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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Why curl of Dirac string attached to Dirac monopole is zero?

So let we have a magnetic field which is $$B_\mu=\frac{1}{2}\frac{x_\mu}{|x|^3}-2\pi\delta_{3\mu}\theta(x_3)\delta{(x_1)}\delta{(x_2)}$$ where $\theta$ is step function and $\delta{(x_\mu)}$ is dirac ...
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Green’s function of $\delta(t_1 - t_2) \frac{d}{d t_2}$?

Does anyone know how to find the Green function for this operator? $$\delta(t_1 - t_2) \frac{d}{d t_2}.$$ Where we should have something like $$\int dt_2 \delta(t_1 - t_2) \frac{d}{d t_2} G(t_2,t_3) = ...
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Mathematical problem in 1D bosonization

I am reading the following article on bosonization : https://arxiv.org/abs/cond-mat/9805275 and I encountered the following set of equalities. $$\begin{align} [\phi_\eta (x),\partial_{x'}\phi_{\eta'}(...
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25 views

Differential form of Gauss's law from Coulomb's law in spherical coordinates [duplicate]

Coulomb's law for the static electric field of a point charge is given by $$\overrightarrow{E}=\frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{r}$$ Now if we take the divergence of both sides of the above ...
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38 views

Dirac delta and covariance [duplicate]

Is there a covariant form of the Dirac delta function? And how to build a covariant form of an identity that contains Dirac delta? To be more precise, what I am looking for is Some distribution that ...
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1answer
111 views

What does a 4+1D wave look like at the light cone?

I need help making sense of a few comments from under this answer. I think it’s best if I reproduce the comments below: “The Green's function for the wave equation in even spatial dimensions is ...
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35 views

Green function for Laplace's equation in complex three dimensional space

The Green function for Laplace's equation in three dimensions for a source at the origin is $$ \nabla^2 G(\mathbf{r})= \delta(\mathbf{r}) =\delta(x)\delta(y)\delta(z) $$ where is $\mathbf{r}=\mathbf{...
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29 views

Treating the delta potential in a Schroedinger equation in 1D

It is a standard problem in quantum mechanics. For the equation $$ -\psi'' + g \delta(x) \psi = E \psi ,$$ we integrate from $-\epsilon$ to $+\epsilon$ and thus get the boundary condition $$ g \psi(0) ...
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2answers
68 views

Quantum mechanics Dirac delta representation with integral

So I’m doing QM and found bunch of problems for beginners and I’m struggling with this one: $$\lim_{a\rightarrow 0}\int^{\infty}_{-\infty}e^{\frac{ip x}{\hbar}-a x^2}dx=2\pi\hbar\delta(p).$$ If I swap ...
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Commutation rules of field operators and Dirac delta

My question concerns the commutation rules between bosonic fields operators in the case in which the bosons can assume only discrete positions. I have organised this post in two sections, in the ...
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1answer
69 views

4-dimensional Fourier transform of $(k\cdot v)^{-1}$

I have been trying to compute, without much success, the following Fourier transform in 4-dimensional Minkowski space $$ I=\frac{1}{(2\pi)^4}\int d^4 k \,\frac{e^{ik\cdot x}}{k\cdot v}, $$ where $v^\...
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B.C. for time-decaying delta barrier inside an infinite well

So let's say I have an infinite well with walls up at $x=-L$ and $x=L$. Suppose that inside the well, there is a time-dependent potential $$ V(x,t)= \alpha_0\delta{(x)}f(t) $$ where $f(t)$ is a ...
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35 views

Total charge from a charge density

I was trying to calculate the total charge from a charge density that has a very strange form involving delta functions. The charge density is $$\rho(\vec r) = - \vec d . \nabla \delta(\vec r)$$ $d$ ...
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Integral involving Dirac delta function over a finite interval

During the course of a textbook problem, I obtain the following (simplified to keep only important elements) : $$\int^{b}_{-b}dy\int^{b}_{-b}dy' \space exp\{A(y^{2}-y'^{2})\} \space \delta(y-y')$$ ...
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Renormalization and regularization operators for ultracold atoms

When dealing with s-wave scattering in ultracold atoms physics people usually work with the pseudopotential $U = g_0 \delta^{(3)}(r)\frac{\partial}{\partial r}(r\cdot)$. On the other hand one (...
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104 views

Dirac delta function as an inner product

In Shankar's principles of quantum mechanics, the dirac delta function is introduced for generalizing inner products to infinite dimensional spaces. The dirac delta function is such that $$δ(x-x’) = ⟨...
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158 views

Is divergence of electric field always going to give you Dirac delta function?

We all know that when we calculate the divergence of point charge at origin, it turns out that it's zero at all points except origin and infinite at origin, which is called Dirac delta function. refer ...
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Continuity equation for a charged particle [duplicate]

I am trying to prove the continuity equation for a charged particle moving with some speed v. So, I start with the charge density and current density as, \begin{align} \rho(x,t) & = q\delta(x-vt) \...
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Lorentz transformation of the Dirac function

Consider a Lorentz transformation that takes the $(x, y, z, t)$ coordinates of a point in Minkowski space into $(x', y', z', t')$. An electrically charged object at rest in the first reference frame ...
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Differential equation [closed]

I am trying to solve the following differential equation; $$\frac{d^2 x}{d t^2}=-\omega^2 x \delta(t-t^\prime).$$ I know this is of the form $$x(t)= A \sin(\omega t) + B \cos(\omega t).$$ However this ...
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How does the delta prime boundary conditions behave when the jump factor approaches infinity?

I recently came across the concept of a $\delta'(x)$ (delta prime) potential, which is basically a potential which imposes the boundary condition: $\frac{\partial\psi}{\partial x}$ is 'continuous' at ...
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2answers
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Eigenstate of position operator after collapse of the wave-function [duplicate]

I am studying basic quantum mechanics in undergrad and have hit a wall. I understand that if a measurement is made for position, the wave function collapses into one of the eigenstates of position, i....
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Confusion about quantum field in AQFT

As far as I known, quantum field is defined by operator-valued distribution mathematically. If I understand correctly, in AQFT, we use self-adjoint elements of $C$* algebra to describe algebra of ...
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1answer
47 views

How does dirac's delta function appear in transition rate in fermi's golden rule?

In the context of time dependent perturbation theory as in 8.06, video's code L 11.2 from mit ocw, I can't see any Dirac delta function appear anywhere. When I read about "Fermi's Golden Rule&...
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681 views

How can I compute the derivative of delta function using its Fourier definition?

I am wondering if it's possible to compute the derivative of the Dirac Delta function using the definition obtained from Fourier transformation: $\delta(x-x')=\frac{1}{\sqrt{2\pi}}\int e^{-ik(x-x')}dk$...
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1answer
46 views

How does the orthogonality relation for Bloch states look in the case when $\boldsymbol{k}$-space is assumed to be continuous?

In solid state physics, it is helpful for some analytical results to assume that $\boldsymbol{k}$-space is a continuum and perform the replacement $$ \sum_{\boldsymbol{k}} f\left(\boldsymbol{k}\right) ...
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56 views

Mistake in Walter Greiner's “Quantum Mechanics” Special chapters

I am going through section 2.4 and 2.5 of Walter Greiner's book "Quantum Mechanics: Special Chapters". In section 2.4, there is a detailed analysis of the elastic scattering of a free ...
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How to apply even symmetry condition at an interface?

Suppose we have an infinite potential box with zero potential from $0\le x\le a$ and infinite boundaries at 0 and a. we have delta function in between the potential box (i.e. at $\frac{a}{2}$) with ...
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1answer
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Dirac delta function and stochastic processes

It is given to us some white noise as $A z(t)$ and the autocorrelation of $A z(t)$ is given as $\phi(t)= A^2 \delta(t)$ where $\delta(t)$ is the Dirac delta function Now one signal with $y(t)= B \cos(...
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56 views

A proof involving derivatives of Dirac delta functions

Let us define $$\tag{1} Q=i\nabla_\mathbf{k}\delta(\mathbf{k-k'})\left[\rho_{nm}(\mathbf{k'})-\rho_{nm}(\mathbf{k})\right], $$ where $\delta(\mathbf{k})$ is a Dirac delta, and $\rho_{nm}(\mathbf{k}...
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1answer
39 views

Schrodinger equation for fermi pseudo potential

This may very well be a very basic question, but I am somewhat confused by a version of the Schrodinger equation I have encountered studying quantum scattering. Let us assume we have some potential ...
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Mean value of observable in non-normalizable state

If $|\psi\rangle$ is a normalizable state in the Hilbert space of a quantum system and if ${\cal O}$ is some observable we can always evaluate the mean value of ${\cal O}$ on the state $|\psi\rangle$ ...
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Charge distribution and electric field on a conductive sheet

I have a sheet of paper, clad with a half-circle shape of conductive material. The half circle is not filled to the center. The inner radius is about 8cm, and the outer radius about 12cm, not that the ...
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3answers
78 views

Multiplying $X$ and $P$ operators in quantum mechanics using delta functions (on the $X$ basis)

Alright so I'm trying to figure out how to find the operator $XP$ in the $x$ basis, knowing that the elements of $X$ and $P$ are $x \delta(x-x')$ and $-ih \delta'(x-x')$ respectively. I know how to do ...
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51 views

Asymmetric delta potential well modelling wave function [closed]

I'm trying to model the wave function for an asymmetric delta function potential well, in which the left side of the well is at a potential of $0$, however the right side of the well has been moved ...
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2answers
79 views

What happens if I change the integration limits of the Fourier transform of $1$?

The Fourier transform of $1$ is the (one-dimensional) Dirac delta function: $$\delta(x) = \frac{1}{2\pi} \int_{-\infty}^\infty dp\ e^{-i p x}. \tag{1}$$ Now I would like to replace the RHS with: $$\...
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15 views

Calculation of a 2D scattering length with different masses along x and y - contact interaction

I want to calculate the 2D scattering length of two particles with equal (but anisotropic) masses interacting via a pseudo contact potential. In a relative coordinate system, the Schrödinger equation ...
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1answer
72 views

What is the meaning of Schrodinger equation solution for bound state of delta potential well?

Let's assume that we have delta potential well with $V = -\lambda\delta(x)$, where $\lambda >0$. Now if we solve Schrodinger equation, we get one eigenvalue $E_b=-\frac{m\lambda^2}{\hbar^2}$ with ...
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41 views

Potential energy of particle in delta function potential

What is the potential energy of a particle in the single bound state $\psi_b(x)=\frac{\sqrt{m\alpha}}{\hbar}e^{-\frac{m\alpha}{\hbar^2}|x|}$ of the Dirac-delta potential well $$V(x) = -\alpha \delta(x)...
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2answers
60 views

Solving 1d poisson equation with point source and periodic boundary condition

How can I solve poisson equation with two point charge sources in periodic 1D domain analytically. $$\dfrac{d^2\phi}{dx^2}=-\dfrac{q}{\epsilon_0}\left(\delta(x-x')-\delta(x+x')\right)\,,$$ where $...
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1answer
44 views

Equations for 3D waves from an Impulsive Point Source

I think there should be two ways of writing the equation for the impulsive spherical wave from an impulsive point source at the origin, say $\delta(t) \delta(r)$: $$(4\pi ct)^{-1} \delta(r-ct) \tag{1}...
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1answer
57 views

Issues with Feynman parameters

As a sanity check, I have tried to evaluate a Feynman parameter integral, and have been unable to reproduce the textbook result. I wish to verify the identity $$\frac{1}{ABC} = \int\limits_0^1\int\...
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1answer
56 views

Approximation from discrete Kronecker Delta to continuum Dirac Delta

I am working on second quantization of the Dirac field with discrete momentum I was asked to compute the creation/annihilation anticommutator by imposing the anticommutators on $\psi$ i.e. $$ \{\...
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1answer
59 views

How to find the matrix elements of $ \hat{P}^2 $ in the $X$ basis?

In a resolution of a question in Shankar's book (https://www.physicspages.com/pdf/Shankar/Shankar%20Exercises%2005.01.02.pdf), the derivation of the matrix elements of $ P^2 $ is obtained as follows $...
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1answer
56 views

Interpretation and applications of Sokhotski–Plemelj theorem in physics

Sokhotski–Plemelj theorem states: $$ \tag{1} \frac{1}{x + i0} = \text{P}\frac 1 x - i \pi\delta(x) $$ I have seen this theorem being used in QFT and in non relativistic QM (collision theory, Green ...
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60 views

Quantum Tunneling in Dirac Delta potential

In quantum mechanics phenomenon of tunneling is well understood ; we know that there is some finite probability to find the particle in classically forbidden region but potential of this forbidden ...
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45 views

$\text{div} \ \vec{F} = \vec{0}$ for a conservative force? [duplicate]

I saw from "Advanced Engineering Mathematics, 10th Edition" by Kreyszig, p. 400, that the solution $V$ of the Laplace's equation, $$\nabla^2 V = \frac{\partial^2V}{\partial x^2}+\frac{\partial^2V}{\...
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22 views

Prove the given equation [duplicate]

While solving for the curl of the magnetic field($\vec \nabla\times \vec B = \mu_0 \vec J$), I got one formula which is written as $$\vec \nabla \cdot \frac{\hat r}{r^2} = 4\pi \delta^3 \vec r \tag{1}$...
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1answer
59 views

Density of observable is expected value of Dirac delta

I am currently studying Statistical Mechanics and already have a background in probability and statistics. However, there are still things that remain unclear to me. So far I understand that time ...
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102 views

Gauss law in vector form [duplicate]

The electric field strength in a region is given by $\vec{E}=\dfrac {x\widehat{i} +y\widehat {j}}{x^{2}+y^{2}}$. In order to calculate the net charge inside a sphere of radius $a$ centred at origin, ...

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