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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184 views

Mean value with delta function [closed]

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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3answers
2k views

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
87 views

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
193 views

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
318 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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0answers
256 views

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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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 ...
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1answer
167 views

Step involving delta-function in the Klein-Gordon equation solving

The solution to the equation \begin{equation} \int d^3k \; e^{i\mathbf{k}.\mathbf{x}}(k^2-m^2)\phi(\mathbf{k})=0 \end{equation} (which appears in the Klein-Gordon equation solving) is said to be \...
3
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1answer
217 views

Massless $m=0$ 4D Fourier transform of $(p^2 + i \epsilon)^{-2}$

This question is related to this one. I'm assuming that we're in or on the the light-cone $s \leq 0$ in what follows. Suppose I'm interested in computing the following Fourier transform, in the ...
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1answer
321 views

Solution of differential equation (Dirac delta function)

I have been given the following: $$y''(x)+\omega^2y(x)=s(x),$$ $$s(x)= \delta(x)-\delta\left(x-\frac{1}{2}\right)$$ for $-\frac{1}{4}<x<\frac{3}{4}$. (Periodically repeating for $x$ ...
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1answer
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Physical meaning of the Jacobian in relation to Dirac delta function

Is there a physical meaning to the equation $$\delta(x-a)=\dfrac{\delta(\xi-\alpha)}{|J|} \, ?$$ In non-rectangular coordinate systems where the transformation is non-singular, what is the ...
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1answer
140 views

Ambiguous Use of Dirac Delta Function [closed]

Shankar (in his book Principle of Quantum Mechanics book,page 64) mentions that instead of integrating with respect to dx' in $$\int \delta '(x-x') f(x')dx'=\frac{df(x)}{dx},$$ where $$ \delta '(x-x'...
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157 views

$D$-dimensional Schrodinger's equation with a Dirac delta potential

I know that for $D\geq 2$, there is no bound state for a Dirac potential $V=-\alpha \delta(\textbf{x})$ unless we use an ultraviolet cutoff $k_{max}=1/a$. I showed this by solving the Schrodinger's ...
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1answer
93 views

Using integrals to expand a vector in continuous basis

I am new to quantum mechanics. I have been trying to understand why when we want to represent a function $$\psi(x)$$ as a ket in continuous basis |x> we us the integral: $$\vert \psi(x)\rangle =\int\...
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1answer
477 views

Normalization of eigenfunction to Dirac-delta function

In the first chapter of Principles of Quantum Mechanics by R. Shankar, he describes finding the eigenvalues and eigenfunctions of the operator $K=-iD=-i\frac{d}{dx}$. For context, he does this: What ...
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1answer
178 views

Wigner function of position eigenket?

I am trying to derive the Wigner function of position eigenket $\rho = |x'\rangle \langle x'|$. One method is to use the formal expression for the Wigner function and then solve: $$ W(q,p) = \frac{1}...
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1answer
159 views

Delta function potential and continuity of the derivative

In Gasiorowicz Quantum Mechanics, 3rd ed, pg.81, he finds the bound states for a delta function potential the following way: I have the following delta potential: $V(x)=-\frac{\hbar^2\lambda}{2ma}\...
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1answer
141 views

Scalar commutation relations with equal times enforced by a delta function

(I'm following these notes by Vadim Kaplunovsky titled "Feynman Propagator of a Scalar Field," specifically asking about equation 10.) When calculating the time derivative of the Feynman scalar ...
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1answer
134 views

Regarding the commutator of ladder operator in QFT

I am trying to verify the computation of the commutator of the ladder operator for Klein-Gordon solutions, but it seems like I am unable to do it properly. Here is what I do: For, $$ \varphi(x^\mu)=\...
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1answer
314 views

Normalization of states and bracket notation

In Peskin & Schroeder's QFT, if we set $$|p\rangle = \sqrt{2E_p} a^{\dagger}_p|0\rangle \tag{2.35}$$ as in equation 2.35, then how do I get to the next equation 2.36: $$\langle p|q\rangle = 2E_p (...
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1answer
81 views

Is the static electric field $\vec{E}$ defined and finite where charges are located?

Suppose you have a continuous charge distribution, e.g. a wire, a disk, or a plane, represented by a subset $D\subset \mathbb{R^3}$. The charge density is $\rho: \mathbb{R^3}\rightarrow \mathbb{R}$. ...
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0answers
376 views

Relationship between Green's function and impulse response

In my field, electrical engineering, we frequently study linear time-invariant systems of the following form: $$ a_n\frac{d^ny}{dt^n} + a_{n-1}\frac{d^{n-1}y}{dt^{n-1}} + \ldots + a_1\frac{dy}{dt} + ...
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2answers
105 views

Basic confusion with quantum mechanical operators

Given a classical observable, $a(x,p),$ Weyl quantization gives the correspondent QM observable as: $$\langle x | \hat{A} | \phi \rangle=\hbar^{-3}\int \int a \left(\frac{x+y}{2},p\right)\phi(y) e^{2\...
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1answer
359 views

Off-diagonal elements of momentum operator in position representation

In another Phys.SE question, I've proposed the next-cited proof of this statement: the momentum matrix elements in position representation, $\langle x'|\hat{p}|x\rangle$, are all not null I'm ...
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1answer
650 views

What are the real life examples of Double Dirac-Delta Potential barrier/well?

My question is do we see in nature any potential which is close to Double Delta Potential barrier/well? If yes then which are those? Thanks in advance.
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2answers
159 views

Physics terminology about smeared and unsmeared fields

Let $M$ be a smooth manifold and denote $C^\infty_0(M)$ the space of smooth functions with compact support. In Mathematics a distribution is defined to be a continuous linear functional $\phi : C^\...
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232 views

Delta function potential, bound state energies

A lot of books find the bound state energies of the single delta function potential centred at $x=0$ by integrating the Schrodinger equation around $x=0$, using symmetric limits and making the width ...
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1answer
215 views

The charge density $\rho_{\infty}$ of the sphere

I have two charged hemispheres (which are very close to each other, we consider them now as a sphere) with the charge density given as $\rho = \frac {Q}{\pi a^3 \frac {4}{3+n}} (\frac{r}{a})^n$, ...
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3answers
603 views

Heat equation: Heat Kernel as $t\to0$

Consider heat flow on an infinite, 1D wire. The temperature T(x,t) obeys the diffusion equation, $$ \frac{∂T}{∂t} = D \frac{∂^2T}{∂x^2} $$ with initial condition $T(x,0) = δ(x)$. The heat kernel is ...
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0answers
258 views

Density of a thin disk by delta Dirac function

How to write a volume (charge for example) density using delta Dirac function in spherical coordinates in this case: There are thin charged disk (radius $a$) with surface charge density $\sigma$. The ...
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0answers
159 views

Classical field theory: marriage of differential geometry to functional analysis?

I have always wondered (and thoroughly - but perhaps not enough - searched for literature) how the purely geometrical formulation of classical field theory (take for example a hardcore text on this ...
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1answer
449 views

Units of a dirac delta function in quantum mechanics

In quantum mechanics the eigenfunctions of the position are dirac delta functions, $A\delta(x-x_0)$, where $A$ is some constant. Eigenfunctions of the position are usually normalized with a "Delta ...
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1answer
207 views

position state as a sum of momentum states

Leonard Susskind, in his lecture (minute 41) about QFT states that when a field acts on a vacuum state it gives a position state- $$\Psi^\dagger(x)=\sum_{k}e^{-ikx}a^\dagger(k)\left|0\right>=\sum_{...
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1answer
452 views

How to express Dirac delta-function in functional form?

In the paper written by Dmitry Bagrets, Alexander Altland and AlexKamenev (Sachdev–Ye–Kitaev model as Liouville quantum mechanics: http://www.sciencedirect.com/science/article/pii/S0550321316302206?...
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3answers
143 views

Obtaining the charge from the charge density using distribution theory

In electrostatics, for several reasons, it seems that the correct way to understand the charge density $\rho$ isn't as a function $\rho : \mathbb{R}^3\to \mathbb{R}$, but rather as a distribution $\...
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3answers
798 views

Generalising a Dirac Delta function formula in General Relativity

I'm currently stuck on a problem where I have to integrate on a particular set defined through a dirac delta function. If I understood correctly it all boils down to using the curved analogous of $$ ...
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1answer
106 views

Understanding the Formalism of wave function collapse [duplicate]

For example, a perfect measurement ($\hat A=\hat x$) on position: $$\hat x\psi=x_0\psi$$ The eigenvalue $x_0$ is the result of measurement. The eigenfunction $\psi$ is the wavefunction after ...
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1answer
58 views

Why is it physically resonable to consider a function as a regular distribution?

In "Mathematical Methods in Physics" by Blanchard and Brüning the introduction of regular distributions and then general distributions is motivated by the following idea: When we think about the ...
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1answer
165 views

Normalizable free particle wave on 1D

I am trying to solve a free 1D particle non-relativistic Schrodinger equation with initial wavefunction $\psi(x,0)=\delta(x)$, where $$\delta(x)=\lim_{a\to0}(a/2)|x|^{(a-1)}.$$ Here is my approach: ...
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224 views

Derivatives of distributions in general relativity

I am having some trouble when trying to reproduce some calculations involving the description of distributions (mostly used in spacetime junction conditions). I am trying to reproduce the ...
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0answers
180 views

Sommerfeld expansion for temperature dependent spectral function

Consider the integral \begin{align} I(\beta)=\int_{-\infty}^{\infty}\frac{\text{d}\epsilon}{\pi}\frac{\rho(\epsilon,\beta)}{1+e^{\beta \epsilon}} \end{align} where $\rho(\epsilon,\beta)$ is a generic ...
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3answers
277 views

Trouble with position operator in quantum mechanics

I'm having some trouble with understanding the derivation of the action of the $X$ operator. It seems to be a result of the notation used and not a property of itself. The usual argument is to ...
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1answer
118 views

Expression for Delta Function in Quantum Mechanics

For a free particle, we can derive the following well-known relation: $$\langle k|k'\rangle = \frac{1}{2\pi} \int_{-\infty}^{+\infty} e^{-i(k-k')x} \, dx = \delta(k-k'). \tag{1.10.33}$$ Reference: ...
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2answers
491 views

Commutator of scalar field and its spatial derivative

Consider the usual commutation relations of two scalar fields $$\left[\phi_{m}\left(t,\boldsymbol{x}\right),\pi_{n}\left(t,\boldsymbol{y}\right)\right]=\boldsymbol{i}\delta_{mn}\delta\left(\...
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3answers
143 views

Why include a constant in the delta potential $\alpha\delta(x)$?

Why do we have to multiply a proportionality constant in the delta potential $V(x)=-\alpha \delta(x)$ where $\alpha$ is some positive constant? Isn't that $V(x)=\pm \delta(x)$ already enough to ...
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0answers
109 views

Under what circumstances is the functional derivative (of an action functional) an actual function?

In general, for a functional $F[\phi]$, the functional derivative is $$\frac{\delta F[\phi]}{\delta \phi} [f(x)] = \lim_{\varepsilon \to 0} \frac{F[\phi + \varepsilon f ] - F[\phi]}{\varepsilon}$$ ...
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4answers
211 views

Representing dimensions in Dirac delta function results

The Dirac delta function can be defined as $$\delta(x)=\frac{1}{2\pi}\int_{-\infty}^\infty e^{itx}dt$$ From this we see that the dirac function has units of $x^{-1}$. How do we represent the units ...
3
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2answers
141 views

$\delta(0)=\int_{-\infty}^\infty |x_1(x)|^2dx$?

In non-relativistic quantum mechanics: By definition $$\langle x_1|x_1\rangle=\int_{-\infty}^\infty |x_1(x)|^2dx.$$ On the other hand, $$\langle x_1|x_2\rangle=\delta(x_2-x_1).$$ Where $x_1$ and ...
2
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1answer
92 views

Free Fermion and Dirac Delta

Taking this paper (Zinn-Justin: Six-Vertex, Loop and Tiling Modles: Integrability and combinatorics) as reference (chapter 1), I would like to ask a question. First of all some fixed points: The ...
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1answer
326 views

Differentiating the electric field in Gauss's law, I get zero charge density. Can anyone help me where am I going wrong?

$$\mathbf E = \frac{1}{4\pi\epsilon_0}\frac{Q}{r^2} \mathbf e_r$$ $$\nabla \cdot\mathbf E = \frac{\rho_V}{\epsilon}$$ \begin{align} \implies \nabla\cdot\mathbf E & = \frac{1}{r^2}\frac{\partial}{\...