Questions tagged [dirac-delta-distributions]

Distributions are generalized functions, such as, e.g., the Dirac delta function. DO NOT USE THIS TAG for statistical probability distributions, profiles, graphs, plots, etc.

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Commutator of scalar field and its spatial derivative

Consider the usual commutation relations of two scalar fields $$\left[\phi_{m}\left(t,\boldsymbol{x}\right),\pi_{n}\left(t,\boldsymbol{y}\right)\right]=\boldsymbol{i}\delta_{mn}\delta\left(\...
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3answers
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Why include a constant in the delta potential $\alpha\delta(x)$?

Why do we have to multiply a proportionality constant in the delta potential $V(x)=-\alpha \delta(x)$ where $\alpha$ is some positive constant? Isn't that $V(x)=\pm \delta(x)$ already enough to ...
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0answers
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Under what circumstances is the functional derivative (of an action functional) an actual function?

In general, for a functional $F[\phi]$, the functional derivative is $$\frac{\delta F[\phi]}{\delta \phi} [f(x)] = \lim_{\varepsilon \to 0} \frac{F[\phi + \varepsilon f ] - F[\phi]}{\varepsilon}$$ ...
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Representing dimensions in Dirac delta function results

The Dirac delta function can be defined as $$\delta(x)=\frac{1}{2\pi}\int_{-\infty}^\infty e^{itx}dt$$ From this we see that the dirac function has units of $x^{-1}$. How do we represent the units ...
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$\delta(0)=\int_{-\infty}^\infty |x_1(x)|^2dx$?

In non-relativistic quantum mechanics: By definition $$\langle x_1|x_1\rangle=\int_{-\infty}^\infty |x_1(x)|^2dx.$$ On the other hand, $$\langle x_1|x_2\rangle=\delta(x_2-x_1).$$ Where $x_1$ and ...
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1answer
99 views

Free Fermion and Dirac Delta

Taking this paper (Zinn-Justin: Six-Vertex, Loop and Tiling Modles: Integrability and combinatorics) as reference (chapter 1), I would like to ask a question. First of all some fixed points: The ...
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1answer
335 views

Differentiating the electric field in Gauss's law, I get zero charge density. Can anyone help me where am I going wrong?

$$\mathbf E = \frac{1}{4\pi\epsilon_0}\frac{Q}{r^2} \mathbf e_r$$ $$\nabla \cdot\mathbf E = \frac{\rho_V}{\epsilon}$$ \begin{align} \implies \nabla\cdot\mathbf E & = \frac{1}{r^2}\frac{\partial}{\...
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202 views

Physical picture of Dirac's delta impulses [closed]

In linear response theory, the response $A(t)$ is related to the impulse $g(t)$ by $$A(t) = \int_{-\infty}^{\infty}\chi(t-t^\prime)g(t^\prime)\, dt^\prime $$ A typical example is the case where $g(...
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1answer
161 views

Quantum filter, potential barrier

I have been trying to teach myself quantum mechanics for quite a time now and I need help with a probably simple problem. We are looking at the Schrödinger equation of particles of mass $m$ in one ...
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1answer
502 views

The explicit solution of the time-dependent Schrödinger equation for a free particle that starts as a delta function

A previous thread discusses the solution of the time-dependent Schrödinger equation for a massive particle in one dimension that starts off in the state $\Psi(x,0) = \delta(x)$. This can easily be ...
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246 views

What is the definition of $\delta^{m|n}$ and $\delta^{m}$?

I am reading the paper. What is the definition of $\delta^{m|m}$ and $\delta^{m+k}$ in (1.1) and (1.3) on pages 2,3? Are they some kind of delta function?
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393 views

Integral representation of Dirac distribution

The Fourier transform of the Dirac distribution is given by $$\tilde \delta(\vec{k}) = \frac{1}{(\sqrt{2 \pi})^3} \int_{\Bbb R^3} \delta(\vec{r})e^{-i \vec{k} \vec{r}} d^3r = \frac{1}{(\sqrt{2 \pi})^...
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2answers
126 views

How to do the integrals over the multivariate delta function?

How to do this integration? $$ \int_{-\infty}^{\infty}dq\int_{-\infty}^{\infty} dp \; \delta(E-\frac{p^2}{2m}-\frac{k}{2}q^2)= 2\pi\sqrt{\frac{m}{k}}$$ I obtained the result using Mathematica, I am ...
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1answer
196 views

Commutation relation in derivative of creation operator?

I am basically trying to get the BMS hard charge for the subleading soft theorem. I have the usual commutation relation $[a(\omega',z',\overline{z'}),a^\dagger(\omega,z,\overline{z})]=\delta(\omega'-\...
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1answer
212 views

How to evaluate the coordinate transformation of Dirac delta function?

I have come across the following calculation while evaluating path integrals.How do one determine the coordinate transformation of delta function? For instance,If the delta function $\delta(X_{N}-X_{f}...
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406 views

Why is $\langle x| x' \rangle=\delta(x-x')$? [duplicate]

I've tried to find any solution or proof for $$\langle x| x' \rangle=\delta(x-x'),$$ but I only came to this post: Wave function and Dirac bra-ket notation So I got the information, that the vector $|...
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1answer
196 views

Why does the finite difference script for solving Poisson equation not work for delta function charge? How to fix it?

I know the charge density at the plates of a capacitor, from which the applied voltage and potential profile is to be calculated. Using the finite difference method in MATLAB script I solved the ...
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5answers
652 views

Simple question about Dirac delta-function [closed]

I can not undestand this mathematical formula: $$ \large \int_{a-\epsilon}^{a+\epsilon}f(x)\delta(x-a)dx=f(a) $$ I understand that it is the derivative of an integral evaluated in a, but still can ...
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1answer
44 views

How do I understand this symmetry of the Dirac delta function? [closed]

So I was just studying classical electromagnetic theory and I just learnt about the dirac delta (density) function. Out of curiosity, I just looked up why we choose to write the dirac delta function ...
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133 views

Can Maxwell's Equations in differential form be viewed as equalities of measures?

What I have in mind is the following - Suppose we choose to model the universe as a 3 dimensional flat Euclidean space $\mathbb{R}^3$ equipped with the standard topology and the Borel-sigma algebra. ...
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Why does the square integral of a Dirac-Delta-function blow up to infinity?

In Griffiths Introduction to Quantum Mechanics on page 102 it is shown, that the eigenfunctions of the position operator $\hat{x}=x$ are not normalizable. $$\int_{-\infty}^{\infty}g_{\lambda}\left(x\...
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1answer
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This vector potential gives a magnetic monopole field, what's wrong with it?

$\mathbf{A} = \frac{g(1-\cos\theta)}{r\sin\theta} \mathbf{\hat\phi} \Rightarrow \mathbf{B} = g \mathbf{\hat r}/r^2$ But yet the existence of $\mathbf{A}$ itself hinges on the fact that there are no ...
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1answer
512 views

Relativistic normalisation of delta functions

So in general with a Fourier transform you have something like: $$ \int d^3x\ e^{i\vec{q}\cdot\vec{x}}\ e^{-i\vec{p}\cdot\vec{x}} = \int d^3x\ e^{-i(\vec{p}-\vec{q})\cdot\vec{x}} = (2\pi)^3\ \delta^3(...
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1answer
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Usage of Dirac delta function in physics

I know that the Dirac delta function is not really a function but a distribution. It means that when we write $\int \delta(x-x_0) f(x) dx$ we want to say in fact $<\delta_{x_0},f>$. So the ...
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3answers
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Dirac Delta and Sloppy Notation

I am an undergraduate neuroscientist and recently I have been studying probability distributions in relation to information theory, and came across the definition of the Dirac Delta as a singular ...
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159 views

Counting experiment and standard error

I have a counting experiment: Let's say I have N identical bees. I take one of them, expose it to $\gamma$-radiation and look if it has died or survives. If it survives, I count it as 1 count. If it ...
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1answer
332 views

How to handle a Dirac delta at $r = 0$ for a first Born approximation?

I have been trying to get my hands around a scattering problem all day but I can't wrap my head around the idea. It's a scattering problem with First Born Approximation and the potential is a Dirac ...
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1answer
342 views

Problem with electric potential involving a delta function [closed]

Consider $V(r)=\dfrac{Ae^{-\lambda r}}{r}$ with $\lambda, A$ constants with appropriate units. Calculate the electric field, the charge density $\rho$ and the total charge $Q_{tot}$. (HINT: In a ...
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1answer
456 views

Exchange Integral and Derivative respect to a parameter of a Dirac delta-function

I'm trying to solve the 6.2 problem of Jackson's Classical Electrodynamics textbook. At some point, to get the desired solution, I have to exchange a derivative applied to a Dirac delta-function with ...
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1answer
97 views

Use of the delta function in the cancellation of real and virtual corrections

In light of the not so well defined integral $\int_a^b \delta(x-a) dx$ and from David Z's comment at the end of this Math.SE post, consider the following equation, which I've come across in many ...
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473 views

Why is the Dirac delta function used as model of impulse inputs?

I'm currently studying Stability of Flight, and there's a topic in this course that treats aircraft responses to certain maneuvers of the pilot. Consider for example, the impulsive deflection of the ...
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2answers
444 views

Dirac delta function property in a scattering proof

I'm studying the proof for the decoherence of the off diagonal elements of a density matrix through scattering with the environment and I'm stuck at a certain point: My problem is A1.14 relation. (A1....
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1answer
71 views

Distributive properties of quantum field theory

In the Quantum Field Theory(QFT), we work in the distributional sense, that the normalization of vacuum is \begin{equation}\langle0|0\rangle=2E(\vec{0})(2\pi)^3\delta(\vec{0})\end{equation} This fact ...
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1answer
136 views

Fourier transforming the wave equation twice

The wave equation $$\nabla^2 u(r,t)-\frac{1}{c^2}\frac{\partial^2 u}{\partial t^2}(r,t)=0$$ can be Fourier transformed with respect to time, using $\frac{\partial}{\partial t}=i\omega$, to obtain the ...
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1answer
267 views

Dirac Delta Function

In the image above the second faint part is the answer to the question c). The answer starts with the word evidently, and I am confused why it is evident. Also, since we are considering a shell, ...
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3answers
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Dirac delta function and volume charge density

I just got introduced to the Dirac delta function and one of the questions was to express volume charge density $\rho({\bf r})$ of a point charge $q$ at origin. I saw that the answer is related to ...
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1answer
849 views

Delta function potential in an infinite square well

I have a question concerning delta function potentials inside an infinite well, like this one to which we add the following delta function potential: $V_2(x)=\frac{\hbar^2}{2m}\frac{\beta}{a}\delta (...
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1answer
191 views

Using Delta Dirac function as a mathematical tool in Green's functions

So, I was studying green's functions and in general I understood that if I have an operator $\mathscr{O}$ that acts of a function $h_1(\vec{r})$ such that $$\mathscr{O}h_1(\vec{r})=h_2(\vec{r})$$ ...
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1answer
123 views

Towards a matrix element definition of PDF

In Schwartz's book, 'Quantum Field Theory and the Standard Model' P.$696$, he starts to derive an expression for a parton distribution function in terms of matrix elements evaluated on the lightcone. ...
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568 views

Group theory, character, representations, delta function, unit element

I am reading a physics paper (hep-th/9204083 v1 above eq (4.15)) where there is a nice formula that reads $$ \delta(U-\mathbb{1}) = \sum\limits_{r} \dim(r)\, \mathrm{tr}_r(U) $$ Here $\:U\:$ is a ...
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1answer
190 views

Singularity of $B$-field in a Dirac String

I was assigned this question related to Dirac strings: Given a vector potential $\vec{A}= \frac{1-\cos(\theta)}{r\sin(\theta)}\hat{\phi}$, show that there is a singularity in the B field ...
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1answer
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Charge density as Dirac delta function

I little bit confused while I'm trying to convert a cylindrical charge distribution to spherical. The question is: A uniformly charged thin disk of radius $a$ and surface charge density $\sigma$ ...
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82 views

How is $\sum_{q}{e^{-i\ {q} \cdot(x-y)}}=\delta(x-y)$?

Evening all, I'm working through some practise problems at the very beginning of a QFT text book, the question I'm having trouble with is as follows: Given $[\hat a_{\bar p} ,\hat a_{\bar q}^\dagger]...
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1answer
190 views

What is the Fourier transform of an exponential function with a delta function as part of its argument?

Specifically, I'm wondering if it is possible to integrate something like this $$ x(t) = \int\limits_{-\infty}^{\infty} X(f) \ e^{i2\pi f t} \ df $$ where $$ X(f) = \exp\left\{\frac{\alpha_0}{2}\...
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486 views

Role of molecular mass in the Maxwell-Boltzmann distribution

Let's take into consideration the Maxwell-Boltzmann distribution of molecular speeds in one dimension, say $$P(v_x)dv_x=\left(\frac{m}{2\pi K_BT}\right)^{1/2}e^{-mv^2_x/{2K_BT}}dv_x$$ As we can see, ...
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1answer
370 views

How can I recover the delta potential $V(x)\propto-\delta(x)$ as a limit of a finite square well?

If we consider the potential $V= - a V_0 \delta (x)$, it satisfies the bound state energy $$E= - \frac{m a^2 V_0^2 }{2 \hbar^2}.$$ I have done this solving Schrodinger's equation outside the potential ...
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4answers
209 views

Problem with physical application of Dirac Delta

Consider the problem of projectile motion in 2 dimensions. Launch angle is constant. Range of projectile, $x$, then depends only on launch speed, $v$, and is given by \begin{equation} x=v^2, \quad v\...
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0answers
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Second-order functional dervative of the Yang-Mills action by DeWitt

DeWitt calculated a second order functional dervative of a Yang-Mills action in Table I p. 1201 in his paper PR 162 (1967) 1195. The action is the usual one, $$S=-(1/4)\int d^4x F_{a\mu\nu}F^{a\mu\nu}...
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77 views

Field operator anti-commutator relation

For the field operators (fermions) $$\hat{\Psi}^\dagger_\sigma(x) = \dfrac{1}{\sqrt{V}}\sum_k e^{-ikx}~\hat{a}^\dagger_{k,\sigma}$$ $$\hat{\Psi}_\sigma(x) = \dfrac{1}{\sqrt{V}}\sum_k e^{ikx}~\hat{a}...
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What is “white light” ? Uniform wavelengths or uniform frequencies ?

Suppose you have a mixture of electromagnetic waves of wavelengths spreaded on the visible spectrum only (from $\lambda_{\text{min}} \sim 400 \, \text{nm}$ to $\lambda_{\text{max}} \sim 700 \, \text{...