Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Revista Brasileira de Ensino de Física

Unusual situations that arise with the Dirac delta function and its derivative

Francisco Antônio Bezerra Coutinho, Yukihisa Nogami, F. Masafumi Toyama

Published Dec 1, 2009• 22 citations
Publisher

Abstract

There is a situation such that, when a function ƒ( ) is combined with the Dirac delta function δ( ), the usual formula does not hold. A similar situation may also be encountered with the derivative of the delta function δ'( ), regarding the validity of . We present an overview of such unusual situations and elucidate their underlying mechanisms. We discuss implications of the situations regarding the transmission-reflection problem of one-dimensional quantum mechanics.

DOI
10.1590/s1806-11172009000400004

Related papers

Transmission–reflection problem with a potential of the form of the derivative of the delta function

F. Masafumi Toyama, Yukihisa Nogami

Regarding the quantum mechanical transmission–reflection problem in one dimension with a potential of the form of the derivative of the Dirac delta function δ'(x) = dδ(x)/dx, Christiansen et al recently found that, depending on how δ'(x) is interpreted, there can be a resonance which leads to partial transmission. This is in contrast to the earlier consensus that such a potential allows no transmission. The δ'(x) can be regarded as the narrow-width limit of a certain function Δ'(x) of a finite range. Christiansen et al assumed a rectangular function for Δ'(x). We examine various other forms and how the resonance depends on the shape of Δ'(x). We also present some general observations related to the 'threshold anomaly'.

Journal of Physics A Mathematical and Theoretical•Jul 3, 2007•48 citations

Physical interpretations of the differentiation rule for the Dirac delta function

Denys I. Bondar, Robert R. Lompay, Wing‐Ki Liu

The canonical commutation relation between the quantum mechanical operators of momentum and coordinate and the Ehrenfest theorem in the free particle case are derived solely from the rule of differentiation of the delta function [$x\delta'(x) = - \delta(x)$].

•Jul 24, 2010•0 citations

Modified Dirac delta function and modified dirac delta potential in the quantum mechanics

Won Sang Chung, HASSAN HASSANABADI

The European Physical Journal Plus•Jan 1, 2022•2 citations

Dirac ?-function

Based on the definition of Dirac-delta function δ(x),no definition at x=0 for δ(x)is explained.The δ(x) possesses a dimension of the reciprocal of x and the physical significance of x operator's eigenfunction.Some ambiguities in quantum mechanics textbooks are clarified.

Institute of Physics eBooks•Nov 26, 2004•0 citations

Did you really miss a Delta function here?

Marcos Amaku, Francisco Antônio Bezerra Coutinho, P.C.C. dos Santos +1 more

This is the third article in a series dedicated to examining the use of the delta function in the physics literature. Here we discuss the appearance of a delta function in the second derivative of the wave function if its derivative is discontinuous. When this happens it is often claimed in the literature that the proposed wave function must be eigenfunction of an operator that contains a delta function and, in some cases, it is rejected. We study in this paper what we believe are alternatives ways to deal with this situation.

Revista Brasileira de Ensino de Física•Jan 1, 2024•0 citations

Scattering Study of Fermions due to Double Dirac Delta Potential in Quaternionic Relativistic Quantum Mechanics

HASSAN HASSANABADI, Hadi Sobhani, Won Sang Chung

We have studied the scattering problem of relativistic fermions from a quaternionic double Dirac delta potential. We have used Dirac equation in the presence of the scalar and vector potentials in the quaternionic formalism of relativistic quantum mechanics to study the problem. The wave functions of different regions have been derived. Then, using the reflection coefficient, transmission coefficient, and the continuity equation, the scattering problem has been investigated in detail. It has been shown that we have faced some fluctuations in the reflection and transmission coefficients.

arXiv:1709.09941•Advances in High Energy Physics•Jan 1, 2018•18 citations
There is a situation such that, when a function ƒ( ) is combined with the Dirac delta function δ( ), the usual formula does not hold. A similar situation may also be encountered with the derivative of the delta function δ'( ), regarding the validity of . We present an overview of such unusual situations and elucidate their underlying mechanisms. We discuss implications of the situations regarding the transmission-reflection problem of one-dimensional quantum mechanics.
Regarding the quantum mechanical transmission–reflection problem in one dimension with a potential of the form of the derivative of the Dirac delta function δ'(x) = dδ(x)/dx, Christiansen et al recently found that, depending on how δ'(x) is interpreted, there can be a resonance which leads to partial transmission. This is in contrast to the earlier consensus that such a potential allows no transmission. The δ'(x) can be regarded as the narrow-width limit of a certain function Δ'(x) of a finite range. Christiansen et al assumed a rectangular function for Δ'(x). We examine various other forms and how the resonance depends on the shape of Δ'(x). We also present some general observations related to the 'threshold anomaly'.
The canonical commutation relation between the quantum mechanical operators of momentum and coordinate and the Ehrenfest theorem in the free particle case are derived solely from the rule of differentiation of the delta function [xδ′(x)=−δ(x)x\delta'(x) = - \delta(x)xδ′(x)=−δ(x)].
This is the third article in a series dedicated to examining the use of the delta function in the physics literature. Here we discuss the appearance of a delta function in the second derivative of the wave function if its derivative is discontinuous. When this happens it is often claimed in the literature that the proposed wave function must be eigenfunction of an operator that contains a delta function and, in some cases, it is rejected. We study in this paper what we believe are alternatives ways to deal with this situation.
We have studied the scattering problem of relativistic fermions from a quaternionic double Dirac delta potential. We have used Dirac equation in the presence of the scalar and vector potentials in the quaternionic formalism of relativistic quantum mechanics to study the problem. The wave functions of different regions have been derived. Then, using the reflection coefficient, transmission coefficient, and the continuity equation, the scattering problem has been investigated in detail. It has been shown that we have faced some fluctuations in the reflection and transmission coefficients.