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arXiv:1209.1398 · Physical Review A

Symmetries of three harmonically trapped particles in one dimension

Nathan L. Harshman

Published Nov 30, 2012• 62 citations
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Abstract

We present a method for solving few-body problems for trapped particles and apply it to three equal-mass particles in a one-dimensional harmonic trap, interacting via a contact potential. By expressing the relative Hamiltonian in Jacobi cylindrical coordinates, i.e., the two-dimensional version of three-body hyperspherical coordinates, we discover an underlying ${C}_{6v}$ symmetry. This symmetry simplifies the calculation of energy eigenstates of the full Hamiltonian in a truncated Hilbert space constructed from the trap Hamiltonian eigenstates. Particle superselection rules are implemented by choosing the relevant representations of ${C}_{6v}$. We find that the one-dimensional system shows nearly the full richness of the three-dimensional system, and can be used to understand separability and reducibility in this system and in standard few-body approximation techniques.

DOI
10.1103/physreva.86.052122

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We present a method for solving few-body problems for trapped particles and apply it to three equal-mass particles in a one-dimensional harmonic trap, interacting via a contact potential. By expressing the relative Hamiltonian in Jacobi cylindrical coordinates, i.e., the two-dimensional version of three-body hyperspherical coordinates, we discover an underlying C6v{C}_{6v}C6v​ symmetry. This symmetry simplifies the calculation of energy eigenstates of the full Hamiltonian in a truncated Hilbert space constructed from the trap Hamiltonian eigenstates. Particle superselection rules are implemented by choosing the relevant representations of C6v{C}_{6v}C6v​. We find that the one-dimensional system shows nearly the full richness of the three-dimensional system, and can be used to understand separability and reducibility in this system and in standard few-body approximation techniques.
This paper introduces a model for a few repulsively interacting particles trapped in a one-dimensional harmonic well and provides exact solutions for the three-particle case. This model is a hybrid of two other well-known systems, the Calogero model and the contact-interaction model, and coincides with them in limiting cases. However, those models have purely two-body interactions whereas this model has intrinsically few-body interactions. Comparing these three models provides clarifying distinctions among the properties of symmetry, separability, and integrability. The model's analytic solutions provide a useful basis to improve approximation schemes, especially near the unitary limit of hard-core contact interactions.
This is the first in a pair of articles that classify the configuration space and kinematic symmetry groups for NNN identical particles in one-dimensional traps experiencing Galilean-invariant two-body interactions. These symmetries explain degeneracies in the few-body spectrum and demonstrate how tuning the trap shape and the particle interactions can manipulate these degeneracies. The additional symmetries that emerge in the non-interacting limit and in the unitary limit of an infinitely strong contact interaction are sufficient to algebraically solve for the spectrum and degeneracy in terms of the one-particle observables. Symmetry also determines the degree to which the algebraic expressions for energy level shifts by weak interactions or nearly-unitary interactions are universal, i.e. independent of trap shape and details of the interaction. Identical fermions and bosons with and without spin are considered. This article sequentially analyzes the symmetries of one, two and three particles in asymmetric, symmetric, and harmonic traps; the sequel article treats the NNN particle case.
Abstract Strongly interacting particles in one dimension subject to external confinement have become a topic of considerable interest due to recent experimental advances and the development of new theoretical methods to attack such systems. In the case of equal mass fermions or bosons with two or more internal degrees of freedom, one can map the problem onto the well-known Heisenberg spin models. However, many interesting physical systems contain mixtures of particles with different masses. Therefore, a generalization of the recent strong-coupling techniques would be highly desirable. This is particularly important since such problems are generally considered non-integrable and thus the hugely successful Bethe ansatz approach cannot be applied. Here we discuss some initial steps towards this goal by investigating small ensembles of one-dimensional harmonically trapped particles where pairwise interactions are either vanishing or infinitely strong with focus on the mass-imbalanced case. We discuss a (semi)-analytical approach to describe systems using hyperspherical coordinates where the interaction is effectively decoupled from the trapping potential. As an illustrative example we analyze mass-imbalanced four-particle two-species mixtures with strong interactions between the two species. For such systems we calculate the energies, densities and pair-correlation functions.
The Bethe-Peierls asymptotic approach which models pairwise short-range forces by contact conditions is introduced in arbitrary representation for spatial dimensions less than or equal to 3. The formalism is applied in various situations and emphasis is put on the momentum representation. In the presence of a transverse harmonic confinement, dimensional reduction toward two-dimensional (2D) or one-dimensional (1D) physics is derived within this formalism. The energy theorem relating the mean energy of an interacting system to the asymptotic behavior of the one-particle density matrix illustrates the method in its second quantized form. Integral equations that encapsulate the Bethe-Peierls contact condition for few-body systems are derived. In three dimensions, for three-body systems supporting Efimov states, a nodal condition is introduced in order to obtain universal results from the Skorniakov--Ter-Martirosian equation and the Thomas collapse is avoided. Four-body bound state eigenequations are derived and the 2D ${}^{\ensuremath{'}}3+{1}^{\ensuremath{'}}$ bosonic ground state is computed as a function of the mass ratio.
The configuration interaction (CI) method for calculating the exact eigenstates of a quantum-mechanical few-body system is problematic when applied to contact interactions between the particles. In two and three dimensions, the approach fails due to the pathology of the Dirac delta-potential, making it impossible to reach convergence by gradually increasing the size of the Hilbert space. However, for practical applications this problem may be cured in a rather simple manner, by renormalizing the STRENGTH of the contact potential, which must be diagonalized in a TRUNCATED Hilbert space. The procedure relies on the comparison of CI energies and wave functions with those obtained by the exact solution of the two-body Schrodinger equation for the regularized contact interaction. The rather simple scheme, while keeping the numerical procedures still elementary, nevertheless provides both cut-off-independent few-body physical observables and an estimate of the error of the CI calculation.