Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

0
votes
2answers
66 views

Dispersion Relations in Particle Physics [on hold]

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

Infinite Subcritical Reactor with a plane Source; Deriving analytical solution with boundary conditions [migrated]

this might aswell be a mathematics question since I am only looking for a way to solve a basic model of the neutrondistribution in an infinite plane subcritical reactor. So the model for the ...
1
vote
0answers
31 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 ...
0
votes
1answer
28 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 ...
4
votes
1answer
59 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\...
1
vote
2answers
64 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{...
0
votes
0answers
28 views

Where does the “-”sign come from in formula (22) of Diracs principles of Quantum Mechanics, section 75?

When inserting (21) into the defining formula (20) of Delta, I don't see why there should be a minus sign. Formula (22) is used several times in section 75, so it should be correct; but I can`t see ...
0
votes
2answers
64 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 ...
0
votes
1answer
10 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 ...
2
votes
0answers
49 views

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 ...
0
votes
1answer
35 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)\;. ...
1
vote
2answers
60 views

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 $...
0
votes
1answer
68 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
votes
1answer
81 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
votes
1answer
101 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 ...
0
votes
1answer
41 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?
0
votes
0answers
136 views

Probability Density Function vs Radial Distribution for 1s Orbital

What is the difference between the probability density function for the 1s orbital of hydrogen and the radial distribution function? I know that the radial distribution function is 4πr2(Rnl(r))2 but ...
0
votes
0answers
48 views

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 ...
0
votes
1answer
38 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(...
1
vote
0answers
103 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 ...
1
vote
1answer
113 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) &...
0
votes
1answer
64 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}) = \...
0
votes
0answers
31 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 ...
0
votes
1answer
57 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}...
0
votes
3answers
181 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 ...
0
votes
0answers
49 views

Does it matter whether or not $\delta(x)$ is a valid wave function for a particle on the real line?

We model the wave-functions of a particle on the line by vectors $\psi \in L^2(\mathbb{R})$, and the position operator $X:D(X) \rightarrow L^2(\mathbb{R})$ as the operator such that $X\psi(x) = x\psi(...
1
vote
2answers
96 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 ...
2
votes
2answers
193 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 ...
2
votes
0answers
72 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
1
vote
1answer
79 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 ...
2
votes
0answers
44 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 ...
3
votes
1answer
129 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 ...
-1
votes
1answer
34 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 ...
-1
votes
1answer
81 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 ...
0
votes
1answer
31 views

Non-resting initial value problem with impulsive input

Consider a hypothetical model of an extended mechanical system (in which a derivatives of higher order than acceleration may exist d) as bellow: $$\sum_{n=0}^N {a_n x^{(n)}}= f_0 \delta(t-t_0)$$ ...
3
votes
2answers
70 views

Multiplying Distributions in finite-temperature Keldysh/Thermo-field field theory

In the real-time finite temperature formalisms (Schwinger-Keldysh or Thermo-field), the free propagators are often defined with terms like: $$ \mathrm{Dirac\ Delta}\ \times \ \mathrm{Thermal\ ...
1
vote
0answers
23 views

Shifting the derivative outside the integral [closed]

In page 62 of Shankar's Principles of Quantum Mechanics, the author conveys the following: $$\int \delta'( x-x') f(x') dx' = \int \frac{d\delta(x-x')}{dx}f(x')dx'= \frac{d}{dx} \int \delta(x-x') f(x') ...
0
votes
0answers
112 views

What is the normalized Maxwell-Boltzmann velocity distribution for 2D lattices?

The Maxwell Boltzmann velocity distribution is most of times deduced in 3D on books. How does it looks like in 2D?
0
votes
1answer
40 views

Quantum mechanics perturbation and the orthogonality of energy states [closed]

Consider the following question and its solution: My question is concerning the solution of $a_{nm}$. Surely if the energy eigenstates are orthogonal then $a_{nm}$ must be equal to zero. WHy is this ...
0
votes
2answers
21 views

probability distribution of dependent random variables

If we have dependent random variables, then what how is the distribution the pdf look like? Can it be a normal distribution? For example, additive white Gaussian noise (AWGN) has a normal distribution,...
0
votes
0answers
24 views

Use advection diffusion equation to define an initial-boundary problem via programming

Considering the advection-diffusion equation: $$ u_t=-au_x+\mu u_{xx} $$ for $x\in[-\infty,\,\infty]$, $0\leq t\leq T$ and $u\left(t=0,\,x\right)=u_0\left(x\right)$. A solution to the initial value ...
0
votes
0answers
80 views

Spherical Harmonics: Physical Meaning behind the Mathematics: Dirac

Hello Physics Exchange, I have a series of concept questions. Mostly to prove or disprove my understanding. This will be my first post, so if you have any suggestions to improve my formatting - I ...
0
votes
0answers
116 views

Scattering states in Dirac Delta potential

I was reading the solution to the Dirac delta potential well in Griffiths and for the case of scattering, the book mentioned is possible to generate normalizable functions with this eigenstates, but ...
0
votes
1answer
71 views

Integration over phase space for a one dimensional harmonic oscillator

The problem asks for a proof of the following equation, and I have no idea on how to approach this: $\int dx dp \delta(E-\frac{p^2}{2m}-\frac{m \omega^2 x^2}{2})f(E) = \frac{2\pi}{\omega}f(E)$ , for ...
0
votes
0answers
107 views

Dirac-delta orthonormality

Is my understanding that eigenfunctions of position operator and momentum operator exhibit, Dirac-delta orthonormality. I wanted to ask if all self-adjoint operators in quantum mechanics exhibit this ...
0
votes
1answer
69 views

1D delta potential hamiltonian

I have two concerns regarding the delta potential hamiltonian: Is my understanding that in quantum mechanics we use self-adjoint operators. However I cannot figure it out if the hamiltonian that ...
1
vote
0answers
69 views

is it true that $\lim\limits_{\epsilon \to 0^{+}} \ln( x \pm i \epsilon ) = \mathscr{P}\ln|x| \pm i \pi \Theta(-x)$?

Is the following statement true? $$ \lim\limits_{\epsilon \to 0^{+}} \ln( x \pm i \epsilon ) = \mathscr{P}\ln|x| \pm i \pi \Theta(-x) $$ where $\mathscr{P}$ is the Cauchy principal value. The above ...
2
votes
2answers
54 views

Calculating $\langle p|x\rangle$ and $\langle x|\hat{p}|x'\rangle$ - does one result from the other?

In showing that $$\langle x|\hat{p}|x'\rangle = -i \hbar \frac{dδ(x-x')}{dx}$$ I've seen many solutions doing something similar to Can I replace eigenvalue of p operator with position space ...
-1
votes
1answer
138 views

Dirac Delta potential

As we know a particle in attractive Dirac delta potential has discontinuity in the derivative of its wavefunction. I have two questions in this regard: Can a second order differential equation be ...
1
vote
1answer
292 views

Solution of Schrödinger equation for Dirac delta potential $V(x) = \sum_{i=1}^P \sigma_i\delta(x-x_i)$

So, I am trying to solve Schrödinger Equation for Dirac delta potential. The Schrödinger equation: $$ -\frac{\hbar^2}{2m}\frac{d^2\Psi(x)}{dx^2} + V(x)\Psi(x) = E\Psi(x) $$ And, the potential looks ...