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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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How can we evaluate the following integral using the tricks of delta functions? [migrated]

I am trying to teach myself the statistical field theory formulation of statistical mechanics. Not part of a class, just self study in my free time. I appreciate any help here. I am starting with ...
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What is the relationship between wave function and classical distribution function?

The question is a bit weird since the wave function is quantum mechanical and the distribution function in phase space is really something classical. But I would still like to know if I take the ...
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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^{'...
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Mathematical rigorous definition for an electrical dipole

I've been reading Laurent Schwartz's Mathematics for the physical sciences, and in the chapter on distributions he makes many cool examples of ways to define in a mathematical rigorous way certain ...
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Position representation of an operator

$$\langle\ x\rvert M\lvert\ x'\rangle=M(x)\langle\ x\lvert\ x'\rangle=M(x)\delta(x-x')$$ I know this is true for if $M$ is a momentum operator or position operator, is this is true for a general ...
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Itzykson and Zuber: Reduced $T$-matrix

Could someone help me understand the reduced $T$-matrix mentioned in Itzykson and Zuber, eq. $$\langle{f}| T|p_1p_2\rangle=(2\pi)^4\delta^4(P_f-p_1-p_2)\langle f|\mathcal{T}|p_1p_2\rangle. \tag{5-7}$...
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Is it true that any electrostatic charge density can be represented by Dirac Delta function? Give me a general example

My college professor said that any charge density can be replaced by Dirac Delta function but how I want to know.
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Finding normalization constant of a wave function with definite momentum

I try to read Sakurai's Modern Quantum Mechanics but I stuck at this point, $$\delta(x^{'}-x^{''})=|N|^{2}\int dp^{'}\exp\Biggl({ip^{'}(x^{'}-x^{''})2\pi\over h}\Biggr)$$ This is an expression for ...
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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{...
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1answer
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Why do Candelas and Howard say that $\sum_{n=1}^\infty \cos\left( n \kappa \epsilon \right) \ = \ - \frac{1}{2}$?

In the paper Vacuum $\langle \phi^2 \rangle$ in Schwarzschild Spacetime by Candalas and Howard, they say that for each non-zero $\epsilon$ it is true that $$ \sum_{n=1}^\infty \cos\left( n \kappa \...
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Problem in the continuum limit of a Kronecker delta

I am having troubles in understanding how to correctly perform the continuum limit of a double sum containing a Kronecker delta. Imagine to integrate a function depending on $t$ and $t'$, both ...
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Factorising a 4D Dirac delta function in a rest frame

I'm working through a QFT problem and at one stage in the solutions we have this step: $$\delta^{(4)}(p - q_1 - q_2) = \delta(E_1 +E_2 - M)\delta^{(3)}(\bf{q_1} - \bf{q_2}).$$ We are working in the ...
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Properties of Dirac delta function in Integral

I was reading commutation relation of canonical momentum in KG Field from Lectures of Quantum Field Theory by Ashok Das. In page 179, He has used Integration to derive the result where he expressed ...
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What is the volume charge density (in spherical coordinates)?

What is the volume charge density (in spherical coordinates) of a uniform, in finitesimally thin spherical shell of radius $R$ and total charge $Q$, centered at the origin? Give your answer in terms ...
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$\delta^{(2)}$ convention

In this note: https://arxiv.org/abs/hep-th/0410165 at page 12 there is a delta-function constraint written as: \begin{align} \delta \left( ^ { U } M \right) = \prod _ { i < j } \delta ^ { ( 2 ) } \...
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A question about $\delta(x) \star \delta(p)$

Using the Moyal product between two delta functions in $(x,p)$-space one gets $$ \delta(x) \star \delta(p) = \frac{1}{\pi} e^{2ixp}. $$ However, $\delta(-x)=\delta(x)$ and last time I checked $e^{...
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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 ...
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1answer
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Dirac delta function mathematical expression proof

In a discussion of the second order transitions in graphene this mathematical expression is used. $$ \left|\frac{1}{\varepsilon + i \Gamma/2}\right|^2 = \frac{2\pi}{\Gamma}\delta(\epsilon) $$ And I'm ...
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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, \...
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1answer
64 views

Delta function eigenstate for non-zero potential

Consider the potential $V(x)=\frac{2}{x^2}$ and let $\frac{\hbar^2}{2m}=1$ for convenience. Now consider the function $\psi(x)=\delta(x)$. According to Griffiths (electrodynamics book) problem 1.45(a),...
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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{\...
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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*} \...
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1answer
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Anomalous magnetic moment of the electron - integration problem

In Schwartz's QFT book (eqn 17.31), to find the anomalous magnetic moment of the electron from the form factors, near the end of the calculation the following integral needs to be evaluated: $$ F_{2}(...
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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 ...
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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 ...
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Explanation of the identities $\rho=\rho(\delta)$ ($\delta$ function) and $\rho=q\,\delta(\bar{r})$

Poisson's equation is $$\boldsymbol{\nabla}^{2}\varphi=-4\pi k_{e}\,\rho, \tag{*}$$ that in the case of a point charge $q$, already with spherical symmetry, has as solution \begin{equation} \varphi(...
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2answers
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Curl and circulation of a vector field that is ill-defined at the origin: any interesting physical effects?

In the cylindrical polar $(\rho,\phi,z)$ coordinate, suppose the velocity field in a liquid is given by $$\vec{v}=\frac{K}{\rho}\hat{e}_{\phi}, \qquad K=\text{constant}.\tag{1}$$ It can be easily ...
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1answer
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Divergence of $ \frac{1}{s}\hat{s}$ in cylindrical coordinates

In Griffiths' electrodynamics, the divergence of $\frac{1}{r^{2}}\hat{r}$ is evaluated in spherical coordinates to be $4\pi\delta(r)$. I encounter the same problem in case of $\frac{1}{s}\hat{s}$ ...
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A formula for delta function in quantum mechanics

I met a formula for delta function in a QM book ( not in English). The formula is used in the scattering theory. Its form is $$\lim_{\alpha\rightarrow \infty}\exp[i\alpha x]=2i\delta(x), ~~(x\geq 0).$...
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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{\...
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2answers
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Wilson-Sommerfeld Quantization of Dirac delta in Infinite Square Well (ISW)

I am curious to find the energies of Dirac delta potential inside the ISW (walls at $x=0,L$) $$ H = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V_0\delta(x-L/2) $$ using Wilson-Sommerfeld ...
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If one puts a delta-function spike inside an infinite square well, is the resulting potential analytically solvable?

It was recently floated in chat that a particle in a box with a delta-function spike inside it, with hamiltonian $$ H = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V_0\delta(x-a) $$ and with ...
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Integrating Laplace's equation over a sphere

The Wikipedia page on Laplace's equation states that if the Laplacian of $u$ is integrated over any volume that encloses the source point, $$\iiint_V \nabla \cdot \nabla u \, d^3V =-1.$$ I can'...
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Is the DOS (density of states) wrong for degenerated case?

The density of states (DOS) is defined as $$\mathcal{N}\left(\lambda\right)=\sum_{n=1}^{M}\delta\left(\lambda-\lambda_{n}\right).$$ We can then get $$\int d\lambda\mathcal{N}\left(\lambda\right)=M,$$ ...
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Mathematical expression of impulsive forces during a collision

I'm facing with problem regarding collision between bodies. Specifically, I need to understand the impulsive forces that arises during a collision. Sometime ago, I posted two questions (this and this) ...
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Kac-Moody algebra from WZW model via Poisson brackets

In 'Non-abelian Bosonization in Two Dimensions', Witten shows that the Poisson brackets of the currents that generate the $G\times G$ symmetry of the WZW model give rise to a Kac-Moody algebra upon ...
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$\delta$ potential has highest probability for highest potential

I can't understand this intuitively. Figure 2.9 in Griffith's QM says that the wavefunction at $x=0$ for a delta function potential is $\sqrt{\kappa}$, and to the right it decays like $\psi_+=\sqrt{\...
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Square root of Dirac delta Function? [duplicate]

I've puzzled by the appeareance and manipulation of a Square root of a Dirac Delta function. The article is https://www.researchgate.net/publication/...
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How can we argue $\nabla\cdot v = 0$ when ${\mathscr r} \ne 0$ on this vector function?

I am dealing with the vector field: $$v = \dfrac{\hat{\mathscr r} }{{\mathscr r}^2}$$ And I am studying its divergence. If we compute it we get: $$\nabla\cdot\left(\dfrac{\hat{\mathscr r} }{{\...
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1answer
68 views

A computation problem on reciprocal lattice

I am reading David Tong’s lecture notes on Application of Quantum Mechanics. My confusion is about the following paragraph: Consider a function $f(\vec{x}) $, suppose $f(\vec{x})=f(\vec{x}+\vec{r})$...
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Trying to first understand position and momentum bases in Quantum Mechanics

In my lectures, I am told: $$\langle x \mid \psi \rangle = \psi (x)$$ Which can only be valid if the overlap integral is: $$\langle x \mid \psi \rangle = \int_{-\infty}^{\infty} \delta (x-x') \ \...
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Hamiltonian of quantum harmonic oscillator with $\psi(x)=\delta(x)$: comparison to classical mechanics

I was just reading the question Why can't $\psi(x)=\delta(x)$ in the case of a harmonic oscillator? The accepted answer says that $\psi(x)=\delta(x)$ is a mathematically valid state, though it's not ...
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Why can't $\psi(x) = \delta(x)$ in the case of Harmonic oscillator?

In the analysis of Harmonic Oscillator, it is claimed that $\langle\hat H\rangle$ cannot be zero, why is it so? I mean $\hat H = \frac{ \hat p^2 }{2m } + \frac12 k \hat x^2$, and $$\left<x^2\...
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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. ...
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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 ...
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1answer
68 views

Greiner's Green's function for diffusion

I am reading Greiner's "Quantum Electrodynamics". In example 1.5 he derives the Green's function for diffusion. I am stuck on a step in the derivation. He has the defining differential equation as $$ ...
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175 views

Infinite square well: wall with infinitesimal thickness

Given an infinite square well, it doesn't matter how thick the wall is, the particle is trapped inside the two walls. If we make the wall of arbitrarily small but finite thickness, the particle is ...
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Dirac Delta Function and Position [duplicate]

How does one prove that the Dirac Delta distribution is the eigenfunction of the position operator $\hat{x}$? In math, why does $\langle x’|x\rangle = \delta(x’-x)$?
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Numerical approximation of the wavefunction in a delta-potential [closed]

I am trying to approximate the wavefunction of a particle in a delta potential $U(x) = -U_0 \delta(x)$ with $V_0 \gt 0$. I am using the following formula to calculate the wavefunction: $\psi(x+\Delta ...
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1answer
67 views

Divergence of a displacement vector field multiplied by delta function

I'm trying to work out why $$ \boldsymbol{\nabla\cdot[u}\,\delta^3(\mathbf{r})]=0, $$ where $\boldsymbol{u}$ is the displacement field of a source of stress, $\boldsymbol{\nabla\cdot u}\ne 0$, and $\...