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arXiv:cond-mat/9906258 · Journal of Physics B Atomic Molecular and Optical Physics

Equivalent linear two-body method for many-body problems

Yeong E Kim, Alexander L Zubarev

Published Dec 23, 1999• 24 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/0953-4075/33/1/306

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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 present a theoretical formulation of trapped, dilute Bose-Einstein condensates (BEC's) at zero temperature based on ordinary Schr\"odinger quantum mechanics. By a judicious choice of coordinates and of a variational trial wave function we reduce the many-atom problem to a linear Schr\"odinger equation that is easier to handle and interpret than the usual nonlinear Schr\"odinger equation of BEC theory. Ordinary quantum mechanics then reproduces, semiquantitatively, many of the main features of zero-temperature BEC, including the critical number of atoms in a condensate with negative scattering length. The procedure is similar in results, but completely different in spirit, to recent variational approaches to solving the nonlinear Schr\"odinger equation. Moreover, the present procedure represents a step in a systematic alternative method for computing quantitatively accurate wave functions for trapped bosonic atoms.
We study the ground-state properties of a system of NNN harmonically trapped bosons of mass mmm interacting with two-body contact interactions, from small to large scattering lengths. This is accomplished in a hyperspherical coordinate system that is flexible enough to describe both the overall scale of the gas and two-body correlations. By adapting the lowest-order constrained-variational method, we are able to semiquantitatively attain Bose-Einstein condensate ground-state energies even for gases with infinite scattering length. In the large-particle-number limit, our method provides analytical estimates for the energy per particle ${E}_{0}/N\ensuremath{\approx}2.5{N}^{1/3}\ensuremath{\hbar}\ensuremath{\omega}$ and two-body contact ${C}_{2}/N\ensuremath{\approx}16{N}^{1/6}\sqrt{m\ensuremath{\omega}/\ensuremath{\hbar}}$ for a Bose gas on resonance, where $\ensuremath{\omega}$ is the trap frequency.
We consider an analytic way to make the interacting N -body problem tractable by using harmonic oscillators in place of the relevant two-body interactions. The two-body terms of the N -body Hamiltonian are approximated by considering the energy spectrum and radius of the relevant two-body problem which gives frequency, centre position, and zero point energy of the corresponding harmonic oscillator. Adding external harmonic one-body terms, we proceed to solve the full quantum mechanical N -body problem analytically for arbitrary masses. Energy eigenvalues, eigenmodes, and correlation functions like density matrices can then be computed analytically. As a first application of our formalism, we consider the N -boson problem in two and three dimensions where we fit the two-body interactions to agree with the well-known zero-range model for two particles in a harmonic trap. Subsequently, condensate fractions, spectra, radii, and eigenmodes are discussed as a function of dimension, boson number N , and scattering length obtained in the zero-range model. We find that energies, radii, and condensate fraction increase with scattering length as well as boson number, while radii decrease with increasing boson number. Our formalism is completely general and can also be applied to fermions, Bose–Fermi mixtures, and to more exotic geometries.
We study a physically motivated model for a trapped dilute gas of Bosons with repulsive pairwise atomic interactions at zero temperature. Our goal is to describe aspects of the excited many-body quantum states of this system by accounting for the scattering of atoms in pairs from the macroscopic state. We start with an approximate many-body Hamiltonian, H a p p \mathcal {H}_{\mathrm {app}} , in the Bosonic Fock space. This H a p p \mathcal {H}_{\mathrm {app}} conserves the total number of atoms. Inspired by Wu [J. Math. Phys. 2 (1961), 105–123], we apply a non-unitary transformation to H a p p \mathcal {H}_{\mathrm {app}} . Key in this procedure is the pair-excitation kernel, which obeys a nonlinear integro-partial differential equation. In the stationary case, we develop an existence theory for solutions to this equation by a variational principle. We connect this theory to a system of partial differential equations for one-particle excitation (“quasiparticle”-) wave functions derived by Fetter [Ann. Phys. 70 (1972), 67–101], and prove existence of solutions for this system. These wave functions solve an eigenvalue problem for a J J -self-adjoint operator. From the non-Hermitian Hamiltonian, we derive a one-particle nonlocal equation for low-lying excitations, describe its solutions, and recover Fetter’s energy spectrum. We also analytically provide an explicit construction of the excited eigenstates of the reduced Hamiltonian in the N N -particle sector of Fock space.
A variational method is presented which is applicable to N-particle boson or fermion systems with two-body interactions. For these systems the energy may be expressed in terms of the two-particle density matrix: Γ(1, 2 | 1′, 2′)=(Ψ |a2+a1+a1′a2′| Ψ). In order to have the variational equation: δE/δΓ = 0 yield the correct ground-state density matrix one must restrict Γ to the set of density matrices which are actually derivable from N-particle boson (or fermion) systems. Subsidiary conditions are presented which are necessary and sufficient to insure that Γ is so derivable. These conditions are of a form which render them unsuited for practical application. However the following necessary (but not sufficient) conditions are shown by some applications to yield good results: It is proven that if Γ(1, 2 | 1′, 2′) and γ(1 | 1′) are the two-particle and one-particle density matrices of an N-particle system [normalized by tr Γ = N(N − 1) and trγ = N] then the associated operator: G(1,2 | 1′,2′)=δ(1−1′)γ(2 | 2′)+σ Γ(1′,2 | 1,2′)−γ(2 | 1)γ(1′ | 2′) is a nonnegative operator. [Here σ is + 1 or − 1 for bosons or fermions respectively.]