Analytic harmonic approach to the N-body problem

J. R. Armstrong, N. T. Zinner, D. V. Fedorov, A. S. Jensen

Research output: Contribution to journalArticle

29 Citations (Scopus)

Abstract

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.

Original languageEnglish (US)
Article number055303
JournalJournal of Physics B: Atomic, Molecular and Optical Physics
Volume44
Issue number5
DOIs
StatePublished - Mar 14 2011

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many body problem
harmonics
bosons
radii
harmonic oscillators
condensates
formalism
two body problem
zero point energy
scattering
energy spectra
eigenvalues
fermions
traps
interactions
energy
geometry

ASJC Scopus subject areas

  • Atomic and Molecular Physics, and Optics
  • Condensed Matter Physics

Cite this

Analytic harmonic approach to the N-body problem. / Armstrong, J. R.; Zinner, N. T.; Fedorov, D. V.; Jensen, A. S.

In: Journal of Physics B: Atomic, Molecular and Optical Physics, Vol. 44, No. 5, 055303, 14.03.2011.

Research output: Contribution to journalArticle

Armstrong, J. R. ; Zinner, N. T. ; Fedorov, D. V. ; Jensen, A. S. / Analytic harmonic approach to the N-body problem. In: Journal of Physics B: Atomic, Molecular and Optical Physics. 2011 ; Vol. 44, No. 5.
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