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Separable Structure of Many-Body Ground-State Wave Function

Yeong E. Kim, Er L. Zubarev

Published Jan 1, 2000• 3 citations
OpenAlex

Abstract

We have investigated a general structure of the ground-state wave function 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 wave function can be written in a separable form. As an example of its applications, this form is used to obtain the groundstate wave function describing collective dynamics for N trapped bosons interacting via contact forces.

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We have investigated a general structure of the ground-state wave function 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 wave function can be written in a separable form. As an example of its applications, this form is used to obtain the groundstate wave function 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.
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.
Standard analytical construction of the many-body wave function of interacting particles in one dimension, beyond mean-field theory, is based on the Jastrow approach. The many-body interacting ground state is build up from the ground state of the non-interacting system and the product of solutions of the corresponding interacting two-body problem. However, this is possible only if the center-of-mass motion is decoupled from the mutual interactions. In our work, based on the general constraints given by contact nature of the atom-atom interactions, we present an alternative approach to the standard construction of the pair-correlation wave function. Within the proposed ansatz, we study the many-body properties of trapped bosons as well as fermionic mixtures and we compare these predictions with the exact diagonalization approach in a wide range of particle numbers, interaction strengths, and different trapping potentials.
Standard analytical construction of the many-body wave function of interacting particles in one dimension, beyond mean-field theory, is based on the Jastrow approach. The many-body interacting ground state is build up from the ground state of the non-interacting system and the product of solutions of the corresponding interacting two-body problem. However, this is possible only if the center-of-mass motion is decoupled from the mutual interactions. In our work, based on the general constraints given by contact nature of the atom-atom interactions, we present an alternative approach to the standard construction of the \textit{pair-correlation} wave-function. Within the proposed ansatz, we study the many-body properties of trapped bosons as well as fermionic mixtures and we compare these predictions with the exact diagonalization approach in a wide range of particle numbers, interaction strengths, and different trapping potentials.
Knowledge of the ground state of a homogeneous quantum many-body system can be used to find the exact ground state of a dual inhomogeneous system with a confining potential.We first identify the complete family of one-dimensional parent Hamiltonians with a ground state of Bijl-Jastrow form in free space.For each instance in this family, the dual trapped system is shown to include a one-body harmonic potential and two-body long-range interactions.The extension to anharmonic potentials and quantum solids with Nosanov-Jastrow ground-state wave functions is also presented.We apply this exact mapping to construct eigenstates of trapped systems from free-space solutions for different choices of the pair correlation functions entering the Bijl-Jastrow form.This construction is user friendly and allows one to readily rederive known instances of quasisolvable and integrable models, as well as to introduce unexplored examples.Among the latter, we describe a Lieb-Liniger-Coulomb model whose exact ground state is a McGuire soliton solution embedded in a harmonic trap.