Abstract The on‐shell approximation is investigated in the context of s‐wave scattering for ultracold two‐body collisions. This analysis systematically covers spatial dimensions , with the aim of identifying the regimes in which the approximation remains valid when applied to commonly used model interaction potentials. Specifically, the square well and delta shell potentials are focused, both of which admit analytical solutions for the s‐wave scattering problem in all dimensions considered. By employing the exact analytical expressions for the s‐wave scattering phase shift, a direct comparison is performed between the exact on‐shell matrix element of the interaction potential and their corresponding approximations across a range of collision momenta. Particular attention is given to the low‐energy regime. These findings indicate that, although the on‐shell approximation generally improves with increasing momentum, its accuracy also improves for weaker potentials. Remarkably, in the limit of weak interactions, it is demonstrated that the on‐shell approximation becomes exact at leading order. In this regime, the approximation offers a controlled means of deriving the low‐momentum expansion of the potential and may serve as a useful tool in constructing effective interactions for quantum field theories.
Abstract The on‐shell approximation is investigated in the context of s‐wave scattering for ultracold two‐body collisions. This analysis systematically covers spatial dimensions , with the aim of identifying the regimes in which the approximation remains valid when applied to commonly used model interaction potentials. Specifically, the square well and delta shell potentials are focused, both of which admit analytical solutions for the s‐wave scattering problem in all dimensions considered. By employing the exact analytical expressions for the s‐wave scattering phase shift, a direct comparison is performed between the exact on‐shell matrix element of the interaction potential and their corresponding approximations across a range of collision momenta. Particular attention is given to the low‐energy regime. These findings indicate that, although the on‐shell approximation generally improves with increasing momentum, its accuracy also improves for weaker potentials. Remarkably, in the limit of weak interactions, it is demonstrated that the on‐shell approximation becomes exact at leading order. In this regime, the approximation offers a controlled means of deriving the low‐momentum expansion of the potential and may serve as a useful tool in constructing effective interactions for quantum field theories.
We investigate the scattering theory of two particles in a generic D-dimensional space. For the s-wave problem, by adopting an on-shell approximation for the T-matrix equation, we derive analytical formulas which connect the Fourier transform $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}(k)$ of the interaction potential to the s-wave phase shift. In this way we obtain explicit expressions of the low-momentum parameters ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{0}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{2}$ of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}(k)={\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{0}+{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{2}{k}^{2}+\ensuremath{\cdots}$ in terms of the s-wave scattering length as and the s-wave effective range rs for D=3, D=2, and D=1. Our results, which are strongly dependent on the spatial dimension D, are a useful benchmark for few-body and many-body calculations. As a specific application, we derive the zero-temperature pressure of a two-dimensional uniform interacting Bose gas with a beyond-mean-field correction which includes both scattering length and effective range.
Abstract More than 65 years ago, Jost and Kohn (1952 Phys. Rev. 87 , 977) derived an explicit expression for a class of short-range model potentials from a given effective range expansion with the s -wave scattering length a s being negative. For a s > 0, they calculated another class of short-range model potentials (Jost and Kohn 1953 Dan. Mat. Fys. Medd 27 , no. 9) using a method based on an adaptation from Gel’fand−Levitan theory (Gel’fand and Levitan 1951 Dokl. Akad. Nauk. USSR 77 , 557-60) of inverse scattering. We here revisit the methods of Jost and Kohn in order to explore the possibility of modeling resonant finite-range interactions at low energy. We show that the Jost−Kohn potentials can account for zero-energy resonances. The s -wave phase shift for positive scattering length is expressed in an analytical form as a function of the binding energy of a bound state. We show that, for small binding energy, both the scattering length and the effective range are strongly influenced by the binding energy; below a critical binding energy the effective range becomes negative provided the scattering length is large. As a consistency check we carry out some simple calculations to show that Jost−Kohn potentials can reproduce the standard results of contact interaction in the limit of the effective range going to zero.
To clarify the relation of energy shifts to scattering phase shifts in one-body and many-body problems, we examine their relation in a number of different situations. We derive, for a particle in a container of arbitrary shape with a short-range scattering center, a general result for the energy eigenvalues in terms of the s-wave scattering phase shift and the eigenstates in the absence of the scatterer. We show that, while the energy shifts for a spherical container are proportional to the phase shift over large ranges, those for a cubic container have a more complicated behavior. We connect our result to the description of energy shifts in terms of the scattering T-matrix. The general relation is extended to problems of particles in traps with smoothly varying potentials, including, e.g., the interaction of a small neutral atom with a Rydberg atom. We then consider the many-body problem for particles with a two-body interaction and show that the free energy change due to the interaction is proportional to an average of a generalized phase shift that includes the effects of the medium. Finally, we discuss why, even though individual energy levels are very sensitive to boundary conditions, the energy of a many-body system is not.
Scattering processes are a fundamental way of experimentally probing distributions and properties of systems in several areas of physics. Considering two-body scattering at low energies, when the de Broglie wavelength is larger than the range of the potential, partial waves with high angular momentum are typically unimportant. The dominant contribution comes from l=0 partial waves, commonly known as s-wave scattering. This situation is very relevant in atomic physics, e.g. cold atomic gases, and nuclear physics, e.g. nuclear structure and matter. This manuscript is intended as a pedagogical introduction to the topic while covering a numerical approach to compute the desired quantities. We introduce low-energy scattering with particular attention to the concepts of scattering length and effective range. These two quantities appear in the effective-range approximation, which universally describes low-energy processes. We outline a numerical procedure for calculating the scattering length and effective range of spherically symmetric two-body potentials. As examples, we apply the method to the spherical well, modified Pöschl-Teller, Gaussian, and Lennard-Jones potentials. We hope to provide the tools so students can implement similar calculations and extend them to other potentials.
Studies of cold atom collisions and few-body interactions often require the energy dependence of the scattering phase shift, which is usually expressed in terms of an effective-range expansion.We use accurate coupled-channel calculations on 6 Li, 39 K, and 133 Cs to explore the behavior of the effective range in the vicinity of both broad and narrow Feshbach resonances.We show that commonly used expressions for the effective range break down dramatically for narrow resonances and near the zero crossings of broad resonances.We present an alternative parametrization of the effective range that is accurate through both the pole and the zero crossing for both broad and narrow resonances.However, the effective-range expansion can still fail at quite low collision energies, particularly around narrow resonances.We demonstrate that an analytical form of an energy and magnetic-field-dependent phase shift, based on multichannel quantum defect theory, gives accurate results for the energy-dependent scattering length.
The interaction between atoms behaves as $\ensuremath{-}\ensuremath{\alpha}{/r}^{n}$ at large distances and, owing to the large reduced mass \ensuremath{\mu} of the collision pair, allows a semiclassical treatment within the potential well. As a result, the low-energy scattering is governed by two large parameters: the asymptotic parameter $\ensuremath{\gamma}=\sqrt{2\ensuremath{\mu}\ensuremath{\alpha}}/\ensuremath{\Elzxh}\ensuremath{\gg}{a}_{0}^{(n\ensuremath{-}2)/2}$ (a0 is the Bohr radius) and the semiclassical zero-energy phase $\ensuremath{\Phi}\ensuremath{\gg}1.$ In our previous work [Phys. Rev. A 48, 546 (1993)] we obtained an analytical expression for the scattering length a, which showed that it has 75% preference for positive values for n=6, characteristic of collisions between ground-state neutral atoms. In this paper we calculate the effective range and show that it is a function of a, ${r}_{e}{=F}_{n}\ensuremath{-}{G}_{n}{/a+H}_{n}{/a}^{2},$ where Fn,Gn, and Hn depend only on \ensuremath{\gamma}. Thus, we know the s-phase shift at low momenta $k\ensuremath{\ll}{\ensuremath{\gamma}}^{\ensuremath{-}2/(n\ensuremath{-}2)}$ from the expansion $k\mathrm{cot}{\ensuremath{\delta}}_{0}\ensuremath{\simeq}\ensuremath{-}1/a+\frac{1}{2}{r}_{e}{k}^{2}.$ At $k\ensuremath{\gg}{\ensuremath{\gamma}}^{\ensuremath{-}2/(n\ensuremath{-}2)}$ the phase shift is obtained semiclassically as ${\ensuremath{\delta}}_{0}=\ensuremath{\Phi}+\ensuremath{\pi}/4\ensuremath{-}{I}_{n}{\ensuremath{\gamma}}^{2/n}{k}^{(n\ensuremath{-}2)/n},$ where ${I}_{n}=[n/(n\ensuremath{-}2)]\ensuremath{\Gamma}((n\ensuremath{-}1)/n)\ensuremath{\Gamma}((n+2)/2n)/\sqrt{\ensuremath{\pi}}.$ Therefore, \ensuremath{\gamma} and \ensuremath{\Phi} determine the s-wave atomic scattering in a wide range of momenta, as well as the positions of upper bound states of the diatomic molecule.