Construction of Dirac Delta Function from the Discrete Orthonormal Basis of the Function Space

2017 ◽  
pp. 54-57
Author(s):  
Hari Prasad Lamichhane
Keyword(s):  
Function Space ◽  
Orthonormal Basis ◽  
Delta Function ◽  
Hamiltonian Operator ◽  
Dirac Delta Function ◽  
Basis Set ◽  
One Dimensional ◽  
Dirac Delta ◽  
Simple Basis ◽  
Infinite Potential Well

Orthonormal basis of the function space can be used to construct Dirac delta function. In particular, set of eigenfunctions of the Hamiltonian operator of a particle in one dimensional infinite potential well forms a non-degenerate discrete orthonormal basis of the function space. Such a simple basis set is suitable to study closure property of the basis and various properties of Dirac delta function in Physics graduate lab.The Himalayan Physics Vol. 6 & 7, April 2017 (54-57)

Dirac Delta Function from Closure Relation of Orthonormal Basis and its Use in Expanding Analytic Functions

2020 ◽  
Vol 6 (2) ◽  
pp. 158-163
Author(s):  
B. B. Dhanuk ◽  
K. Pudasainee ◽  
H. P. Lamichhane ◽  
R. P. Adhikari
Keyword(s):  
Analytic Functions ◽  
Orthonormal Basis ◽  
Special Functions ◽  
Single Point ◽  
Functional Space ◽  
Delta Function ◽  
Hamiltonian Operator ◽  
Dirac Delta Function ◽  
Closure Relation ◽  
Dirac Delta

One of revealing and widely used concepts in Physics and mathematics is the Dirac delta function. The Dirac delta function is a distribution on real lines which is zero everywhere except at a single point, where it is infinite. Dirac delta function has vital role in solving inhomogeneous differential equations. In addition, the Dirac delta functions can be used to explore harmonic information’s imbedded in the physical signals, various forms of Dirac delta function and can be constructed from the closure relation of orthonormal basis functions of functional space. Among many special functions, we have chosen the set of eigen functions of the Hamiltonian operator of harmonic oscillator and angular momentum operators for orthonormal basis. The closure relation of orthonormal functions  used to construct the generator of Dirac delta function which is used to expand analytic functions log(x + 2),exp(-x2) and x within the valid region of arguments.

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

2009 ◽  
Vol 31 (4) ◽  
pp. 4302-4308 ◽  
Author(s):  
F.A.B Coutinho ◽  
Y Nogami ◽  
F.M Toyama
Keyword(s):  
Quantum Mechanics ◽  
Delta Function ◽  
Dirac Delta Function ◽  
One Dimensional ◽  
Dirac Delta ◽  
Underlying Mechanisms ◽  
Usual Formula

There is a situation such that, when a function ƒ(<img src="/img/revistas/rbef/v31n4/a04x.gif" align="absmiddle">) is combined with the Dirac delta function δ(<img src="/img/revistas/rbef/v31n4/a04x.gif" align="absmiddle">), the usual formula <img src="/img/revistas/rbef/v31n4/a04form01.gif" align="absmiddle">does not hold. A similar situation may also be encountered with the derivative of the delta function δ'(<img src="/img/revistas/rbef/v31n4/a04x.gif" align="absmiddle">), regarding the validity of <img src="/img/revistas/rbef/v31n4/a04form02.gif" align="absmiddle">. 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.

scholarly journals Variational approach to the Schrödinger equation with a delta-function potential

2021 ◽  
Author(s):  
Francisco Marcelo Fernandez
Keyword(s):  
Variational Method ◽  
Delta Function ◽  
Basis Set ◽  
One Dimensional ◽  
Dirac Delta ◽  
Introductory Course ◽  
Relevant Role ◽  
Secular Equations ◽  
The One ◽  
Computer Algebra Software

Abstract We obtain accurate eigenvalues of the one-dimensional Schr\"{o}dinger equation with a Hamiltonian of the form $H_{g}=H+g\delta (x)$, where $\delta (x)$ is the Dirac delta function. We show that the well known Rayleigh-Ritz variational method is a suitable approach provided that the basis set takes into account the effect of the Dirac delta on the wavefunction. Present analysis may be suitable for an introductory course on quantum mechanics to illustrate the application of the Rayleigh-Ritz variational method to a problem where the boundary conditions play a relevant role and have to be introduced carefully into the trial function. Besides, the examples are suitable for motivating the students to resort to any computer-algebra software in order to calculate the required integrals and solve the secular equations.

Dirac delta regularization

2020 ◽  
Author(s):  
Matheus Pereira Lobo
Keyword(s):  
Delta Function ◽  
Dirac Delta Function ◽  
Dirac Delta ◽  
Finite Result

I present a finite result for the Dirac delta "function."

scholarly journals Electromagnetic Waves Through Disordered Systems: Comparison of Intensity, Transmission and Conductance

2001 ◽  
Vol 694 ◽  
Author(s):  
Fredy R Zypman ◽  
Gabriel Cwilich
Keyword(s):  
Probability Distribution ◽  
Disordered Systems ◽  
Electromagnetic Waves ◽  
Delta Function ◽  
Rayleigh Distribution ◽  
Dirac Delta Function ◽  
Universal Form ◽  
Dirac Delta ◽  
Diffusive Regime ◽  
Analytical Expressions

AbstractWe obtain the statistics of the intensity, transmission and conductance for scalar electromagnetic waves propagating through a disordered collection of scatterers. Our results show that the probability distribution for these quantities x, follow a universal form, YU(x) = xne−xμ. This family of functions includes the Rayleigh distribution (when α=0, μ=1) and the Dirac delta function (α →+ ∞), which are the expressions for intensity and transmission in the diffusive regime neglecting correlations. Finally, we find simple analytical expressions for the nth moment of the distributions and for to the ratio of the moments of the intensity and transmission, which generalizes the n! result valid in the previous case.

All about the dirac delta function(?)

Resonance ◽  
2003 ◽  
Vol 8 (8) ◽  
pp. 48-58 ◽  
Author(s):  
V Balakrishnan
Keyword(s):  
Delta Function ◽  
Dirac Delta Function ◽  
Dirac Delta
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