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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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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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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 $\partial^2$ ...
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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 ...
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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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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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67 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 ...
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44 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 ...
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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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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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106 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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54 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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29 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
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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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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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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(...
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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
160 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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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
54 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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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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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
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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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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 ...
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30 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: $$\sum_{n=0}^N {a_n x^{(n)}}= f_0 \delta(t-t_0)$$ ...
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2answers
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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\ ...
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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') ...
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103 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?
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33 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 ...
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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,...
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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 ...
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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 ...
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92 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 ...
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1answer
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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 ...
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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 ...
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66 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 ...
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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 ...
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2answers
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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 ...
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117 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 ...
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1answer
250 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 ...
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symmetric and antisymmetric solution for delta potential in periodic boundaries

I got the following question in which I got a little confused about: Assume we got periodic boundaries like 1-dimensional "ring" in the length of $a$ with the following potential: $v(x)=g\delta(x)$...
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What's the acceleration of an object if we applied delta dirac function?

If we applied a delta Dirac function as a force, how can we obtain the acceleration of that object? I know that this is called impulse that changes the velocity, but since there is a change in ...
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1answer
51 views

Continuous limit of discrete position basis

Say we have a $1D$ lattice with spacing $a$ between two sites. How does one formally map the discrete position basis of the lattice to a continuous one in the limit $a\to 0$. For instance how does ...
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Mean value with delta function

How do I compute this matrix element $$\langle 1|\delta(\hat x-b)|1\rangle$$ that models a 1-D harmonic oscillator? I have done the same for the ground state (by seting delta function as the Fourier ...
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Divergence of Electric Field Due to a Point Charge

I am trying to formally learn electrodynamics on my own (I only took an introductory course). I have come across the differential form of Gauss's Law. $$ \nabla \cdot \mathbf E = \frac {\rho}{\...
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1answer
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Inhomogeneous wave equation by fourier in analysis

$$\nabla^2\psi_\omega+\frac{\omega^2}{c^2}\psi_\omega=-g\omega,\tag{14-16}$$ which is similar to Poisson's equation. We may synthesize the solution of Eq. (14-16) by the superposition of unit ...
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1answer
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Why position operator is non-degenerated?

In quantum mechanics one can assume position operator $\hat{X}$ must have continuous spectrum, as experiments say it is possible to find a quantum particle at any point of the space. The question is ...
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1answer
184 views

How do you solve the Schrödinger equation with a position space delta function potential in momentum space? [closed]

I am solving the Schrodinger equation in position space with an attractive delta function potential energy, $$ -\frac{h^2}{2m} \frac{d^2}{dx^2} \psi(x)-\lambda \delta(x) \psi(x)=E \psi(x), $$ for a ...
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Kronecker delta commutation relations for QFT

Setup: In many textbook treatments of canonical quantization (e.g., Peskin and Schroeder), one imposes canonical (equal time) Dirac delta commutation relations on the conjugate field operators. e.g., ...
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Product of two one dimensional Dirac delta function of two functions

I had a question pertaining to the product of two one dimensional Dirac delta function of two functions. Let two there be two functions $h(x, \chi, \eta)$ and $g(y, \chi, \eta)$ and a product of two ...
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32 views

Confusions in discretizing a momentum delta integral

I have an integral of the following form: $$\int dk_{x}dk_{y}dk_{z}\frac{1}{(2\pi)^{3}}\delta\left(\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}-p\right)f(k)$$ There are two ways to convert it into discrete ...