The utility of lattice discretization technique is demonstrated for solving nonrelativistic quantum scattering problems and specially for the treatment of ultraviolet divergences in these problems with some potentials singular at the origin in two- and three-space dimensions. This shows that the lattice discretization technique could be a useful tool for the numerical solution of scattering problems in general. The approach is illustrated in the case of the Dirac delta function potential.

arXiv:hep-th/9707231Journal of Physics A Mathematical and GeneralNov 21, 199616 citations

The Dirac Equation- Old and New

Narasimhaiengar Mukunda

Jan 1, 19821 citation

Brazilian Journal of Physics

Some Problems with the Dirac Delta Function: Divergent Series in Physics

Marcos Amaku, Francisco Antônio Bezerra Coutinho, O. J. P. Éboli, Eduardo Massad

Published Jul 2, 2021• 7 citations

Publisher

DOI: 10.1007/s13538-021-00916-5

Related papers

The Dirac Delta Function

Ajoy Kumar Ghatak, S. Lokanathan

Springer eBooksJan 1, 20044 citations

An Effective Field Theory Approach to the One-Dimensional Scattering Problem

Kevin M. Berg

We demonstrate the effective field theory (EFT) principle by approximating the coefficient of reflection resulting from a one-dimensional finite square barrier potential with a series of Dirac delta functions up to second order in derivative. The explicit conditions for energy, E, finite square barrier potential height, V0, and finite square barrier potential width, a, wherein the approximation is valid are provided. We show that the inclusion of the second order derivative with the Dirac delta function in the series improves the agreement of the result. Generally, at any order the agreement between the approximation and the exact result deteriorates as E or the dimensionless quantity V₀a increases. In addition, at a given E or V₀a value the agreement improves as the order increases. Our results suggest that a more accurate approximation will occur with the addition of further derivatives of the Dirac delta function to the approximating potential, which is in line with the underlying assumptions behind the use of EFTs.We demonstrate the effective field theory (EFT) principle by approximating the coefficient of reflection resulting from a one-dimensional finite square barrier potential with a series of Dirac delta functions up to second order in derivative. The explicit conditions for energy, E, finite square barrier potential height, V0, and finite square barrier potential width, a, wherein the approximation is valid are provided. We show that the inclusion of the second order derivative with the Dirac delta function in the series improves the agreement of the result. Generally, at any order the agreement between the approximation and the exact result deteriorates as E or the dimensionless quantity V₀a increases. In addition, at a given E or V₀a value the agreement improves as the order increases. Our results suggest that a more accurate approximation will occur with the addition of further derivatives of the Dirac delta function to the approximating potential, which is in line with the underlying assumptions behind the use of EFTs.

May 1, 20120 citations

The super Dirac $\delta$ function and its applications

Yakir Aharonov, Tomer Shushi

Quantum Studies Mathematics and FoundationsMay 25, 20222 citations

Perturbative Renormalization in Quantum Few-Body Problems

Sadhan K. Adhikari, T. Frederico, Iuda Dawid Goldman

As in quantum field theory, for some singular two-body potentials, the momentum-space perturbative treatment of the three-dimensional quantum $N$ -body problem is shown to exhibit ultraviolet divergences. Renormalization of these models leads to a scale and finite observables. For a Dirac delta potential we carry on this renormalization for two- and three-body systems. A new divergence, and hence a new scale, emerges for each added particle. For $N\ensuremath{\ge}3$ there are no divergences in one and two dimensions. The physical implications of these results are discussed.As in quantum field theory, for some singular two-body potentials, the momentum-space perturbative treatment of the three-dimensional quantum $N$-body problem is shown to exhibit ultraviolet divergences. Renormalization of these models leads to a scale and finite observables. For a Dirac delta potential we carry on this renormalization for two- and three-body systems. A new divergence, and hence a new scale, emerges for each added particle. For $N\ensuremath{\ge}3$ there are no divergences in one and two dimensions. The physical implications of these results are discussed.

Physical Review LettersJan 23, 199595 citations

Lattice discretization in quantum scattering

Sadhan K Adhikari, T Frederico, R M Marinho

arXiv:hep-th/9707231Journal of Physics A Mathematical and GeneralNov 21, 199616 citations

The Dirac Equation- Old and New

Narasimhaiengar Mukunda

Jan 1, 19821 citation

As in quantum field theory, for some singular two-body potentials, the momentum-space perturbative treatment of the three-dimensional quantum

N

-body problem is shown to exhibit ultraviolet divergences. Renormalization of these models leads to a scale and finite observables. For a Dirac delta potential we carry on this renormalization for two- and three-body systems. A new divergence, and hence a new scale, emerges for each added particle. For $N\ensuremath{\ge}3$ there are no divergences in one and two dimensions. The physical implications of these results are discussed.