Universality of the energy spectrum for two interacting harmonically trapped ultra-cold atoms in one and two dimensions

Abstract

Motivated by the recent article of P. Shea et al. [Am. J. Phys. 77 (6), 2009] we examine the exactly solvable problem of two harmonically trapped ultra-cold bosonic atoms interacting via a short range potential in one and two dimensions. A straightforward application in one dimension shows that the energy spectrum is universal, provided that the range of the potential is much smaller than the oscillator length, in addition to clearly illustrating why regularization is not required in the limit of zero range. The two dimensional problem is less trivial, requiring a more careful treatment as compared to the one dimensional case. Our two dimensional analysis likewise reveals that the low-energy physics is also universal, in addition to providing a simple method for obtaining the appropriately regularized two dimensional pseudopotential.

Many-particle systems in one dimension in the harmonic approximation

“…The external field, ω 0 , defines the length scale of our system, l 2 = /(mω 0 ). For two particles, (1) has been solved in, e.g., [18,19], and their results of two-body energies and wave functions for a given scattering length are used to determine the parameters of our model. We consider only the bound molecular branch of the system, i.e., a > 0.…”

Exact Diagonalization Study of Bose-Condensed Gas with Finite-Range Gaussian Interaction

“…Unfortunately, such an interaction potential in 2D is not selfadjoint [13] and cannot be used in a manner analogous to its one dimensional counterpart. Indeed a regularized 2D contact potential is analytically tractable [14] but harder to handle numerically. Further, as the number of particles is increased beyond two, the analytical treatment becomes intractable, leaving one only with a numerical recourse.…”

s-wave scattering and the zero-range limit of the finite square well in arbitrary dimensions

“…In more technical papers, similar results are developed in the context of self-adjoint extensions, and Green's function techniques and are given the name "pseudo-potentials". 13,15,[19][20][21][22] Finally, it is our hope that the presentation used in this paper may serve as a basis for introducing a more general treatment of the FSW in undergraduate quantum mechanics. To this end, we suggest the following useful exercise.…”

“…We feel that students may benefit from this presentation, particularly those not yet exposed to the analytic properties of the S(k)-matrix . 7,13,15 Following our treatment of the d-dimensional FSW, we examined its b → 0 limit by extending the 1D analysis 1 to arbitrary dimensions. Our main result is that the zero-range limit of the FSW is given by…”