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.
484
questions
0
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36 views
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 ...
0
votes
1answer
30 views
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
$$\int dt_2 \delta(t_1 - t_2) \frac{d}{d t_2} G(t_2,t_3) = ...
1
vote
0answers
36 views
Mathematical problem in 1D bosonization
I am reading the following article on bosonization :
https://arxiv.org/abs/cond-mat/9805275
and I encountered the following set of equalities.
$$\begin{align}
[\phi_\eta (x),\partial_{x'}\phi_{\eta'}(...
-1
votes
1answer
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 ...
0
votes
0answers
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 ...
3
votes
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 ...
0
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0answers
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{...
1
vote
1answer
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
$$ g \psi(0) ...
1
vote
2answers
68 views
Quantum mechanics Dirac delta representation with integral
So I’m doing QM and found bunch of problems for beginners and I’m struggling with this one:
$$\lim_{a\rightarrow 0}\int^{\infty}_{-\infty}e^{\frac{ip x}{\hbar}-a x^2}dx=2\pi\hbar\delta(p).$$
If I swap ...
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0answers
47 views
Commutation rules of field operators and Dirac delta
My question concerns the commutation rules between bosonic fields operators in the case in which the bosons can assume only discrete positions.
I have organised this post in two sections, in the ...
1
vote
1answer
69 views
4-dimensional Fourier transform of $(k\cdot v)^{-1}$
I have been trying to compute, without much success, the following Fourier transform in 4-dimensional Minkowski space
$$
I=\frac{1}{(2\pi)^4}\int d^4 k \,\frac{e^{ik\cdot x}}{k\cdot v},
$$
where $v^\...
0
votes
1answer
16 views
B.C. for time-decaying delta barrier inside an infinite well
So let's say I have an infinite well with walls up at $x=-L$ and $x=L$. Suppose that inside the well, there is a time-dependent potential
$$ V(x,t)= \alpha_0\delta{(x)}f(t) $$
where $f(t)$ is a ...
0
votes
1answer
35 views
Total charge from a charge density
I was trying to calculate the total charge from a charge density that has a very strange form involving delta functions. The charge density is
$$\rho(\vec r) = - \vec d . \nabla \delta(\vec r)$$
$d$ ...
0
votes
2answers
60 views
Integral involving Dirac delta function over a finite interval
During the course of a textbook problem, I obtain the following (simplified to keep only important elements) :
$$\int^{b}_{-b}dy\int^{b}_{-b}dy' \space exp\{A(y^{2}-y'^{2})\} \space \delta(y-y')$$
...
5
votes
2answers
144 views
Renormalization and regularization operators for ultracold atoms
When dealing with s-wave scattering in ultracold atoms physics people usually work with the pseudopotential $U = g_0 \delta^{(3)}(r)\frac{\partial}{\partial r}(r\cdot)$.
On the other hand one (...
0
votes
2answers
104 views
Dirac delta function as an inner product
In Shankar's principles of quantum mechanics, the dirac delta function is introduced for generalizing inner products to infinite dimensional spaces. The dirac delta function is such that
$$δ(x-x’) = ⟨...
2
votes
3answers
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 ...
0
votes
1answer
39 views
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) \...
0
votes
1answer
44 views
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 ...
-2
votes
1answer
57 views
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 ...
1
vote
0answers
52 views
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 ...
2
votes
2answers
94 views
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....
5
votes
2answers
120 views
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 ...
0
votes
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&...
11
votes
1answer
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: $\delta(x-x')=\frac{1}{\sqrt{2\pi}}\int e^{-ik(x-x')}dk$...
2
votes
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
$$ \sum_{\boldsymbol{k}} f\left(\boldsymbol{k}\right) ...
0
votes
1answer
56 views
Mistake in Walter Greiner's “Quantum Mechanics” Special chapters
I am going through section 2.4 and 2.5 of Walter Greiner's book "Quantum Mechanics: Special Chapters". In section 2.4, there is a detailed analysis of the elastic scattering of a free ...
0
votes
0answers
38 views
How to apply even symmetry condition at an interface?
Suppose we have an infinite potential box with zero potential from $0\le x\le a$ and infinite boundaries at 0 and a. we have delta function in between the potential box (i.e. at $\frac{a}{2}$) with ...
0
votes
1answer
46 views
Dirac delta function and stochastic processes
It is given to us some white noise as $A z(t)$ and the autocorrelation of $A z(t)$ is given as
$\phi(t)= A^2 \delta(t)$ where $\delta(t)$ is the Dirac delta function
Now one signal with $y(t)= B \cos(...
0
votes
1answer
56 views
A proof involving derivatives of Dirac delta functions
Let us define
$$\tag{1}
Q=i\nabla_\mathbf{k}\delta(\mathbf{k-k'})\left[\rho_{nm}(\mathbf{k'})-\rho_{nm}(\mathbf{k})\right],
$$
where $\delta(\mathbf{k})$ is a Dirac delta, and $\rho_{nm}(\mathbf{k}...
1
vote
1answer
39 views
Schrodinger equation for fermi pseudo potential
This may very well be a very basic question, but I am somewhat confused by a version of the Schrodinger equation I have encountered studying quantum scattering.
Let us assume we have some potential ...
4
votes
0answers
54 views
Mean value of observable in non-normalizable state
If $|\psi\rangle$ is a normalizable state in the Hilbert space of a quantum system and if ${\cal O}$ is some observable we can always evaluate the mean value of ${\cal O}$ on the state $|\psi\rangle$ ...
0
votes
1answer
12 views
Charge distribution and electric field on a conductive sheet
I have a sheet of paper, clad with a half-circle shape of conductive material. The half circle is not filled to the center. The inner radius is about 8cm, and the outer radius about 12cm, not that the ...
0
votes
3answers
78 views
Multiplying $X$ and $P$ operators in quantum mechanics using delta functions (on the $X$ basis)
Alright so I'm trying to figure out how to find the operator $XP$ in the $x$ basis, knowing that the elements of $X$ and $P$ are $x \delta(x-x')$ and $-ih \delta'(x-x')$ respectively. I know how to do ...
2
votes
0answers
51 views
Asymmetric delta potential well modelling wave function [closed]
I'm trying to model the wave function for an asymmetric delta function potential well, in which the left side of the well is at a potential of $0$, however the right side of the well has been moved ...
0
votes
2answers
79 views
What happens if I change the integration limits of the Fourier transform of $1$?
The Fourier transform of $1$ is the (one-dimensional) Dirac delta function:
$$\delta(x) = \frac{1}{2\pi} \int_{-\infty}^\infty dp\ e^{-i p x}. \tag{1}$$
Now I would like to replace the RHS with:
$$\...
0
votes
0answers
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 ...
1
vote
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 ...
2
votes
1answer
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
$$V(x) = -\alpha \delta(x)...
1
vote
2answers
60 views
Solving 1d poisson equation with point source and periodic boundary condition
How can I solve poisson equation with two point charge sources in periodic 1D domain analytically.
$$\dfrac{d^2\phi}{dx^2}=-\dfrac{q}{\epsilon_0}\left(\delta(x-x')-\delta(x+x')\right)\,,$$
where $...
0
votes
1answer
44 views
Equations for 3D waves from an Impulsive Point Source
I think there should be two ways of writing the equation for the impulsive spherical wave
from an impulsive point source at the origin, say $\delta(t) \delta(r)$:
$$(4\pi ct)^{-1} \delta(r-ct) \tag{1}...
3
votes
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
$$\frac{1}{ABC} = \int\limits_0^1\int\...
1
vote
1answer
56 views
Approximation from discrete Kronecker Delta to continuum Dirac Delta
I am working on second quantization of the Dirac field with discrete momentum I was asked to compute the creation/annihilation anticommutator by imposing the anticommutators on $\psi$ i.e.
$$ \{\...
1
vote
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
$...
5
votes
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 ...
0
votes
1answer
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 ...
0
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0answers
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,
$$\nabla^2 V = \frac{\partial^2V}{\partial x^2}+\frac{\partial^2V}{\...
0
votes
0answers
22 views
Prove the given equation [duplicate]
While solving for the curl of the magnetic field($\vec \nabla\times \vec B = \mu_0 \vec J$), I got one formula which is written as $$\vec \nabla \cdot \frac{\hat r}{r^2} = 4\pi \delta^3 \vec r \tag{1}$...
1
vote
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 ...
0
votes
0answers
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, ...
