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arXiv:1806.05726 · Physica Scripta

Generalization of the Fermi pseudopotential

Trang T. Le, Zach Osman, D K Watson, Martin Dunn, B A McKinney

Published Feb 18, 2019• 5 citations

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.

DOI: 10.1088/1402-4896/ab0811

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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.

The theory of zero-range potentials is investigated in an arbitrary number of dimensions. Except for the trivial one-dimensional case the zero-range potentials are described by nonlocal operators called Fermi pseudopotentials. It is shown that in odd dimensions the Fermi pseudopotentials involve a very simple regularization operator. In even dimensions with the help of dimensional regularization, explicit formulas for the Fermi pseudopotentials are derived. The Green's functions, the propagators, and the exact solutions of the Lippmann-Schwinger equations are derived in explicit forms. In odd dimensions d=1 and 3 the Fermi pseudopotentials can be applied to describe multiphoton processes of atoms and molecules with very short-range interactions. In even dimension d=2 the Fermi pseudopotential can be applied to describe tunneling from laser-driven quantum wells. Physical applications involving higher d are also possible.

We derive a generalized zero-range pseudopotential applicable to all partial wave solutions to the Schrödinger equation based on a delta-shell potential in the limit that the shell radius approaches zero. This properly models all higher order multipole moments not accounted for with a monopolar delta function at the origin, as used in the familiar Fermi pseudopotential for s-wave scattering. By making the strength of the potential energy dependent, we derive self-consistent solutions for the entire energy spectrum of the realistic potential. We apply this to study two particles in an isotropic harmonic trap, interacting through a central potential, and derive analytic expressions for the energy eigenstates and eigenvalues.

In this paper we calculate the energy levels of two trapped atoms interacting through a spherical two-body potential

V(r)

under external confinement quantum mechanically. We then replace

V(r)

by either a regularized energy-independent or an energy-dependent \ensuremath{\delta}-function potential. A comparison shows that the use of an energy-dependent pseudopotential improves significantly upon the use of an energy-independent pseudopotential. The accuracy of the pseudopotential approximation is approximately determined by the ratio ${\ensuremath{\beta}}_{6}{/a}_{\mathrm{ho}},$ where ${\ensuremath{\beta}}_{6}$ and

{a}_{\mathrm{ho}}

denote the characteristic length scale of the two-body interaction potential

V(r)

and of the trapping potential, respectively.

Abstract We determine the conditions for bound states (E < 0) for arbitrary Cartesian dimension d using a shape-independent regularized pseudopotential with scattering length a for two cold particles in a harmonic trap. It is known for d ≤ 3 that the regularized pseudopotential supports one bound state for positive scattering length but does not support bound states for negative scattering length. We find that the usual (d ≤ 3) positive scattering length bound states rule holds for certain higher odd dimensions d = 4n + 3 (n = 0, 1, …), but the existence of pseudopotential bound states at other odd dimensions requires a negative scattering length. Specifically, bound states are allowed in higher dimensions d = 4n + 1 (n = 1, 2, …) but they require a negative scattering length, which suggests a universe in these dimensions might lead to different chemistry than d = 3. We derive analytical approximations for bound state (E < 0) and scattering (E > 0) energies for a harmonic trap perturbed by the pseudopotential in arbitrary dimensions.

The zero-range potential approach is extended for the description of situations where two-body scattering is resonant in arbitrary partial waves. The formalism generalizes the Fermi pseudopotential which can be used only for

s

-wave broad resonances. In a given channel, the interaction is described either in terms of a contact condition on the wave function or with a family of pseudopotentials. We show that it is necessary to introduce a regularized scalar product for wave functions obtained in the zero-range potential formalism (except for the Fermi pseudopotential). This metrics shows that the geometry of these Hilbert spaces depends crucially on the interaction.

We use first-order perturbation theory near the fermionic limit of the δ-function Bose gas in one dimension (i.e. a system of weakly interacting fermions) to study three situations of physical interest. The calculation is done using a pseudopotential which takes the form of a two-body δ''-function interaction. The three cases considered are the behaviour of the system with a hard wall, with a point where the strength of the pseudopotential changes discontinuously, and with a region of finite length where the pseudopotential strength is non-zero (this is sometimes used as a model for a quantum wire). In all cases, we obtain exact expressions for the density to first order in the pseudopotential strength. The asymptotic behaviour of the densities is in agreement with the results obtained from bosonization for a Tomonaga–Luttinger liquid, namely, an interaction dependent power-law decay of the density far from the hard wall, a reflection from the point of discontinuity and transmission resonances for the interacting region of finite length. Our results provide a non-trivial verification of the Tomonaga–Luttinger liquid description of the δ-function Bose gas near the fermionic limit.