Symmetries of three harmonically trapped particles in one dimension

N. L. Harshman

Phys. Rev. A 86, 052122 – Published 30 November 2012

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 $C_{6 v}$ 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 $C_{6 v}$ . 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.

Received 6 September 2012

DOI:https://doi.org/10.1103/PhysRevA.86.052122

Authors & Affiliations

N. L. Harshman^*

Department of Physics, 4400 Massachusetts Ave. NW, American University, Washington, DC 20016-8058, USA

^*harshman@american.edu

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Issue

Vol. 86, Iss. 5 — November 2012

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Images

Figure 1
This figure demonstrates that the transformation to Jacobi coordinates can be represented as a rotation in configuration space. The ${\hat{q}}_{i}$ positive particle axes are solid arrows; the Jacobi Cartesian positive axes are dashed arrows. The $\hat{z}$ direction points into the middle of the solid angle bounded by the all-positive octant and the $\hat{x}$ and $\hat{y}$ are orthogonal directions in the relative plane, depicted as a gray disk for $z = 0$ , i.e., the center of mass at the bottom of the potential well. This Jacobi rotation takes place around the yellow (light gray, unlabeled) vector and is about 1.22 radians.Reuse & Permissions
Figure 2
This figure depicts the relative plane in Jacobi coordinates. The magenta (dashed) lines have several interpretations. First, they are the projections of the particle axes on the relative plane (see Fig. 1). For example, the line with $φ = π / 6$ corresponds to increasing separation of particle 1 (depicted as a red circle) from particles 2 (green square) and 3 (blue triangle), with particle 1 on the positive side and the other two colocated on the negative side. Reflections across the magenta (dashed) lines also represent two-particle exchanges, e.g., the reflection through the line $φ = π / 6$ corresponds to the ${\hat{σ}}_{23}$ 2-cycle. Rotations of $\pm 2 π / 3$ are 3-cycles, i.e., three-particle exchanges. A rotation of $π$ corresponds to a reversal of parity $\hat{π}$ . Reflections across the cyan (dotted) lines are a combination of a two-particle exchange and a parity inversion, e.g., $\hat{π} {\hat{σ}}_{23}$ .Reuse & Permissions
Figure 3
Solid black lines are the 40 energy levels with symmetry corresponding to the totally symmetric, positive parity $A_{1}$ representation of $C_{6 v}$ for ${\tilde{N}}_{max} = 30$ . Energies are measured in units of $ℏ ω$ . The dashed red lines are the exact result for universal states in the limit $g / σ \to \infty$ and correspond to the energy levels and degeneracies of the noninteracting $FFF +, A_{2}$ representation. This representation contains the ground state of the BBB, BB $X$ , and $X Y Z$ configurations that is a trimer in the case of attractive interactions. See text for more details.Reuse & Permissions
Figure 4
Solid black lines are the 40 energy levels with symmetry corresponding to the totally symmetric, negative parity $B_{2}$ representation of $C_{6 v}$ for ${\tilde{N}}_{max} = 30$ . Energies are measured in units of $ℏ ω$ . The dashed red lines are the exact result for universal states in the limit $g / σ \to$ and correspond to the energy levels and degeneracies of the noninteracting $FFF -, B_{1}$ representation.Reuse & Permissions
Figure 5
Solid black lines are the 80 energy levels with symmetry corresponding to the two-component fermion with negative parity in the $E_{1}$ / $B_{2}$ representation of $C_{6 v}$ / $C_{2 v}$ for ${\tilde{N}}_{max} = 30$ . Energies are measured in units of $ℏ ω$ . This is the ground-state sector for the two-component fermion. The dashed red lines are the exact result for universal states for both the $FFF +, A_{2}$ and the $FFF -, B_{1}$ representations, showing the emergence of energy levels corresponding to “anomalous” parity states in the unitary limit.Reuse & Permissions

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