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.

64 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
11
votes
0answers
224 views

Derivatives of distributions in general relativity

I am having some trouble when trying to reproduce some calculations involving the description of distributions (mostly used in spacetime junction conditions). I am trying to reproduce the ...
5
votes
0answers
157 views

$D$-dimensional Schrodinger's equation with a Dirac delta potential

I know that for $D\geq 2$, there is no bound state for a Dirac potential $V=-\alpha \delta(\textbf{x})$ unless we use an ultraviolet cutoff $k_{max}=1/a$. I showed this by solving the Schrodinger's ...
5
votes
0answers
440 views

Free path distribution

I'm studying statistical mechanics, and I'm trying to resolve some problem known from my thermodynamics course. So I want to calculate mean free path for particles with a concentration $n$ and ...
4
votes
0answers
270 views

Laplacian and Dirac Delta function

Although I find it mathematically dubious, we said that $$\Delta \frac{1}{r} ~=~ -4\pi \delta^3({\bf r}).$$ Now, I was wondering is there a similar relation to the delta function if we look at $$\...
3
votes
0answers
182 views

Sommerfeld expansion for temperature dependent spectral function

Consider the integral \begin{align} I(\beta)=\int_{-\infty}^{\infty}\frac{\text{d}\epsilon}{\pi}\frac{\rho(\epsilon,\beta)}{1+e^{\beta \epsilon}} \end{align} where $\rho(\epsilon,\beta)$ is a generic ...
2
votes
0answers
19 views

Expected value of the current density operator

In Ullrich's TD-DFT book, the paramagnetic current-density operator is defined as $$\hat{\mathbf{j}}(\mathbf{r})=\frac{1}{2i}\sum_{a=1}^{N}\left[\nabla_a\delta(\mathbf{r}-\mathbf{r}_a)+\delta(\mathbf{...
2
votes
0answers
83 views

How can the propagator be written in the below integral form?

I’am finding it difficult to understand as to how the delta function is written as a product of many delta functions with the integral
2
votes
0answers
84 views

Half of Dirac Delta within spectral integration

I was taking a course in Quantum Information along this last semester, and apart from some mathematical details it all made sense to me. One of these mathematical tricks that I haven't been able to ...
2
votes
0answers
159 views

Classical field theory: marriage of differential geometry to functional analysis?

I have always wondered (and thoroughly - but perhaps not enough - searched for literature) how the purely geometrical formulation of classical field theory (take for example a hardcore text on this ...
2
votes
1answer
58 views

Why is it physically resonable to consider a function as a regular distribution?

In "Mathematical Methods in Physics" by Blanchard and Brüning the introduction of regular distributions and then general distributions is motivated by the following idea: When we think about the ...
2
votes
0answers
109 views

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 $$\frac{\delta F[\phi]}{\delta \phi} [f(x)] = \lim_{\varepsilon \to 0} \frac{F[\phi + \varepsilon f ] - F[\phi]}{\varepsilon}$$ ...
2
votes
1answer
122 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. ...
2
votes
0answers
71 views

Are vacuum expectation values distributions?

In PCT, Spin, Statistics and all That (1964) on page 106 the vacuum expectation value is introduced in the following way. The question concerns the highlighted area, that is how can a tempered ...
2
votes
0answers
677 views

Integration by parts with Dirac Delta function

I am having some hard time trying to understand the following "heuristic" integral, involving integration by parts with the Dirac's Delta. We start with the following relation $$ f(x) = \int_{-\infty}^...
2
votes
0answers
841 views

How to calculate the second functional derivative of the action of a one-particle system?

Given the Lagrangian $$L(q,\dot{q})=m\dot{q}^2/2-V(q)$$ and the corresponding action $$S[q]\equiv\int_0^t dt' (m\dot{q}^2/2-V(q)),$$ I need to be able to evaluate the second functional derivative $\...
2
votes
0answers
78 views

energy distribution of electrons from the heated cathode in magnetic field

I have a very specific question which is troubling me. I use a heated disk cathode as an electron emitter. I know that the energy distribution of the electrons emitting from the cathode is $g(E)=\...
2
votes
0answers
101 views

Poisson-like green functions

How can I verify that equation $$\nabla ^2 f (r) = - \frac{e}{4 \pi \epsilon ^2} \delta (r-\epsilon)$$ in 3D has a solution of the form $$f (r) = a - \frac{e}{4 \pi r} \theta (r-\epsilon) -\frac{e}{...
2
votes
0answers
261 views

Calculation of the Poisson bracket of a (Classical) Yang-Mills generator

This question might be too technical or minute, but I believe someone can give me the right advise. What I want to calculate is a Poisson bracket algebra of classical YM gauge generators, \begin{...
2
votes
0answers
244 views

Finding the moments of the Boltzmann/Gibbs Distribution

I am trying to compute the moments of the Boltzmann distribution using a moment generating function, by taking the Fourier transform of the distribution and then taking derivatives to find the ...
1
vote
0answers
30 views

Interpretation of induced force between two Dirac delta potential wells

This problem is based on MIT OCW course 8.04 problem set 6 question 5(e). https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2013/assignments/MIT8_04S13_ps6.pdf Consider two Dirac ...
1
vote
2answers
56 views

Different definitions of Functional Derivative

In studying QFT and General Relativity, I came across two different definitions of Functional Derivative, and I'd like to know if they are equivalent. Firstly, in Wald's book General Relativity, as ...
1
vote
0answers
99 views

Perturbation theory development for the ground state of the QM particle in the box with a centered dirac-delta spike

In the course of a discussion in the chat there emerged an interesting problem, namely a particle in an infinite well with an additional Dirac-delta function spike of scalable hight: $$ H = -\frac{\...
1
vote
0answers
88 views

Delta function constraint as Lagrange multiplier in SYK model calculation?

In eq. (112) of these lecture notes the author is introducing a 1 into an integral in the following way This looks like an integral representation of the delta function $\delta(x)\sim \int dy\, e^{i ...
1
vote
0answers
167 views

How can a Dirac delta function that does not occur under an integral be used to describe a transition rate?

In his excellent notes (found here), Mark Tuckerman shows that the transition rate of absorption between quantum states i and f, coupled by operator B, can be expressed as the fourier transform of the ...
1
vote
0answers
91 views

is it true that $\lim\limits_{\epsilon \to 0^{+}} \ln( x \pm i \epsilon ) = \mathscr{P}\ln|x| \pm i \pi \Theta(-x)$?

Is the following statement true? $$ \lim\limits_{\epsilon \to 0^{+}} \ln( x \pm i \epsilon ) = \mathscr{P}\ln|x| \pm i \pi \Theta(-x) $$ where $\mathscr{P}$ is the Cauchy principal value. The above ...
1
vote
0answers
377 views

Relationship between Green's function and impulse response

In my field, electrical engineering, we frequently study linear time-invariant systems of the following form: $$ a_n\frac{d^ny}{dt^n} + a_{n-1}\frac{d^{n-1}y}{dt^{n-1}} + \ldots + a_1\frac{dy}{dt} + ...
1
vote
0answers
259 views

Density of a thin disk by delta Dirac function

How to write a volume (charge for example) density using delta Dirac function in spherical coordinates in this case: There are thin charged disk (radius $a$) with surface charge density $\sigma$. The ...
1
vote
0answers
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. ...
1
vote
0answers
151 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 ...
1
vote
1answer
187 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 ...
1
vote
0answers
90 views

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}...
1
vote
0answers
2k views

Green's function for the free particle Hamiltonian

For a free particle in 1D, the equation for its Green function (in atomic units) is $$\left(E+\frac{1}{2}\frac{d^2}{dx^2}\right)G(x,x',E) = \delta(x-x')$$ The textbook I'm following says the solution ...
1
vote
0answers
29 views

Derivation of generation of time between two subsequent particle enters into computational domain

I have problem with understanding derivation of one equation in following problem. You have 1D computational domain (it is not 1D but because it is symmetrical and we are watching only radial ...
1
vote
0answers
235 views

Notation - d.o.f.'s for Grassmann delta functions in a SUSY field theory amplitude

I was reading the following paper http://arxiv.org/pdf/1306.2962v1.pdf as I stumbled upon an issue concerning counting and assigning the Grassmann degrees of freedom that appear in grassmann delta ...
1
vote
0answers
1k views

Delta correlated white noise

I am studying Brownian motion, specifically Langevin equation. This equation includes a force expressed by a white noise, say $\xi(t)$. One of the hypothesis is that it is $\delta$-correlated (since ...
1
vote
0answers
197 views

Particle in a higher-dimensional box with an attractive delta potential

Suppose you have a particle in the box $[0,L]^d$, with an attractive Dirac delta potential $-\delta_{\vec w}(x)$ at $\vec w$. How do you solve the Schroedinger equation for this system? In the case $...
0
votes
1answer
24 views

Deriving the form of radial distribution function in molecular dynamic simulation?

I am trying to derive the equation 8.16 in the attached image from the definition of the radial distribution function. However, I am unable to derive it with proper constants of $\frac{2}{N(N-1)}$ My ...
0
votes
1answer
36 views

Writing a two variable function$ f(x,t)$ in terms of Dirac-Delta $δ(x)$ function and a function $P(t)$?

How to write a two variable function $f(x,t)$ in terms of Dirac-Delta $\delta(x)$ function and a function $P(t)$? For example; I read something in a book. You can find the following picture. But ...
0
votes
0answers
61 views

Green function derived from the idea of position representation in quantum mechanics

I am trying to study green's function by using a lecture by NPTEL its link is https://www.youtube.com/watch?v=ZJ7v6VZQ32k&t=439s I don't get this step by him at the minute of ten. $$D_x G(x,x^{'...
0
votes
1answer
46 views

What is dipolar charge distribution?

An electric dipole is a system of two opposite point charges when their separation goes to zero and their charge goes to infinity in a way that the product of the charge and the separation remains ...
0
votes
0answers
47 views

What is the symmetry behind this degeneracy?

I was working on a quantum mechanics problem, involving the perturbation of the 3D cubical potential well: Suppose we perturb the infinite cubical well \begin{equation} V(x,y,z)=\begin{cases} 0, \...
0
votes
1answer
65 views

Derivation of the QFT Propagator

I don't understand how we get from the RHS to the last line. \begin{eqnarray} \left[ \hat{H}_x - i \frac{\partial}{\partial t_x} \right] G^+(x,t_x,y,t_y) &=& -i \delta (t_x - t_y) \sum_n{\...
0
votes
0answers
26 views

Bosonic Pair Distribution Function

In Schwabls Book "Advanced Quantum Mechanics" in the chapter for Bosons he calculates the Bosonic pair distribution function for noninteracting bosons. He said the expectation value of \begin{align*} \...
0
votes
1answer
49 views

Dirac delta normalization of electromagnetic fields

Usually, to quantize electromagnetic fields we use box normalization and therefore the normalization constant contains the dimensions of Volume V of the box. But if we perform the Dirac-delta ...
0
votes
0answers
112 views

A Naive Question about Delta Function and Wick Rotation

A delta function can be written as $$\delta(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}dp\,e^{ipx}.$$ I have a very poor understanding of the Wick rotation technique used in quantum field theory. ...
0
votes
1answer
78 views

Curl of magnetic field produced by current carrying wires with infinitesimal small area

Can Magnetic fields produced by thin current carrying wires with infinitesimal area have curl with a delta function in it ?? As area is Zero current density J definitely becomes infinite at where ...
0
votes
2answers
84 views

How to plot the graph of this expression which involves Dirac delta function?

I was doing a problem on electrostatics which required finding the charge density from the given electric field and then plot a graph of the charge density. I was able to find the charge density which ...
0
votes
0answers
70 views

Green's function regularization and delta distribution

I have a free Green's function which is proportional to a $2\times 2$ matrix: $$ G_0 = \frac{1}{E^2-E_k^2}\begin{pmatrix} a & b \\ c & d \end{pmatrix} $$ The total Green's function after ...
0
votes
2answers
29 views

probability distribution of dependent random variables

If we have dependent random variables, then what how is the distribution the pdf look like? Can it be a normal distribution? For example, additive white Gaussian noise (AWGN) has a normal distribution,...
0
votes
1answer
105 views

Integration over phase space for a one dimensional harmonic oscillator

The problem asks for a proof of the following equation, and I have no idea on how to approach this: $\int dx dp \delta(E-\frac{p^2}{2m}-\frac{m \omega^2 x^2}{2})f(E) = \frac{2\pi}{\omega}f(E)$ , for ...