Generalization of the Fermi pseudopotential. (28th March 2019)

• Record Type: Journal Article
• Authors: Lê, Trang T; Osman, Zach; Watson, D K; Dunn, Martin; McKinney, B A
• Is Part Of: Physica scripta. Volume 94:Number 6(2019)
• Publication Date: 2019-03-28
• Subjects: pseudopotential -- fractional calculus -- arbitrary dimension -- two cold trapped atoms -- ultra-cold atoms; Physics -- Periodicals; 530.05
• DOI: 10.1088/1402-4896/ab0811
• Languages: English
• ISSNs: 0031-8949

Abstract

Abstract: Introduced eighty years ago, the Fermi pseudopotential has been a powerful concept in multiple fields of physics. It replaces the detailed shape of a potential by a delta-function operator multiplied by a parameter giving the strength of the potential. For Cartesian dimensions d > 1, a regularization operator is necessary to remove singularities in the wave function. In this study, we develop a Fermi pseudopotential generalized to d dimensions (including non-integer) and to non-zero wavenumber, k . Our new fractional calculus approach has the advantage of circumventing singularities that occur in the wave function at integer values of d while being valid arbitrarily close to integer d . In the limit of integer dimension, we show that our generalized pseudopotential is equivalent to previously derived s -wave pseudopotentials. Our pseudopotential generalizes the operator to non-integer dimension, includes energy ( k ) dependence, and simplifies the dimension-dependent coupling constant expression derived from a Green's function approach. We apply this pseudopotential to the problem of two cold atoms ( k → 0) in a harmonic trap and extend the energy expression to arbitrary dimension.