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The Dirac delta function – a quick introduction

The Dirac delta function – a quick introduction The Dirac delta function i e δ(x) is a very useful object Strictly speaking it is not a function but a distribution - but that won’t make any difference to us One of the simplest ways to try to picture what δ(x) looks like is to consider what happens to the piece-wise function

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DIRAC DELTA FUNCTION IDENTITIES

Simplified production of DIRAC DELTA FUNCTION IDENTITIES Nicholas Wheeler Reed College Physics Department November 1997 Introduction To describe the smooth distribution of (say) a unit mass on the x-axis we introduce distribution function μ(x) with the understanding that μ(x) dx mass element dm in the neighborhood dx of the point x μ(x) dx = 1

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Dirac Delta Function

When may such a concept as the Dirac delta function be useful? Here is an example Let us imagine that we have (in the 3-D space) two molecular charge distributions: ρA (r ) and ρB (r ) Each of the distributions consists of the electronic part and the nuclear part How can such charge distributions be represented mathematically? There is no proble

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DIRAC DELTA FUNCTION

DIRAC DELTA FUNCTION IN ONE DIMENSION One of the weirder bits of mathematics that the physics student will en-counter is the Dirac delta function (x) In one dimension the ’function’ (technically it’s really not a function at all but a distribution) can be defined by saying (x) = 0 for all x 6= 0 but (x) = ¥ at x = 0

What tools did Paul Dirac use?
Paul Dirac introduced some useful formal tools (such as his notation for integrals and operators). One of them is the Dirac delta function δ(x ), an object then unknown to mathematicians, which turned out to be very useful in physics. We may think of it as of a function with the following characteristics1
Why did Paul Dirac win a Nobel Prize in physics?
Dirac shared the Nobel Prize in physics for 1933 with Erwin Schrödinger, "for the discovery of new productive forms of atomic theory." The Dirac delta function (x) is a useful function which was proposed by in 1930 by Paul Dirac in his mathematical formalism of quantum mechanics.
What is a Dirac delta function?
1. DIRAC DELTA FUNCTION IN ONE DIMENSION One of the weirder bits of mathematics that the physics student will en-counter is the Dirac delta function (x). In one dimension, the ’function’ (technically it’s really not a function at all, but a distribution) can be defined by saying (x) = 0 for all x 6= 0 but (x) = ¥ at x = 0.
When did Dirac introduce distribution theory?
Dirac introduced this in his 1930 textbook on quantum mechanics. (It showed up in other forms earlier as well.) Laurent Schwartz introduced the theory of distributions in 1945, which provided a framework for working with the Dirac delta function rigorously. This is kind of like the development of calculus.

An Application of the Dirac Delta Function

When may such a concept as the Dirac delta function be useful? Here is an example. Let us imagine that we have (in the 3-D space) two molecular charge distributions: ρA (r ) and ρB (r ). Each of the distributions consists of the electronic part and the nuclear part. How can such charge distributions be represented mathematically? There is no proble

ZB ,b δ r rb

− . b∈B At the nuclear positions, we put delta functions with “intensities” equal to the nuclear charges. For neutral molecules, ρnucl ,A(r )dr and ρnucl ,B(r )dr have to give +NA and +NB, respec-tively. Indeed, we have ρnucl ,A(r )dr = Z A,a booksite.elsevier.com

ZB b = NB. ,

B ∈ B ∈ Thus, the Dirac delta function enables us to write the total charge distributions and their interactions in an elegant way: ρA(r) = −ρel ,A(r ) + ρnucl ,A(r ) ρB(r ) = −ρel ,B(r ) + ρnucl ,B(r ). To demonstrate the difference, let us write the electrostatic interaction of the two charge distributions both without the Dirac delta functions:

ZB

− ρel dr , A(r ) ∈ B b r + − rb ρel ,A(r )ρel ,B(r ) drdr . r r − The four terms mean, respectively, the following interactions: nuclei of A with nuclei of B, nuclei of A with electrons of B, electrons of A with nuclei of B, electrons of A with electrons of B. With the Dirac delta function, the same expression reads: Einter = ρA(r)ρB r dr dr

Dirac delta function Laplace transform Differential Equations Khan Academy

Laplace transform of the dirac delta function Laplace transform Khan Academy

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