground-state wavefunction describing collective dynamics for N trapped bosons interacting via contact forces.
Exact solutions to many-body interacting systems of both bosonic and fermionic particles confined to harmonic potential in an arbitrary number of dimensions are given. Energy levels and their degeneracies for trapped identical particles interacting via harmonic forces are calculated. This specific form of the interaction allows for analytical solutions. The mutual interaction, attractive or repulsive, modifies significantly the properties of the considered system. For a large number of particles the interaction essentially results in a frequency shift. Statistical properties (e.g., microcanonical and grand canonical partition functions) as well as some illustrative, physically relevant examples are discussed. Our results give an unusual opportunity for further studies of interacting systems in the framework of the exactly soluble model.
We provide an efficient form to express the action of a many-body Hamiltonian of harmonically trapped interacting Fermi particles on wavefunctions built from paired states. The expression is suitable to numerically determine the ground state energy, regardless of the form of the two-body interaction. It takes advantage of the knowledge of the two-particle problem and the inherent properties of the matrix form of the many-body wavefunction. As an example, we evaluate the properties of a system composed of a balanced mixture of two families of fermions confined in a harmonic trap interacting through a short-range exponential potential. Numerical results for N ⩽ 10 and N = 35, 56, 84 and 165 particles of each family are reported. In the strong interacting regime corresponding to an infinite s-wave scattering length, our results give an upper bound to the Bertsch parameter for harmonically trapped systems ( E / E IFG ) 2 = 1 + β ⩽ 0.376 ± 0.008 with E the total energy and E IFG the energy for the analogous ideal Fermi gas. The influence of the harmonic trap and the interaction potential is exhibited in one and two-body correlation functions.
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.
We present a detailed description of the equivalent linear two-body method for the many-body problem, which is based on an approximate reduction of the many-body Schrödinger equation by the use of a variational principle. The method has been applied to the one-dimensional N -body problem with pair-wise contact interactions (McGurie-Yang N -body problem) and to the dilute Bose-Einstein condensation of atoms in harmonic traps at zero temperature for both positive and negative scattering lengths. The ground state energy and wavefunction for a dilute Bose gas are obtained analytically for large value of N and it is shown that the method gives excellent results in the large- N limit.
We consider the energetics and structural properties of a many-particle system in one dimension with pairwise contact interactions confined in a parabolic external potential. To render the problem analytically solvable, we use the harmonic approximation scheme at the level of the Hamiltonian. We investigate the scaling with particle number of the ground-state energies for systems consisting of identical bosons or fermions. We then proceed to focus on bosonic systems and make a detailed comparison with known exact results in the absence of the parabolic external trap for three-body systems. We also consider the thermodynamics of the harmonic model, which turns out to be similar for bosons and fermions due to the lack of degeneracy in one dimension.