The two-body problem of ultra-cold atoms in a harmonic trap

We consider two bosonic atoms interacting with a short-range potential and trapped in a spherically symmetric harmonic oscillator. The problem is exactly solvable and is relevant for the study of ultra-cold atoms. We show that the energy spectrum is universal, irrespective of the shape of the interaction potential, provided its range is much smaller than the oscillator length.Comment: Final version accepted for publication in Am. Journ. Phy

Density functional theory of the trapped Fermi gas in the unitary regime

We investigate a density-functional theory (DFT) approach for an unpolarized trapped dilute Fermi gas in the unitary limit . A reformulation of the recent work of T. Papenbrock [Phys. Rev. A, {\bf 72}, 041602(R) (2005)] in the language of fractional exclusion statistics allows us to obtain an estimate of the universal factor, $\xi_{3D}$, in three dimensions (3D), in addition to providing a systematic treatment of finite-$N$ corrections. We show that in 3D, finite-$N$ corrections lead to unphysical values for $\xi_{3D}$, thereby suggesting that a simple DFT applied to a small number of particles may not be suitable in 3D. We then perform an analogous calculation for the two-dimensional (2D) system in the infinite-scattering length regime, and obtain a value of $\xi_{2D}=1$. Owing to the unique properties of the Thomas-Fermi energy density-functional in 2D our result, in contrast to 3D, is {\em exact} and therefore requires no finite-$N$ corrections

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

We examine the zero-range limit of the finite square well in arbitrary dimensions through a systematic analysis of the reduced, s-wave two-body time-independent Schr\"odinger equation. A natural consequence of our investigation is the requirement of a delta-function multiplied by a regularization operator to model the zero-range limit of the finite-square well when the dimensionality is greater than one. The case of two dimensions turns out to be surprisingly subtle, and needs to be treated separately from all other dimensions

Evaluation of inverse integral transforms for undergraduate physics students

We provide a simple approach for the evaluation of inverse integral transforms that does not require any knowledge of complex analysis. The central idea behind the method is to reduce the inverse transform to the solution of an ordinary differential equation. We illustrate the utility of the approach by providing examples of the evaluation of transforms, without the use of tables. We also demonstrate how the method may be used to obtain a general representation of a function in the form of a series involving the Dirac-delta distribution and its derivatives, which has applications in quantum mechanics, semi-classical, and nuclear physics.Comment: 21 Pages, No Figure