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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How do you know when you need to use distributions to represent charge densities? [on hold]

I tried to solve a problem using Gauss' law in the following way. Let's assume we have a spherical shell of radius $R$ with a charge $Q$ being homogenously distributed on its surface. I am trying to ...
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0answers
37 views

Dirac delta function equation intuition and proof [duplicate]

What is the intuition and where should I find proof of this equation (do not know what its name is). It is used to derive Gauss law from Newton equation. $${\nabla \cdot \Bigg ( ...
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0answers
16 views

Dirac Delta function inverse Fourier transform [migrated]

We know that the Fourier transform of the Dirac Delta function is defined as $$\int_{-\infty}^{\infty} \delta(t) e^{-j\omega t} dt = 1,$$ and if I were to reconstruct the function back in time ...
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0answers
63 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) ...
2
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1answer
60 views

Distributional Extension of a Hilbert Space

This question comes from the Complexification section of Thomas Thiemann's Modern Canonical Quantum General Relativity, but I believe it just deals with the foundations of quantum mechanics. The ...
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1answer
58 views

A boundary term for a Bessel Function?

I am reading 't Hooft lectures on black holes and on p. 35, it is stated, that it is not difficult to show that $$ K^*(\omega,a)=\int_0^{\infty} \frac{ds}{s} e^{-i\omega \ln{s} + ia(s-\frac{1}{s})} = ...
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1answer
106 views

Does regularity of distributions have anything to do with definiteness of their product?

Recently I've gone through some literature concerning causal perturbation theory (CPT). As is well known, it deals with UV divergences in QFT by defining products of (operator-valued) distributions ...
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1answer
54 views

Where does this relativistic relation involving the delta function come from?

\begin{equation} \int\delta(E^2-\mathbf{p}^2-m^2)dE=\frac{1}{2E_\mathbf{p}} \end{equation} Shouldn't integrating the delta function like this just give 1?
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43 views

Normalization of the overlap $\langle x'|p'\rangle$

Let $$\langle x'|p'\rangle = N \exp(\frac{ip'x'}{\hbar})$$ be the overlap between position and momentum space, where $N$ is a normalization constant to be determined. We can then compute $N$ by $$ ...
2
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0answers
33 views

For what values of $\lambda$ is the distribution $(x-i\varepsilon)^\lambda$ positive?

I've been reading the famous unpublished paper by Luescher and Mack "The energy momentum tensor of critical quantum field theories in 1+1 dimensions". In the proof of their main theorem, page 7 of the ...
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0answers
44 views

In Solution of the equation $\Box A_\sigma=0$. I find some mathematical difficulties [closed]

The solution of the equation $$\Box A^\mu(x)=0$$ has the form $$A^\mu(x)=\int\frac{d^4k}{(2\pi)^4} e^{ik^\mu x_\mu}A^\mu(k)\delta(k^2).$$ Where we take $A^\mu(k)$ of the form $$A^\mu(k)=\sum ...
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1answer
54 views

Boltzmann Distribution - why maximum number of microstates?

I've recently started to learn statistical mechanics and I've run into Boltzmann Distribution. I wanted to see how it is derived and found some articles on web, but no one of them explain why the idea ...
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0answers
58 views

Equivalence of delta functions when calculating decay rate [closed]

$\newcommand{\bs}{\boldsymbol}$ Hello, I'm currently working through the lecture notes of my Theoretical Particle Physics course, and there, we are calculating the decay rate of the following process ...
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0answers
49 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 ...
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1answer
84 views

Generalizing a Gaussian distribution

Perhaps this a nonsensical question but hear me out. I have a random variable $x$ whose moments I can calculate. The first moment $<x>$ is zero and the second $<x^2> = X^2$ is something ...
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1answer
116 views

Distributions (generalized functions) over manifolds

I have asked a similar question on the math stackexchange website, but since this type of question might have an answer that is known to physicists better than mathematicians I'm posting the question ...
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2answers
80 views

The delta function as an eigenfunction of the position operator explanation

$\delta (\textbf{r})$ can be interpreted as a wavefunction. [...] It is non-vanishing only for $\textbf{r}=0$. [...] $\delta(\textbf{r})$ is an eigenfunction of the position operator with ...
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51 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 ...
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1answer
43 views

Divergence of a vector field, going through the math [closed]

The example I'm working on has this given identity: $\bigtriangledown \cdot \mathbf{\bar{r}}=3$. The question is: find the divergence of a vector field $\bar{\mathbf{E}}=\frac{\mathbf{r}}{r^{3}}$. ...
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1answer
43 views

Regarding calculations with plane waves

I'm dealing with some basic calculations with plane waves and I'm having some trouble with an idea. It has been said in another question that if you take to momenta like, for example ...
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2answers
60 views

Expectation value with plane waves [duplicate]

Hey guys Im a little confused with the concept of plane waves and how to perform an expectation value. Let me show you by an example. Suppose you have a wave function of the form ...
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4answers
268 views

Fourier Transform of 1 [closed]

Consider the following convention for defining the Fourier transform $\hat{f}(\omega) = \int f(x) e^{-2 \pi i x \omega } d\omega $. Why is the Fourier transform of 1 equal to $\delta(\omega)$. ...
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557 views

How to interpret the derivative of the Dirac delta potential?

I met a Hamiltonian containing the derivative of the Dirac delta potential: It is surprising to see terms like $b \delta'(x)$. How to interpret $ \delta'(x) $ ?
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2answers
511 views

Velocity Maxwell-Boltzmann distribution for dummies

I have a volume with N molecules; I need to assign to each particle a velocity vector: $$|\mathbf{v}_{i}|=[v_{x}, v_{y}, v_{z}]^{T}$$ for the i-th molecule; the velocities must fallow the ...
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1answer
176 views

Green function solutions in electrostatics

I have a conducting plate on $x$-$y$ plane. So I have a boundary condition at $z=0$ $\Phi=0$ but, for $z>0$ I have a point charge at z=a which is expected to create a potential. $$ ...
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1answer
88 views

Phase space measure in positron-electron annihilation calculation

I'm still trying to calculate the cross-section of the $e^- e^+ \rightarrow \mu^- \mu^+$ interaction in first order. This time I'm struggling with the phase space measure. Note that I have two ...
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1answer
115 views

Is it equivalent to derive Gauss's law from discrete and continuous source distributions?

I've seen two derivations for Gauss's law in electrostatics. The first assumes a discrete charge distribution, the second a continuous one: Use superposition $$\vec{E}=\sum_{i=1}^n\vec{E}_i,$$ so ...
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Field Lagrangian <--> Particle Lagrangian

The action-functionals describing the motion $\mathbf{x}:[a,b]\to \mathbb{R}^3$ of a free particle of mass $m$ and the evolution $\varphi:[a,b]\times \Omega\to \mathbb{R}$ of a free scalar field of ...
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1answer
159 views

How do we normalize a delta function position space wave function? [duplicate]

I have a position space wavefunction $$\psi(x) = \delta(x-a) + \delta(x+a).$$ Now the question states to compute the following: The Fourier transform of $\psi(x)$. (Which invariably is the momentum ...
5
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1answer
159 views

Functional derivatives as distributions

I have asked this on math stack exchange, due to its mostly mathemtical content, but aside from one upvote and minimal views it has not garnered any attention, so I am trying here as well. This isn't ...
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2answers
282 views

Triple integral $\iiint_{\mathbb{R}^3} d^{3}q ~\delta^{3}(\vec{q})\frac{(\vec{p}\cdot\vec{q})^2}{q^{2}} $ involving Dirac Delta function

I am trying find $$\iiint_{\mathbb{R}^3} d^{3}q ~\delta^{3}(\vec{q})\frac{(\vec{p}\cdot\vec{q})^2}{q^{2}},$$ where $\vec{p}$ is some fixed vector. The answer should be $\frac{p^2}{3}$. Below is ...
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165 views

Why is $\nabla\cdot(\hat{\bf r}/r^2)$ giving 0 as answer? [closed]

While I was reading I encountered the statement $\nabla\cdot(\hat{\bf r}/r^2)$ (r cap divided by $r$ square) is 0. Can anyone explain proof of the statement why is it giving 0?
4
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1answer
190 views

Second derivative of dirac delta expression

I have come across the expression $$ \int f(x) \delta(x-a) \delta''(x-a) \mathrm dx$$ where the prime represents the derivative. Usually with derivatives of the delta distribution I'd partially ...
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0answers
55 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, ...
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2answers
121 views

Unitary change of X Basis: Shankar Exercise 7.4.9 [closed]

I'm currently working through Shankar's Quantum Mechanics and am stuck on one of his exercises. In Exercise 7.4.9 Shankar would like us to show that: if $\mid x \rangle$ is changed to $\mid ...
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4answers
282 views

Having trouble understanding some stuff about delta functions [closed]

I was going through one of the examples in Griffith's Quantum book and there was a few things in Example 3.3 that I didn't understand that I was hoping to get some clarification on. For instance, we ...
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751 views

Mathematical understanding of Quantum Mechanics

Assuming that $\phi(r) = F (\psi(r))$ for some operator $F$ in Quantum Mechanics. Then, in our lecture today, we said that $$\phi(r) = \langle r|F |\psi\rangle = \int_{\mathbb{R}} \langle r |F| r' ...
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1answer
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Yang-Mills constraints and Poisson brackets

Let's have constraints for Yang-Mills theory: $$ \varphi_{a} = \partial_{i}\pi^{i}_{a} - f_{abc}\pi^{b}_{i}A^{c}_{i}. $$ I have read the statement that $$ \tag 1 [\varphi_{a}(\mathbf x), ...
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1answer
126 views

Is it enough to use only sigma criterion to calculate expected value for ideal Bell's test experiment?

Let's build some custom distribution for particle measurement result in CHSH test. Let's assume ideal particles and detectors. Let's also assume that each particle has hidden parameter $z$. Parameter ...
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2answers
300 views

What is the expectation value of the 3D delta function for the Hydrogen atom ground state?

I'm trying to evaluate the expectation value of some perturbation Hamiltonian $$H=\alpha \delta^3(\vec{r}),$$ where $\alpha$ is a positive constant, for the ground state wavefunction of the hydrogen ...
2
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1answer
97 views

Dirac Notation Question Appearing In a Projection

So I have a part of the energy eigenvalue equation that look like this: $$ \delta(\hat{x})|n\rangle $$ Where n is the energy basis of the Hamiltonian I'm considering. To deal with this, I tried ...
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1k views

What is the square root of the Dirac Delta Function?

What is the square root of the Dirac Delta Function? Is it defined for functional integrals? Can it be used to describe quantum wave functions? \begin{align} \int_{-\infty}^{\infty} ...
6
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1answer
292 views

Mathematical interpretation of Poisson Brackets

Lets say we are working in a classical scalar field theory and we have two functional $ F[\phi, \pi](x)$ and $G[\phi, \pi](x)$. In most of the references, starting with two functional the Poisson ...
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1answer
136 views

Fourier integral form of the delta function?

I'm learning basic maths for physicist and was wondering what do we use the Dirac delta function for? What is the difference between "the Fourier integral form" and the usual way of expressing the ...
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168 views

Eigenstates of a Hermitian field operator

Consider a Hermitian field operator $\phi(x)$ with eigenstates satisfying $$ \phi(x) |\alpha\rangle = \alpha(x) | \alpha \rangle $$ I'm trying to determine the inner product between the eigenstates. ...
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How do you handle a functional input in a Dirac delta function and prove these types of relations?

I have a quadratic relation inside of a Dirac delta function with the following relation \begin{align} \delta((x-x_1)(x-x_2)) = \dfrac{ \delta(x-x_1) + \delta(x-x_2) }{|x_1-x_2|}. \end{align} How do ...
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516 views

The Dirac-Delta function as an initial state for the quantum free particle

I want to ask if it is reasonable that I use the Dirac-Delta function as an intial state ($\Psi (x,0) $) for the free particle wavefunction and interpret it such that I say that the particle is ...
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75 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 ...
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3answers
287 views

Describing a circular current loop as delta functions

It would be really nice to see how Jackson got eqn 5.33 on his example problem for finding the vector potential of a circular current loop $$ ...
2
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1answer
126 views

Where does this delta of zero come from?

It is common when evaluating the partition function for a $O(N)$ non-linear sigma model to enforce the confinement to the $N$-sphere with a delta functional, so that $$ Z ~=~ \int d[\pi] d[\sigma] ~ ...