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

Kevin M. Berg

Published May 1, 2012• 0 citations
OpenAlex

Abstract

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.

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Abstract In this paper, we propose a pedagogical presentation of the Lippmann–Schwinger equation as a powerful tool, so as to obtain important scattering information. In particular, we consider a one-dimensional system with a Schrödinger-type free Hamiltonian decorated with a sequence of N attractive Dirac delta interactions. We first write the Lippmann–Schwinger equation for the system and then solve it explicitly in terms of an N × N matrix. Then, we discuss the reflection and the transmission coefficients for an arbitrary number of centres and study the threshold anomaly for the N = 2 and N = 4 cases. We also study further features like the quantum metastable states and resonances, including their corresponding Gamow functions and virtual or antibound states. The use of the Lippmann–Schwinger equation simplifies our analysis enormously and gives exact results for an arbitrary number of Dirac delta potentials.

European Journal of Physics•Jan 18, 2018•17 citations
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.
In one-dimensional scattering, the portion of particles that is transmitted in general vanishes as the kinetic energy of the incident particles approaches zero. Based upon published results from the literature, the present work shows that for a potential consisting of two Dirac δ functions of arbitrary strength, a finite portion of the incident particles is transmitted at threshold for certain choices of the set of parameters defining the potential. The underlying reasons are shown to be related to the structure of the bound levels of the potential. It is shown that for a potential with a finite range but of general shape the reflection coefficient in general tends to unity near threshold, which is what is usually observed. However, in the presence of a bound level at the onset of the continuum, a finite portion of the incident particles is transmitted at threshold for a potential with a general shape and, furthermore, it can be shown that if the potential V is symmetric with V(x)=V(−x) then resonance transmission is observed, i.e., the reflection coefficient vanishes at threshold.
Particles interacting with sufficiently low energies have dynamics independent of the details of the interaction. This problem is generalized and extended within the context of an effective field theory (EFT). We investigate the case of two such particles in an harmonic trap and construct the inter-particle potential using an EFT expansion. In position space the effective interaction consists of a series of delta-functions and their derivatives. We approximate the energy spectrum in perturbation theory, compare to the results of analytical solutions and discuss the ordering of terms from the potential within the perturbation expansion.
This paper presents several cases in which the effects of the addition of a delta function potential on bound states can be computed exactly. In the case of the one dimensional Schrödinger equation, a Green’s function technique is used to compute an exact implicit expression that gives the effect of the one−dimensional delta function potential on the bound states eigenvalues. It is also shown that the weak coupling limit of the implicit expression agrees with the perturbative treatment of the delta function. It is pointed out that this technique cannot be applied to obtain an exact treatment of the delta function potential δ3(r) in the three−dimensional Schrödinger equation. Instead, an alternate definition of the delta function in three dimensions as a limit of a sequence of square well potentials is used. Unlike the one dimensional case, the exact effect on the eigenvalue does not agree in the limit of weak coupling with the perturbative results. This indicates that the three−dimensional delta function potential should not be used to approximate strongly coupled short range potentials for which perturbation theory is inadequate.
In this paper we will demonstrate the use of Feynman Diagrams for one dimensional scattering in quantum mechanics. We will evaluate the S-Matrix explicitly for the Dirac delta and finite wall potentials by summing the full series of Feynman diagrams, illustrating the spirit of perturbation theory. This technique may be useful in introductory quantum mechanics courses, and provides the student with intuition about conservation laws in the context of scattering problems by connecting Feynman diagrams, free propagation, and conservation of the corresponding observable. It also provides a toy model for calculating S-matrix elements in quantum field theory.
Scattering of non-interacting, identical bosons or fermions by a one-dimensional Dirac delta function barrier potential underlines the importance of the role of statistics (that is, whether the particles obey Fermi–Dirac or Bose–Einstein statistics) in the scattering. We consider an initial wave function for the system that corresponds to one particle incident from the left and one from the right of the potential barrier. For bosons, both particles are scattered either to the left or to the right if the intensity reflection coefficient is 1/2, provided the left and right propagating wave packets fully overlap in the scattering region. For fermions, the particles “pass through” one another, provided the left and right propagating wave packets fully overlap in the scattering region, with zero probability that both particles are scattered to the left or right, consistent with the Pauli exclusion principle.
Abstract In this paper, we propose a pedagogical presentation of the Lippmann–Schwinger equation as a powerful tool, so as to obtain important scattering information. In particular, we consider a one-dimensional system with a Schrödinger-type free Hamiltonian decorated with a sequence of N attractive Dirac delta interactions. We first write the Lippmann–Schwinger equation for the system and then solve it explicitly in terms of an N × N matrix. Then, we discuss the reflection and the transmission coefficients for an arbitrary number of centres and study the threshold anomaly for the N = 2 and N = 4 cases. We also study further features like the quantum metastable states and resonances, including their corresponding Gamow functions and virtual or antibound states. The use of the Lippmann–Schwinger equation simplifies our analysis enormously and gives exact results for an arbitrary number of Dirac delta potentials.