Confined Quantum Systems: Beyond Harmonic Order in Dimensional Perturbation Theory

Abstract

We generalize the harmonic-order N-body DPT method for an isotropically confined quantum system of identical interacting bosons to first-anharmonic order and, in principle, higher orders. We introduce a graphical decomposition of the perturbative expansion of the N-body Hamiltonian. We calculate the Clebsch-Gordon coefficient tensors that couple together 3 irreducible representations of SN analytically, and analytically transformed the graphical basis to collective coordinates. We calculate the N-body wavefunction and density profile in general and have demonstrated agreement with an analytic model. We apply this formalism to the exactly-solvable example of a trapped gas of atoms interacting with a (fully-interacting) harmonic-oscillator "Hooke's law" interaction and compare with the dimensional expansion of the exact ground-state wavefunction and density profile. We report progress on the example of a cold gas BEC (with zero angular momentum).