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arXiv · 1209.1398

Symmetries of Three Harmonically-Trapped Particles in One Dimension

N. L. Harshman
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Abstract

We present a method for solving trapped few-body problems 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 ${\rm C}_{6v}$ 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 ${\rm C}_{6v}$. 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.

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N. L. Harshman. 2012-09-06. Symmetries of Three Harmonically-Trapped Particles in One Dimension. https://doi.org/10.1103/physreva.86.052122

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