Symmetries of Three Harmonically-Trapped Particles in One Dimension

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 ${\mathrm{C}}_{6v}$ 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 ${\mathrm{C}}_{6v}$. 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.

27 pages, 5 figures, 6 tables, 37 references, 4 footnotes, 1 article; v2 has revised introduction and results sections as well as typos corrected

References in corpus (8)

• Analytical solution of the bosonic three-body problem
• No-core shell model in an effective-field-theory framework
• Effective Theory for Trapped Few-Fermion Systems
• Universal correlations of trapped one-dimensional impenetrable bosons
• Level crossing in the three-body problem for strongly interacting fermions in a harmonic trap
• Production of three-body Efimov molecules in an optical lattice
• Three-fermion problems in optical lattices
• Short Range Scaling Laws of Quantum Gases With Contact Interactions