### 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 language | English (US) |
---|---|

Article number | 055303 |

Journal | Journal of Physics B: Atomic, Molecular and Optical Physics |

Volume | 44 |

Issue number | 5 |

DOIs | |

State | Published - Mar 14 2011 |

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### ASJC Scopus subject areas

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

### Cite this

*Journal of Physics B: Atomic, Molecular and Optical Physics*,

*44*(5), [055303]. https://doi.org/10.1088/0953-4075/44/5/055303

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

Research output: Contribution to journal › Article

*Journal of Physics B: Atomic, Molecular and Optical Physics*, vol. 44, no. 5, 055303. https://doi.org/10.1088/0953-4075/44/5/055303

}

TY - JOUR

T1 - Analytic harmonic approach to the N-body problem

AU - Armstrong, J. R.

AU - Zinner, N. T.

AU - Fedorov, D. V.

AU - Jensen, A. S.

PY - 2011/3/14

Y1 - 2011/3/14

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=79951872838&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79951872838&partnerID=8YFLogxK

U2 - 10.1088/0953-4075/44/5/055303

DO - 10.1088/0953-4075/44/5/055303

M3 - Article

AN - SCOPUS:79951872838

VL - 44

JO - Journal of Physics B: Atomic, Molecular and Optical Physics

JF - Journal of Physics B: Atomic, Molecular and Optical Physics

SN - 0953-4075

IS - 5

M1 - 055303

ER -