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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Product of delta distributions

I’m studying Kleinert theory and Delta functions of surfaces and curves, defined as $\boldsymbol{\delta}_S(x)=\int_S \delta^{(3)}(x-y) dy$ Do you know some references about the extension of the Dirac ...
3 votes
0 answers
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Question on limits and delta functions (Coming from work on Matsubara Green's functions and analytic continuation)

I am working with condensed matter field theory with theory along the lines of Altland and Simon. I have some Matsubara Green's functions where I want to do analytic continuation and in that regard I ...
3 votes
1 answer
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Mixed second functional derivative not symmetrical with respect to order of differentiation

I have a functional $E$, which is a functional of two different functions and their gradients: $$ E[\psi_1,\psi_2] = \int d^3\mathbf{x} ~ \varepsilon\{\mathbf{x}\}$$ where I'm using $\varepsilon\{\...
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Trouble proving Wigner function identity [closed]

I am trying to prove $$\int d^2 \alpha W(\alpha)=1$$ where $W(\alpha)$ represents the Wigner funcion. However, I have trouble solving it. I tried solving it as follows but I think I have done some ...
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5 votes
1 answer
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Evolution of a position state in an infinite well potential

Let the potential be $$V = \infty \hspace{3cm}(0>x, x>L)$$ $$V = 0 \hspace{3.7cm}(L>x>0).$$ Now, we measure the position of a particle and discover it is located at $L/4$. What is the ...
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1 vote
1 answer
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Why does QFT require operator-valued distributions? [duplicate]

I am new to QFT, and so far I have only gone through the basics up to defining what a quantum field is, which is an operator valued distribution. I have been struggling so far understanding why ...
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-1 votes
1 answer
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Integration with the Dirac delta function

I have equation like $$I = \int \prod\limits_{i=1}^{N} dr_{i}e^{-\mathrm i\beta \int dr \sum\limits_{i}q_{i}\delta(r-r_{i})\phi(r)}.$$ First, I did integration in the exponent and got \begin{align} I &...
2 votes
0 answers
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Why is there a $2\pi$ in $\int dk ~ e^{ik(x-x')} =2\pi~\delta(x-x')$? [duplicate]

This comes up in Physics from Symmetry by Jakob Schwichtenberg. That's where the factors of $2\pi$ come from. Another way of seeing this is demanding the solutions to form an orthonormal set: $$\int ...
1 vote
1 answer
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Understanding operator valued distributions

Suppose $a(x)$ is an operator valued distribution (i.e. a linear map that associates an operator to each test function). If $f$ is a test function in a Schwartz space, the annihilation operator $A$ is ...
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5 votes
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How does the Dirac delta, written as $\delta(x)=\frac1{2\pi}\int e^{i\omega x}d\omega$, satisfy $\int_{-k}^k \delta(x)dx=1$? [migrated]

We see the Dirac delta representation as follows, $$\delta(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty} e^{i\omega x} d\omega$$ I want to know, how does this satisfy the following? ($k > 0$) $$\int_{-...
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Single vs. Double Dirac Potential

Both single and double Dirac potentials have an even ground state solution. Supposing the 'strengths' $\alpha$ of the single and double Dirac potentials to be the same, i.e., supposing $V_{single} = -\...
4 votes
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Regularisation of supersymmetric two-point function

In four-dimensional Minkowski space, one of the building blocks for two-point functions in CFTs is the squared modulus of the separation between the two points $x_1$ and $x_2$, $$x_{12}^2 = x_{12}^ax_{...
1 vote
2 answers
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Energy and momentum conservation using Dirac delta function

I found in many text of QED dealing with scattering, the scattering matrix $S_{fi} \propto$ $\delta^4(p_f -p_i)$. They say that the $\delta$ function ensures the conservation of momentum and energy. ...
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4 votes
1 answer
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Linear Response theory

I was reading through Lecture of of Prof Patrick Lee on Linear Response Theory. I have found the following relation and could not understand why is it true: $$\Im \left\{\frac{1}{x + i\eta}\right\} = -...
0 votes
1 answer
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Expressing the four-momentum operator in terms of field operators

There are a series of problems in chapter 3 of the book Quantum Field Theory of Point Particles and Strings by Hatfield that lead to a proof of Lorentz invariance in the canonical formulation of ...
14 votes
5 answers
2k views

To what extent can we use the informal version of the Dirac delta function in Physics?

Apparently expressions such as $$ \int \delta (x) f(x)dx = f(0)\tag{1}$$ are widely used in Physics. After a little discussion in the Math SE, I realized that these expression are absolutely wrong ...
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2 votes
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Dirac delta function at singularities in spherical coordinates [migrated]

Background Information Let $\delta^3(\vec x-\vec a)$ represent a point density at $\vec a$. It satisfies $$ f(\vec a)=\int \delta^3(\vec x-\vec a)f(\vec x)|J(\vec x)|\mathrm d^3\vec x, $$ where $f(\...
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Factor of $(2\pi)^4$ in momentum space Feynman rule

I'm trying to figure out the momentum space Feynman rules using Peskin and Schroeder. For simplicity I'll ask about section 4.6 for case of the $\phi^4$ theory. In section 4.5, we have $$\tag{4.72}S=1+...
2 votes
1 answer
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Why two energies look same in the relativistic normalization?

I'm reading Peskin's QFT textbook. In this book, to make normalization of momentum eigenstate Lorentz invariant, we define momentum eigenket as $$\left| \mathbf{p} \right> = \sqrt{2E_{\mathbf{p}}} ...
-1 votes
1 answer
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What is the difference between Dirac delta function orthogonality and Kronecker delta orthogonality?

In the derivation of Bloch Wave, I encountered a problem. First of all this is the definition of Bloch Wave: $$ \psi_{n\mathbf{k}} (\mathbf{r} ) = e^{i\mathbf{k} \cdot \mathbf{r} } u_{n\mathbf{k}} (\...
2 votes
2 answers
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Two identities involving Dirac deltas

In A. Smilga (2001) Lectures in Quantum Chromodynamics (p. 10) the following two identities are presented: $$\delta'(x-y)[f(y)-f(x)] = -\delta'(x-y)(f'(x)(x-y)) = \delta(x-y)f'(x)$$ but no proof is ...
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7 votes
1 answer
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Where does it become apparent in real scalar QFT that the field has to be an operator-valued distribution, as opposed to an operator-valued function?

It its very often stated that in QFT, we don't actually deal with operator valued functions (assign a field operator to each point in space time), but instead with operator valued distributions (in ...
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9 votes
5 answers
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Wave Function Collapse and the Dirac Delta Function

When the wave function of a quantum system collapses, the probability of finding it at some specific point is given depends on $||\Psi||^2$: $$ \int_{\mathbb{R}^3}{d^3 \mathbf x \; ||\Psi||^2} = 1 $$ ...
4 votes
2 answers
413 views

What is the stress-energy tensor in this case?

In case of static point mass we have the following stress energy tensor $$T_{00}=m\delta^3(\vec{r})$$ And other components are zero. What are the components of this tensor in case of moving point mass ...
0 votes
0 answers
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$\delta$-function integral when calculating the differential cross section

I am reading an article and trying to do step-by-step calculations for some other similar work I am working on. The article is about scattering of a photon on a positronium. In the initial state there ...
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1 answer
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Two questions concerning dirac delta function and Hamiltonian

I'm trying to compute to quantities with Hamiltonian and Dirac delta function but I don't how to do it properly. I'm stuck calculating the following quantity $$ \frac{d}{dE} \left[ \theta(E-H(x,p;V)) ...
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2 votes
1 answer
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Metric at the junction

In general relativity, while deriving junction conditions one works with the metric $$g_{\mu\nu} = \Theta(f) g^{+}_{\mu\nu} + \Theta(-f) g^{-}_{\mu\nu} \tag{1}$$ where $f = 0$ is the junction. Now, ...
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1 vote
0 answers
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Intervals between the last drops of liquid

When a bottle is nearly empty, the remaining liquid would come in the form of drops, which are becoming less and less frequent. I am wondering, if there is distribution describing the intervals ...
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2 votes
1 answer
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Delta function squared in Weigand's QFT notes

On page 74 of Timo Weigand's QFT notes, right at the top, the following equality is used: $$\left[(2\pi)^4\delta^{(4)}(p_f-p_i)\right]^2=(2\pi)^4\delta^{(4)}(p_f-p_i)(2\pi)^4\delta^{(4)}(0) \tag{2.167}...
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0 votes
3 answers
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Why does the integral of $E\psi(x)dx$ go to zero around the the delta function? [closed]

My lecturer writes: Firstly, I assume the term with a second derivative is, well, exactly that - a second derivative and therefore intended to be $\frac{d^2\psi(x)}{dx^2}$ and not $\frac{d^2\psi(x)}{...
1 vote
0 answers
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Reference on Curie-Weiss model

I am looking for a reference on the Curie-Weiss model and mean-field approximation. Model. Consider the Curie-Weiss model with the following Hamiltonian: \begin{align*} H = - \frac{J}{2N} \sum_{i \neq ...
3 votes
1 answer
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Curl of a vector field at a single point

I have always imagined the magnetic field of wires, as the superposition of infinitely many curl elements. I, naturally, wanted to see what a function with a single point of curl would look like. The ...
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0 votes
1 answer
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Klein-Gordon solution's Fourier image

I'm solving Klein-Gordon equation in order to get scalar field expression. $$(\partial^2 + m^2)\phi=0$$ I expand solution $\phi$ into Fourier integral in momentum space: $$\phi=\int\frac{d^4p}{(2\pi)^...
0 votes
1 answer
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Three Delta Function Potential -- Possible Nodes [closed]

Let us say you have a potential function of the following form: $$V(x) = -V_o a \sum_{n = -1}^{1} \delta(x-na)$$ where $V_o > 0, a>0$ I am trying to figure out the number of possible nodes that ...
1 vote
0 answers
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How to deal with the appearance of delta functions in differential cross sections?

I've recently started learning quantum field theory, and I'm on chapter 3 of Hatfield's Quantum Field Theory of Point Particles and Strings. I was flipping through the book today, and I came across an ...
2 votes
3 answers
228 views

Delta function singularity in curvature

Are there 3+1D spacetimes that lead to a $\delta$-function in curvature? Are there any examples that one can provide?
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1 answer
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How to deal with the force of the form $\delta (\mathbf{r}) u(\mathbf{r})$, $u(\mathbf{r})$is the interaction potential

Now I have such an expression for potential energy: $$ U_j(\mathbf{r}_j) = \int_{\mathbf{r}_j-\frac{\mathbf{L}^b}{2}}^{\mathbf{r}_j+\frac{\mathbf{L}^b}{2}} d\mathbf{r} \sum_{\mathbf{n}} \sum_{i\ne j}^{...
0 votes
1 answer
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Derivation of the Yukawa potential by integration

I'm trying to solve the Yukawa potential using standard integration methods, but can't seem to be getting the correct result. This derivation is a part of Bellan's "Fundamentals of Plasma Physics&...
2 votes
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What is the wavefunction of definite position? [duplicate]

Reading the quantum mechanics textbook we are told the wave function for a definite position at $a$ is $\psi(x)=\delta(x-a)$. Yet, also we are told that the probability must be $\int|\psi(x)|^2 dx$=1. ...
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6 votes
1 answer
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Integration of Laplacian by parts

I'm trying to solve assignment (1.5) in Bellan's "Fundamentals of Plasma Physics" using Fourier transforms, but I'm stuck integrating the Laplacian. Here's the problem: Equation (1.5) is ...
0 votes
0 answers
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Doubt on delta function [duplicate]

So, I was solving a problem on quantum mechanics from a problem set (Problem 4) provided by MIT OCW. I wasn't able to solve it on my own, so I looked at the solution. I am having a doubt where $$\int_{...
3 votes
3 answers
666 views

What is the difference between Dirac delta vs removing point from space approach?

Let us take for instance E&M, in it when we deal with the failure of divergence theorem to give us the right expression when evaluating the volume integral of divergence of electric field over ...
1 vote
1 answer
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Is there such a thing as infinitesimal electric field?

I am interested in calculating some response properties, namely, susceptibility and polarizability. In principle, susceptibility should be the functional derivative of the electron density to a ...
user avatar
0 votes
1 answer
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Delta function: Intuitive way for boundary conditions

Giving the Schrödinger equation $$-\dfrac{\hbar^2}{2\,m}\,{\partial_x}^2\psi(x)+ V(x)\,\psi(x) = E\,\psi(x)$$ with potential $V(x) = V_0\,\delta(x)$. Solving this equation using an ordinary Ansatz ...
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1 vote
1 answer
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Can a singular function plus a Dirac delta have be non-infinite?

In QFT we sometimes encounter functions of the form: $$K(x-y) = \delta(x-y) + \frac{k}{(x-y)^n} $$ Where $x$ and $y$ are $d$ dimensional vectors and $k$ is a (possibly imaginary) constant. These can ...
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-1 votes
1 answer
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Ordinarily continuous function of the wave function

I just started studying quantum mechanics using the textbook Introduction to Quantum Mechanics by Griffith. Under the section of solving the Shrodinger equation for a Dirac delta potential, he ...
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1 vote
1 answer
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Discretization of derivative of delta function and affine Kac-Moody algebra

In equation (4.16) of https://arxiv.org/abs/1506.06601, a discretization of the (classical) affine Kac-Moody algebra is presented: $$ \frac{1}{\gamma}\left\{J_{m}^{1}, J^2_{n}\right\}=J_{m}^{1} J_{n}^{...
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2 votes
2 answers
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Charge density of Hertzian dipole

I am wanting to find the charge density of an infinitely thin Hertzian dipole, but am struggling evaluating the Dirac delta functions gradient. $$\vec{J} = I_{0}\cos(\omega t) \delta^3(r) \hat k.$$ ...
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0 votes
0 answers
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Model of light distribution along radius

I measured the light intensity of Xe lamp along the radius at different points. By using polyfit function, I have found the function of light distribution along the radius. I want to model the light ...
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0 votes
1 answer
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Loop Integrals and Dimensional Regularization

I want to calculate the divergent part of a Feynman diagram using the Feynman parameters: $$\frac{1}{A_1 A_2 \ldots A_n} = \int_0^1 dx_1 ... dx_n \delta (\Sigma x_i -1) \frac{(n-1)!}{[x_1 A_1 + x_2 ...
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