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

Farrell, Aaron; Zyl, Brandon P. van

doi:10.1139/p10-061

Cited by 7 publications

(8 citation statements)

References 22 publications

(31 reference statements)

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“…As shown above, there exists a unique bound state for a Hamiltonian with a one-dimensional delta potential well. The delta potentials of a higher dimensionality D provide not only a pedagogical model but find their applications in nuclear and condensed matter physics [39,40,41,42]. However, solving a Schrödinger equation with these potentials is a more cumbersome problem since there is no rigorous construction of a self-adjoint realization of the Hamiltonian in Eq.…”

Section: Potential Barrier (τ > 0)mentioning

confidence: 99%

Infinitesimal and infinite numbers as an approach to quantum mechanics

Benci

Baglini

Simonov

2019

Quantum

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Non-Archimedean mathematics is an approach based on fields which contain infinitesimal and infinite elements. Within this approach, we construct a space of a particular class of generalized functions, ultrafunctions. The space of ultrafunctions can be used as a richer framework for a description of a physical system in quantum mechanics. In this paper, we provide a discussion of the space of ultrafunctions and its advantages in the applications of quantum mechanics, particularly for the Schrödinger equation for a Hamiltonian with the delta function potential.Notice that, trivially, every superreal field is non-Archimedean. Infinitesimals allow introducing the following equivalence relation, which is fundamental in all non-Archimedean settings.Definition 2. We say that two numbers ξ, ζ ∈ K are infinitely close if ξ −ζ is infinitesimal. In this case we write ξ ∼ ζ.In the superreal case, ∼ can be used to introduce the fundamental notion of "standard part" 2 . Theorem 3. If K is a superreal field, every finite number ξ ∈ K is infinitely close to a unique real number r ∼ ξ, called the the standard part of ξ.Following the literature, we will always denote by st(ξ) the standard part of any finite number ξ. Moreover, with a small abuse of notation, we also put st(ξ) = +∞ (resp. st(ξ) = −∞) if ξ ∈ K is a positive (resp. negative) infinite number. Definition 4. Let K be a superreal field, and ξ ∈ K a number. The monad of ξ is the set of all numbers that are infinitely close to it,1 Without loss of generality, we assume that Q ⊆ K.2 For a proof of the following simple theorem, the interested reader can check, e.g., [21].

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Section: Potential Barrier (τ > 0)mentioning

confidence: 99%

Infinitesimal and infinite numbers as an approach to quantum mechanics

Benci

Baglini

Simonov

2019

Quantum

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“…In particular, the 1/r divergence of the scattering solution (2) is taken properly into account by the regularization operator ∂ r r, which ensures a finite scattering amplitude f (k) = − a 1+ika with a = µ 2π 2 g as the corresponding s-wave scattering length (4) [56]. Furthermore, in [59,60] it has been shown that the regularized delta interaction emerges, if one considers a proper zero-range limit of a three-dimensional finite square well or a delta-shell potential, respectively.…”

Section: Regularized Delta Interactionmentioning

confidence: 99%

Beyond mean-field dynamics of ultra-cold bosonic atoms in higher dimensions: facing the challenges with a multi-configurational approach

Bolsinger

Krönke

Schmelcher

2017

J. Phys. B: At. Mol. Opt. Phys.

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Exploring the impact of dimensionality on the quantum dynamics of interacting bosons in traps including particle correlations is an interesting but challenging task. Due to the different participating length scales the modelling of the short-range interactions in three dimensions plays a special role. We review different approaches for the latter and elaborate that for multi-configurational computational strategies finite range potentials are adequate resulting in the need of large grids to resolve the relevant length scales. This results in computational challenges which include also the exponential scaling of complexity with the number of atoms. We show that the recently developed The beneficial scaling of our approach is demonstrated by studying the tunnelling dynamics of bosonic ensembles in a double well. Comparing three-dimensional with quasi-one dimensional simulations, we find dimensionality-induced effects in the density. Furthermore, we study the crossover from weak transversal confinement, where a mean-field description of the system is sufficient, towards tight transversal confinement, where particle correlations and beyond mean-field effects are pronounced.

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“…In a fermionic system this zero-ranged interaction acts only between distinguishable particles, with the interaction strength described by a scattering length a. We can capture the full effect of the interaction by imposing the boundary condition [33,34] …”

Section: B 2d Contact Interactionmentioning

confidence: 99%

Pseudopotential for the two-dimensional contact interaction

Whitehead

Schonenberg

Kongsuwan³

et al. 2016

Phys. Rev. A

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We propose a smooth pseudopotential for the contact interaction acting between ultracold atoms confined to two dimensions. The pseudopotential reproduces the scattering properties of the repulsive contact interaction up to 200 times more accurately than a hard disk potential, and in the attractive branch gives a 10-fold improvement in accuracy over the square well potential. Furthermore, the new potential enables diffusion Monte Carlo simulations of the ultracold gas to be run 15 times quicker than was previously possible.

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

Cited by 7 publications

References 22 publications

Infinitesimal and infinite numbers as an approach to quantum mechanics

Infinitesimal and infinite numbers as an approach to quantum mechanics

Beyond mean-field dynamics of ultra-cold bosonic atoms in higher dimensions: facing the challenges with a multi-configurational approach

Pseudopotential for the two-dimensional contact interaction

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