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 a singular function plus a Dirac delta have be non-infinite?

In QFT we sometimes encounter functions of the form: $$K(x-y) = \delta(x-y) + \frac{k}{(x-y)^n} $$ Where $x$ and $y$ are $d$ dimensional vectors and $k$ is a (possibly imaginary) constant. These can ...
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Ordinarily continuous function of the wave function

I just started studying quantum mechanics using the textbook Introduction to Quantum Mechanics by Griffith. Under the section of solving the Shrodinger equation for a Dirac delta potential, he ...
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Discretization of derivative of delta function and affine Kac-Moody algebra

In equation (4.16) of https://arxiv.org/abs/1506.06601, a discretization of the (classical) affine Kac-Moody algebra is presented: $$ \frac{1}{\gamma}\left\{J_{m}^{1}, J^2_{n}\right\}=J_{m}^{1} J_{n}^{...
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Charge density of Hertzian dipole

I am wanting to find the charge density of an infinitely thin Hertzian dipole, but am struggling evaluating the Dirac delta functions gradient. $$\vec{J} = I_{0}\cos(\omega t) \delta^3(r) \hat k.$$ ...
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Model of light distribution along radius

I measured the light intensity of Xe lamp along the radius at different points. By using polyfit function, I have found the function of light distribution along the radius. I want to model the light ...
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Loop Integrals and Dimensional Regularization

I want to calculate the divergent part of a Feynman diagram using the Feynman parameters: $$\frac{1}{A_1 A_2 \ldots A_n} = \int_0^1 dx_1 ... dx_n \delta (\Sigma x_i -1) \frac{(n-1)!}{[x_1 A_1 + x_2 ...
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Momentum integral yielding $\delta$ function

I am reading the paper Asymptotic conditions and infrared divergences in quantum electrodynamics by P. P. Kulish & L. D. Faddeev (the paper is not important for the question I think, but I will ...
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The general form of the distribution function and an example

I am reading Landau & Lifshitz's Statistical Physics. On page 12, section 4, the distribution function for a closed system $$\rho=constant\times\delta(E-E_0)\delta(\mathbf P-\mathbf P_0)\delta(\...
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Showing that dirac delta point charge densities is covariant

Consider a point charge $Q$ with a trajectory of $\textbf{s}(t)$ in frame $O$. The densities are: $$\rho(\textbf{x},t) = Q\delta^{(3)}(\textbf{x} - \textbf{s}(t))$$ $$\textbf{J}(\textbf{x},t) = Q\...
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Unit issues with commutator relations with two dependent variables

Suppose I have the following commutator relation for an operator $a[x,\omega]$ which depends on position $x$ and frequency $\omega$ $$ \left[a[x,\omega],a^{\dagger}[x^{\prime},\omega^{\prime}]\right]=\...
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Expressing radiation flux density through a surface with Dirac delta

What is the general way in which radiation density for a given surface is expressed using Dirac deltas? Consider this surface expressed in cylindrical coordinates (for any $\phi$ and $r_0$ an ...
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WKB and Dirac Delta potential

The semi-classical approximation for using WKB in simple words says that in a region where the potential doesn't vary sharply compared to the wavelength of the wavefunction the momentum (or the ...
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Charge density of line of length $L$ expressed with Dirac delta function in cylindrical coordinates

I want to find the charge density of a line charge of length $L$ in cylindrical coordinates. I suppose charge density is independent of $\phi$. The line charge is only defined for coordinates of $z$ ...
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Rate Dispersion is the distribution of rates on a rate spectrum. What are the quantities physically associated in a Rate Spectrum?

I am working on a project to decipher the "origin of rate dispersion" arriving because of Heterogeneity in our system using correlation function analysis in Python. Rate Dispersion, non-...
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What is $\int_{x_0 - \frac{\Delta x}{2}}^{x_0 + \frac{\Delta x}{2}}\delta^2(x - x_0) dx $ [duplicate]

I'm going through basic quantum mechanics and I've got the below expression. $$\int_{x_0 - \frac{\Delta x}{2}}^{x_0 + \frac{\Delta x}{2}}\delta^2(x - x_0) dx$$ How can I evaluate it?
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Integrating $\int \frac{e^{ikx}}{x} dk$ by parts to get delta function derivative, how to handle undefined boundary terms?

I'm going through Sergio Dutra's Cavity Electrodynamics: The Strange Theory of Light in a Box. In equation (2.31) he computes: $$\begin{aligned}\langle x|\hat{p}|x'\rangle&=i\hbar\int\frac{dk}{2\...
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What is the Cauchy principal value in Sokhotski-Plemelj formula?

I met the Sokhotski-Plemelj formula in a paper: in which $P$ is the Cauchy principal value. But the principal value in wiki is this form: It is a limit of an integration. But the $P$ in the first ...
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Derivative of delta function with respect to first argument

In Shankar's QM book pg. 61, the derivative of the delta function $\delta(x-x')$ with respect to the first argument is $$\delta'(x-x')=\frac{d}{dx}\delta(x-x')=-\frac{d}{dx'}\delta(x-x').$$ I tried ...
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How is it possible to obtain delta function in the degenerate perturbation theory?

$$H = H_0 + V(t), V(0) = 0$$ Let $|i_0\rangle$ be the eigenstates of $H_0$, i.e. $H_0|i_0\rangle = E_{i_0}|i_0\rangle$, and $|i(t)\rangle$ is $|i_0\rangle$ after a time $t$ (with hamiltonian $H$). If $...
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How to obtain Green function for the Helmholtz equation?

all. I am following Jackson's Classical Electrodynamics. At Chapter 6.4, the book introduces how to obtain Green functions for the wave equation and the Helmholtz equation. I have a problem in fully ...
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Intuition behind the divergence of $\frac{\hat{r}}{r^2}$ [duplicate]

This a well-known discussion in Griffths's Electrodynamics book. Cutting to the chase, the paradox between the apparent zero divergence of the vector field $\frac{\hat{\mathbf{r}}}{r^2}$ and the fact ...
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Derivative of Delta fuction? [closed]

In Shankar's Quantum Mechanics book p-64 the last equation reads: $$ \delta'(x'-x) = -\delta'(x-x'); $$ I am confused because if I think of it using the gaussian approximation then since: $$ g(x' -x) ...
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Fourier transform of the Hamiltonian of harmonic oscillator from $k$-space to $x$-space

Conventionally, I have the sum of an infinite number of harmonic oscillators with the free Hamiltonian $H_{0}$ to be $$ \begin{equation} H_{0} = \sum_{k}\hbar\omega_{k}\left(a_{k}^{\dagger}a_{k} + \...
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How do I correctly evaluate this simple expression with delta-commutator operators?

Let $a$ denote a bosonic annihilation operator for the mode $q$. The different-$q$ commutator is $$ [a(q),a^{\dagger}(q')] = \delta(q-q') $$ Then, how do I evaluate this expression correctly using the ...
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Intuition for the charge density of a homogeneously charged disc

The charge distribution $$\rho(\vec{r})=\frac{q}{\pi a^2r}\theta{(a-r)}\delta{(\vartheta-\pi/2)}$$ describes a homogeneously charged disc which makes sense in terms of distributions. But if we plot ...
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Why this boundary term could be ignored for a free relativistic particle?

How can we justify that the boundary integral we get from the following could be ignored, when we want to find the equation of motion? I consider the energy-momentum of a free particle in special ...
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Functional derivative for the action $S$

From Lancaster and Blundell's Quantum Field Theory for the Gifted Amateur, p. 15: Example 1.3 The Lagrangian $L$ can be written as a function of both position and velocity. Quite generally, one can ...
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Functional derivative for $J[f]=\int [f(y)]^p \phi(y)dy$

In QFT for gifted amateur pg. 13, the functional derivative for the functional $$J[f]=\int [f(y)]^p \phi(y)dy$$ is given by $$\frac{\delta J[f]}{\delta f(x)}= \lim_{\epsilon\rightarrow0} \frac{1}{\...
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Lorentz invariant phase Volume [duplicate]

I am trying to derive Lorentz Invariant phase volume, $$\int \frac{d^3p}{2E} = \int d^4p \delta(p^2-m_0^2) \Theta(p_0)$$ $$\int dp_0 \delta(p^2 -m_0^2) \Theta (p_0) = \int dp_0 \delta(p_0^2-E_p^2)\...
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The sign of delta function's gradient when deriving electric displacement [closed]

I got confused when deriving electric displacement $\bf D$ following Wolfgang Nolting's Electrodynamics book. Here is his derivation: My problem is about integral ① in red. Property of derivative of ...
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4 votes
2 answers
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Dirac-delta-distribution charge densitity

Are the charge distributions $$\rho(\vec{r})=\frac{Q}{2\pi R^2}\delta(r-R)\delta(\vartheta-\pi/2)$$ and $$\rho(\vec{r})=\frac{Q}{2\pi r^2\sin(\vartheta)}\delta(r-R)\delta(\vartheta-\pi/2)$$ of a ...
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Wiener process as the integral of a stochastic force

I have seen (in my lecture notes) the following definition for a Wiener process: $$W(t)=\int _0 ^t dt'\eta(t') \tag{1}$$ where $\eta(t)$ is the stochastic force appearing in the Langevin equation for ...
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Dirac Delta function in Brillouin zone - Python Coding Problem

I am confused with the expression of Dirac delta function in the 1st Brillouin zone in python. Suppose we are dealing with a 1D chain with period boundary condition which have 10 sites for instance. ...
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3 votes
1 answer
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Continuum States in QM

In the Hilbert space of QM, in the finite dimensional case, for a complete orthonormal set of basis vectors, one writes the generic state vector as: $\psi=\sum_j(\phi_j,\psi)\phi_j$. When the complete ...
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Epstein-Glaser renormalization and perturbative renormalization

I am trying to understand the differences between perturbative renormalization and Epstein-Glaser renormalization, which conceives of renormalization as the extension of distributions. What is ...
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Solving Wave eq. using fourier transformation

I am trying to understand the derivation of the solution of the wave equation (homogeneous case) while using the Fourier Transformation: $$\left(\frac 1 {c^2} \frac {\partial^2}{\partial t^2}- \...
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Representing $\frac{\mathbf{r-r'}}{|\mathbf{r-r'}|^3}$ in polar coordinates

In his book introduction to electrodynamics, Griffiths uses derives the identity $$\nabla \cdot \frac{\mathbf{\hat{r}}}{r^2} = 4\pi\delta^3(\mathbf{r})$$ Using the formula for divergence in polar ...
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Distributions in QFT

If field operators are really distributions then surely objects like the commutator, any correlation functions or even the free theory lagrangians are all ill-defined since products of distributions ...
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Multiplying distributions for QFT

My understanding is that UV divergences arise due to improperly handling the product of distributions. In what sense is it "improper"? And how does its proper handling relate to the notion ...
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7 votes
2 answers
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Dirac delta identity, and intuition on normalization

I'm working through Quantum Field Theory for the Gifted Amateur by Lancaster and Blundell, and in Chapter 3 one of the problems is written like this: For boson operators satisfying $[\hat{a}_{\mathbf{...
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3 votes
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Harmonic oscillator with delta function

For a one-dimensional Quantum Harmonic oscillator with delta function, the Hamiltonian $$\hat{H}=-\frac{1}{2m}\frac{d^2}{dx^2}+\frac{m\omega^2}{2}x^2+\frac{g}{2m}\delta({x})$$ The potntial is even so ...
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Dirac Delta function expressed in terms of molecular orbital basis set

In the book of Helgaker 1 on page 16 is written that for a complete one-electron molecular orbital basis, the Dirac delta function may be written in the form of: $$\delta(\mathbf{x} -\mathbf{x'})=\...
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Can't understand how Griffiths derives an expression in QM text

I am just dumb, as he provides the equations he uses (highlighted) but I just want to know how he is getting the final expression in 3.31?
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2 votes
1 answer
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Is there time reversal symmetry for delta potentials?

I was looking at problem $4.6$ of Gasiorowicz's Quantum Physics, where he asks to prove that the scattering matrix of a potential of the form $$V(x)=\frac{\hbar^2}{2m}\frac{\lambda}{a}\delta(x-b)$$ is ...
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Getting stuck trying to solve electromagnetic wave equation using Green's function

I've recently learned about Green's function and am trying to derive an equation similar to that of the Biot-Savart law but for the electric field around a wire of changing current using the electric ...
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Propagator of harmonic oscillator at specific times

It is well known that the propagator (kernel) of a simple harmonic oscillator is given by $$ U\left(x_{b},T;x_{a},0\right)=\sqrt{\frac{m\omega}{2\pi i\hbar\sin\omega T}}\exp\left\{ \frac{im\omega}{2\...
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Normalization of One-Particle States for Klein-Gordon Field Quantization

Peskin & Schroeder in their QFT textbook discusses how we may normalize one-particle states $|\textbf{p}\rangle$ for Klein-Gordon field quantization in pages 22-23. The excerpts are given below. ...
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Modeling bouncing using a Dirac delta function [closed]

For a ball that is dropped in the presence of a gravitational field $mg$, with a resistive term $-kv$, how can I find the equation of motion, given that when the ball hits the ground at $y=0$ it ...
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0 votes
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On Newton's $2^{\mathrm{nd}}$ law and singularities

A body acted upon by a force moves in such a manner that the time rate of change of momentum equals the force. $${\bf{F}} = \frac{ {\rm d}\, \mathbf{p}}{{\rm d} \, t}, \qquad {\bf p }= m {\bf v}.$$ ...
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'Normalization' of free particle wave function [duplicate]

I'm trying to obtain the expression for the free particle that is known $$\psi_p(x)=\frac{1}{\sqrt{2\pi\hbar}}e^{i\frac{xp}{\hbar}}$$ Easily you can arrive to the exponential, $$p\langle x|p\rangle=-i\...
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