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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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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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2answers
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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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0answers
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Question about divergence and dirac delta function [closed]

How to derive 1.100? This is from Griffith's electrodynamics p.50.
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43 views

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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2answers
71 views

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
49 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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1answer
66 views

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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4answers
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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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1answer
153 views

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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0answers
74 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. ...
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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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Need help about references for 2D delta “function” [migrated]

I am writing a paper about some numerical methods in the field of electrostatics and I remember from somewhere that the following equation is true: $$\left( \frac{\partial^2}{\partial x^2} + \frac{\...
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1answer
51 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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1answer
117 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 ...
5
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1answer
115 views

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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1answer
62 views

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
52 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 $\...
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1answer
47 views

Fermionic ghost path integral results in $\delta$ function?

This is related to a statement in pg 20 of hep-th/9408074 formula (2.39). Suppose $$\mathcal{L}\sim\frac{i}{\lambda^{\prime}}\bar{\eta}^xg_{ij}U_x{}^i\psi^j+\cdots \tag{2.35}$$where $\bar{\eta}$ to ...
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1answer
50 views

Orthonormality and completeness in infinite dimensions: 2 different definitions [duplicate]

In finite dimensional vector spaces, orthonormality is defined as $\langle x_i|x_j \rangle=\delta_{ij}$ and the completeness relation is given simply by $$I = \sum_i |x_i\rangle\langle x_i|.$$ To me, ...
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0answers
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Orthonormality: from finite ($\delta_{ij}$) to infinite ($\delta(x-y)$) dimensional vector spaces [duplicate]

I've been reading Shankar's book on QM, but I'm unsatisfied with the section on "Generalization to Infinite Dimensions". Given a finite dimensional vector space with a basis $\{x_i\}$, I understand ...
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2answers
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Dispersion Relations in Particle Physics [closed]

Please tell me how to get the identity(2) in this image
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0answers
42 views

Why Gauss divergence theorem isn't working? [duplicate]

$\vec{E}$ is electric field $r$ is distance between source and field points $\hat{r}$ is a unit vector from source point to field point $x,y,z$ are Cartesian coordinates of field point ...
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1answer
34 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 ...
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1answer
65 views

Equation for the field of a magnetic dipole

In my electrodynamics class, my professor derived the equation for the field of the magnetic dipole $$\vec{B}(\vec{r})=\frac{\mu_0}{4\pi}\frac{1}{r^3}[3(\vec{m}\cdot\hat{r})\hat{r}-\vec{m}]+\frac{2\...
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2answers
73 views

Retarded potentials with a dirac delta fail to give Lienard-Wiechert

In the derivation of the Liénard-Wiechert potential the expression for the retarded potential is given $$\varphi(\mathbf{r}, t) = \frac{1}{4\pi \epsilon_0}\int \frac{\rho(\mathbf{r}', t_r')}{|\mathbf{...
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2answers
69 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 ...
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1answer
11 views

Synchrotron emissivity change of variables

I have an expression for an emissivity $j_\nu$ $$j_\nu =a_0 \left(\frac{p}{mc}\right)^2 B^2\; \delta\mathopen{}\left(\nu -a_1 \left(\frac{p}{mc}\right)^2B\right)$$ where $a_0$ and $a_1$ are ...
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0answers
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Why does $\langle x' | x \rangle$ give the Dirac delta distribution? [duplicate]

I'm having difficulty understanding why the following is true: $$ \int_\mathbb{R} \langle x' | x \rangle dx = \int_ \mathbb{R} \delta(x-x')dx$$ where $\delta(x)$ is the delta distribution. Are we ...
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1answer
37 views

Pöschl–Teller free wave solution normalization

I'm considering one-dimentional QM ($\hbar=1$, $m=1$) with the following potential $$ V(x) = - \frac{1}{\cosh^2 x}\;. $$ I know that free-wave solution are $$ ψ_k(x) = e^{\pm i k x}(\tanh x \mp ik)\;. ...
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2answers
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Can $E=\frac{q}{4\pi\epsilon_0 r^2}$ be directly derived from differential form of Maxwell equations?

The electric field of a point charge $q$ is well known to be $$\mathbf E=\frac{q}{4\pi\epsilon_0 |\mathbf r|^3}\hat{\mathbf r}$$ This can be derived easily from integral form of Gauss’s law. Taking $...
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1answer
70 views

How to integrate by parts ghost fields in electrodynamics?

When applying Faddeev-Popov method to electrodynamics in the Lorenz gauge we obtain the ghost action $$S=\int d^4xd^4y\bar\eta(x)\left(\partial^2\delta(x-y)\right)\eta(y),\tag{0}$$ where $\partial^2$ ...
3
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1answer
102 views

How to derive the $\frac{4\pi}{3}\vec{p}\delta^3(\vec{r})$ element for the dipole field, from its potential?

This might be a bit more general question about how to figure out what is the appropriate (delta) expression in singular points, but e.g. for the dipole, we can derive its potential by a taylor ...
3
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1answer
111 views

How do you write the Wightman function $\langle\phi(t_1)\phi(t_2)\rangle$ for a massive scalar field in position space?

For a free real scalar field $\phi(t,\mathbf{x})$, we define the Wightman function as: $$ W(t_1,t_2) \equiv \langle 0 | \phi(t_1,\mathbf{x}_1) \phi(t_2,\mathbf{x}_2) | 0 \rangle $$ I'm suppressing the ...
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1answer
43 views

How can only 2 phases of a 3 phase power system be used to power a load?

When looking at 2 of the 3 phases on a graph, there's a point where they're both positive or both negative. How does one of the phases act as a return path?
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Fixing the Constant in the Schwarzschild Solution Using a Dirac Delta instead of the the Newtonian Limit

I have seen several derivations of the Schwarzschild solution of the Einstein equations, and they all invoke the Newtonian limit $$ g_{00} \approx -1 + \frac{2GM}{r} $$ for large $r$ to fix the ...
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1answer
43 views

How can one evaluate the expression: $\nabla_{i}\nabla_{j}\left(\frac{1}{r}\right)$, such that $i,j = x,y,z$?

I'm familiar with the Laplacian, but I'm unsure how to evaluate $\nabla_{i}\nabla_{j}\left(\frac{1}{r}\right)$, such that $i,j = x,y,z$, with this notation. This is my attempt, assuming $r=\left(...
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0answers
120 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 ...
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1answer
118 views

Under what conditions is a wavefunction $\psi(x)$ equal to the probability amplitudes $a(x)$?

For context, consider a general expansion of a wavefunction into continuous eigenstates of position, $\phi(x_m,x)$, multiplied by continuous probability amplitudes, $a(x_m)$ $$\begin{align}\psi(x) &...
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1answer
78 views

Delta function from poles of Green's function

In quantum mechanical scattering theory, we often use Green's functions which contain poles. For example, in Schroedinger quantum mechanics the free Green's function is given by $$ G_0(\vec{p}) = \...
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0answers
42 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 ...
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1answer
97 views

To prove the Lorentz invariance of density distribution functions for massless particles in phase space

One defines the density distribution function of a collection of $N$ particles in phase space as follows, $$f(\vec{x},\vec{p},t)=\sum_{i=1}^N\delta^{(3)}(\vec{x}-\vec{x}_i)\delta^{(3)}(\vec{p}-\vec{p}...
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3answers
213 views

What is principal value in delta function integral? [closed]

The delta function may have different forms of definition. One related to Fourier transform is shown below, $$\int_{-\infty}^{\infty}\!dt ~e^{i\omega t}~=~2\pi\delta(\omega).$$ then I wonder what if ...
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3answers
211 views

Is $\delta(r-ct)/4\pi r$, the 3D wave equation elementary solution, a transverse or longitudinal wave?

Background: https://en.wikipedia.org/wiki/Longitudinal_wave 'Longitudinal waves are waves in which the displacement of the medium is in the same direction as, or the opposite direction to, the ...
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2answers
540 views

Derivative of delta function

I am reading and following along the appendices of "The Physical Principles Of The Quantum Theory", and trying to learn how he derives Schrödinger's Equation from his Matrix Mechanics, but I have run ...
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0answers
76 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
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1answer
105 views

How to prove that divergence of the current density is equal to the minus time derivative of the charge density?

Namely, in Weinberg's book (Graviation and Cosmology...) on p. 40 after eq. 2.6.5 we see: $$\begin{align}\nabla\cdot \vec J(\vec x,t) = \sum_n e_n \frac{\partial}{\partial x^i} \delta^3(\vec x-\vec ...
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0answers
57 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 ...
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1answer
157 views

Triple Delta Potential in Quantum Mechanics

I am facing a problem of Quantum Mechanics, and I gently need your help in continuing to solve it. The problem is the old usual problem of a particle subject to a potential, which this time has the ...
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1answer
40 views

Collision and impulsive forces: a formal approach

Consider two bodies $m$ and $M$. Suppose that $m$ is moving with constant velocity $v_0 > 0$ along a certain axis (e.g., it is moving on the right on the $x$-axis), and at a certain time, it ...
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1answer
110 views

Wave-Function Normalization in Momentum Space Not Possible

Hello, I just have a question about this passage; specifically, I do not understand why the result of the inner product (the integral of u_k* and u_k') being the delta function defies conventional ...