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Physical Review A

Soluble model of many interacting quantum particles in a trap

Magdalena A. Załuska–Kotur, Mariusz Gajda, Arkadiusz Orłowski, Jan Mostowski

Published Feb 16, 2000• 48 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.61.033613

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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 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.
We consider a small number of identical bosons trapped in a two-dimensional isotropic harmonic potential and also the NNN-boson system when it is feasible. The atom-atom interaction is modeled by means of a finite-range Gaussian interaction. The spectral properties of the system are scrutinized and, in particular, we derive analytic expressions for the degeneracies and their breaking for the lower-energy states at small but finite interactions. We demonstrate that the degeneracy of the low-energy states is independent of the number of particles in the noninteracting limit and also for sufficiently weak interactions. In the strongly interacting regime, we show how the many-body wave function develops holes whenever two particles are at the same position in space to avoid the interaction, a mechanism reminiscent of the Tonks-Girardeau gas in one dimension. The evolution of the system as the interaction is increased is studied by means of the density profiles, pair correlations, and fragmentation of the ground state for N=2N=2N=2, 3, and 4 bosons.
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.
The exactly solvable model of two indistinguishable quantum particles (bosons or fermions) confined in a one-dimensional harmonic trap and interacting via finite-range soft-core interaction is presented and many properties of the system are examined. Particularly, it is shown that independently on the potential range, in the strong interaction limit bosonic and fermionic solutions become degenerate. For sufficiently large ranges a specific crystallization appears in the system. The results are compared to predictions of the celebrated Busch et al. model and those obtained in the Tonks-Girardeau limit. The assumed inter-particle potential is very similar to the potential between ultra-cold dressed Rydberg atoms. Therefore, the model can be examined experimentally.
The authors study a one dimensional model system of fermions which proves to be exactly soluble in the thermodynamic limit. It is shown that the thermodynamic functions for the model system are the same as for a system of noninteracting fermions in a harmonic oscillator well. As a special case of the model system, with a particular choice for the coupling constant, it reduces to a particular two-species system of fermions interacting via pairwise harmonic forces. For this case of the two-species system the free energy per particle, the pressure the specific heat at constant volume, and the isothermal compressibility are calculated in the canonical ensemble in closed form. It is shown that this system does not undergo a phase transition.
We consider systems of a small number of interacting bosons confined to harmonic potentials in one and two dimensions. By exact numerical diagonalization of the many-body Hamiltonian we determine the low-lying excitation energies and the ground-state energy and density profile. We discuss the dependence of these quantities on both interaction strength ggg and particle number NNN. The ground-state properties are compared to the predictions of the Gross-Pitaevskii equation, and the agreement is surprisingly good even for relatively low particle numbers. We also calculate the specific heat based on the obtained energy spectra.