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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40
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3k views

Hilbert space of harmonic oscillator: Countable vs uncountable?

Hm, this just occurred to me while answering another question: If I write the Hamiltonian for a harmonic oscillator as $$H = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 x^2$$ then wouldn't one set of ...
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4answers
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What's wrong with this derivation that $i\hbar = 0$?

Let $\hat{x} = x$ and $\hat{p} = -i \hbar \frac {\partial} {\partial x}$ be the position and momentum operators, respectively, and $|\psi_p\rangle$ be the eigenfunction of $\hat{p}$ and therefore $$\...
24
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What are the units or dimensions of the Dirac delta function?

In three dimensions, the Dirac delta function $\delta^3 (\textbf{r}) = \delta(x) \delta(y) \delta(z)$ is defined by the volume integral: $$\int_{\text{all space}} \delta^3 (\textbf{r}) \, dV = \int_{-...
23
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3answers
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Don't understand the integral over the square of the Dirac delta function

In Griffiths' Intro to QM [1] he gives the eigenfunctions of the Hermitian operator $\hat{x}=x$ as being $$g_{\lambda}\left(x\right)~=~B_{\lambda}\delta\left(x-\lambda\right)$$ (cf. last formula on ...
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How exactly is the propagator a Green's function for the Schrodinger equation

Sakurai mentions that the propagator is a Green's function for the Schrodinger equation because it solves $$\left(H-i\hbar\frac{\partial}{\partial t}\right)K(x,t,x_0,t_0) = -i\hbar\delta^3(x-x_0)\...
17
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4answers
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What is the square root of the Dirac Delta Function?

What is the square root of the Dirac Delta Function? Is it defined for functional integrals? Can it be used to describe quantum wave functions? \begin{align} \int_{-\infty}^{\infty} f(x)\sqrt{\delta(...
16
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4answers
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Divergence of a field and its interpretation

The divergence of an electric field due to a point charge (according to Coulomb's law) is zero. In literature the divergence of a field indicates presence/absence of a sink/source for the field. ...
15
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Can momentum have a complex expectation value?

I'm making examples of wave functions to incorporate in a QM exam. I came up with the following wave function, which gives me some troubles: $$\psi(x,0) = \begin{cases} A(a-x), & -a \leq x \leq a\...
14
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3answers
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Dirac Delta and Sloppy Notation

I am an undergraduate neuroscientist and recently I have been studying probability distributions in relation to information theory, and came across the definition of the Dirac Delta as a singular ...
14
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What is “white light” ? Uniform wavelengths or uniform frequencies ?

Suppose you have a mixture of electromagnetic waves of wavelengths spreaded on the visible spectrum only (from $\lambda_{\text{min}} \sim 400 \, \text{nm}$ to $\lambda_{\text{max}} \sim 700 \, \text{...
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Unfamiliar Notation in Sakurai

In chapter 5 section 9 of Sakurai, 2nd edition, he uses some notation that I am unfamiliar with. This may be suited for Math.se but I figured it could be peculiar physicist notation. Anyways it is ...
13
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Possible ambiguity in using the Dirac Delta function

When doing integration over several variables with a constraint on the variables, one may (at least in some physics books) insert a $\delta\text{-function}$ term in the integral to account for this ...
13
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1answer
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3D Delta Potential Well

The 1D delta potential well $V(x) = -A\delta(x - a)$ always has exactly one bound state. The same is true for the 3D delta potential well $V(\vec{r}) = -A\delta(\vec{r}-\vec{a})$. I can show this for $...
13
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3answers
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Bound states of the $V(x)=\pm \delta'^{(n)}(x)$ potential?

The $\delta(x)$ Dirac delta is not the only "point-supported" potential that we can integrate; in principle all their derivatives $\delta', \delta'', ...$ exist also, do they? If yes, can we look for ...
12
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3answers
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What really is a Dirac delta function?

Yesterday a friend asked me what a Dirac delta function really is. I tried to explain it but eventually confused myself. It seems that a Dirac delta is defined as a function that satisfies these ...
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Laplacian of $1/r^2$ (context: electromagnetism and Poisson equation)

We know that a point charge $q$ located at the origin $r=0$ produces a potential $\sim \frac{q}{r}$, and this is consistent with the fact that the Laplacian of $\frac{q}{r}$ is $$\nabla^2\frac{q}{r}~...
12
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3answers
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The Dirac-delta function as an initial state for the quantum free particle

I want to ask if it is reasonable that I use the Dirac-Delta function as an initial state ($\Psi (x,0) $) for the free particle wavefunction and interpret it such that I say that the particle is ...
11
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4answers
10k views

How is Gauss' Law (integral form) arrived at from Coulomb's Law, and how is the differential form arrived at from that?

On a similar note: when using Gauss' Law, do you even begin with Coulomb's law, or does one take it as given that flux is the surface integral of the Electric field in the direction of the normal to ...
11
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1answer
875 views

Dirac Delta in definition of Green function

For a inhomogeneous differential equation of the following form $$\hat{L}u(x) = \rho(x) ,$$ the general solution may be written in terms of the Green function, $$u(x) = \int dx' G(x;x')\rho(x'),$$ ...
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0answers
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 ...
10
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1answer
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Hilbert space of a free particle: Countable or Uncountable?

This is obviously a follow on question to the Phys.SE post Hilbert space of harmonic oscillator: Countable vs uncountable? So I thought that the Hilbert space of a bound electron is countable, but ...
10
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1answer
364 views

What, exactly, is a “delta function p-form” as used in the theory of branes?

In string theory, when dealing with branes, the following happens: We rewrite a worldvolume action $S = \int_{\Sigma_{p+1}} \omega^{(p+1)}$ of a $D_p$-brane as an integral over the whole $\mathbb{R}^{...
9
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2answers
563 views

Group theory, character, representations, delta function, unit element

I am reading a physics paper (hep-th/9204083 v1 above eq (4.15)) where there is a nice formula that reads $$ \delta(U-\mathbb{1}) = \sum\limits_{r} \dim(r)\, \mathrm{tr}_r(U) $$ Here $\:U\:$ is a ...
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How to interpret the derivative of the Dirac delta potential?

I met a Hamiltonian containing the derivative of the Dirac delta potential: In order to do it we use a method described in [9]. We define a formal Hamiltonian $$ \tag{2}\tilde{H}_{abcd}=-\frac{{\...
8
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How do you do an integral involving the derivative of a delta function?

I got an integral in solving Schrodinger equation with delta function potential. It looks like $$\int \frac{y(x)}{x} \frac{\mathrm{d}\delta(x-x_0)}{\mathrm{d}x}$$ I'm trying to solve this by ...
8
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755 views

Maxwell Equations don't give unique Electric Field?

Consider the class of electric fields given by $$\mathbf{E}=\begin{cases} \ln (Cr)\hat{z} & 0\leq r < R\\ 0 & r> R \end{cases}$$ where $C$ is a constant and $r$ is the polar-distance ...
8
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1answer
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Mathematical interpretation of Poisson Brackets

Lets say we are working in a classical scalar field theory and we have two functional $ F[\phi, \pi](x)$ and $G[\phi, \pi](x)$. In most of the references, starting with two functional the Poisson ...
8
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1answer
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Particle density operator in second quantization form

The particle-density operator is given by $n(\mathbf{x})=\sum_{\alpha}\delta^{(3)}(\mathbf{x}-\mathbf{x}_{\alpha})$, then how to derive its representation in terms of creation and annihilation ...
8
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2answers
369 views

Deriving the expectation of $[\hat X,\hat H]$

For a free particle of mass $m$, with Hamiltonian $$\hat{H} = \frac {\hat{P}^2} {2m},$$ where $$\hat{P} = -i \hbar \frac{\partial} {\partial x}.$$ The commutative relation is given by $$[\hat{X}, \...
7
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3answers
843 views

Mathematical understanding of Quantum Mechanics

Assuming that $\phi(r) = F (\psi(r))$ for some operator $F$ in Quantum Mechanics. Then, in our lecture today, we said that $$\phi(r) = \langle r|F |\psi\rangle = \int_{\mathbb{R}} \langle r |F| r' \...
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3answers
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Multivariable Dirac Delta and Faddeev-Popov Determinant

From this mathstack page and in particular Qmechanic's answer: There exists an $n$-dimensional generalization $$\tag{1} \delta^n({\bf f}({\bf x})) ~=~\sum_{{\bf x}_{(0)}}^{{\bf f}({\bf x}_{(0)})=...
7
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2answers
368 views

Eigenstates of a Hermitian field operator

Consider a Hermitian field operator $\phi(x)$ with eigenstates satisfying $$ \phi(x) |\alpha\rangle = \alpha(x) | \alpha \rangle $$ I'm trying to determine the inner product between the eigenstates. ...
7
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2answers
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Basis in quantum mechanics

My quantum mechanics textbook (Primer of Quantum Mechanics, by Marvin Chester) says that both the momentum space and the position space are basis spaces. It also says that the momentum space is ...
7
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1answer
161 views

Statistical physics: How do I find the number of particles that have energy above/below a level?

Say I have a gas consist of atoms or molecules. How do I find the number of atoms in that ensemble that have energy above/below a specific amount, say E? I mean, what is the function that I'll have to ...
7
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2answers
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Resources for theory of distributions (generalized functions) for physicists

I am looking for tutorials, articles or books containing theory of distributions in context of mathematical physics. Please suggest.
7
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1answer
255 views

Functional derivatives as distributions

I have asked this on math stack exchange, due to its mostly mathemtical content, but aside from one upvote and minimal views it has not garnered any attention, so I am trying here as well. This isn't ...
7
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1answer
871 views

The virial theorem and a delta function potential

So the virial theorem tells us that: $2\langle T\rangle = \langle \textbf{r}\cdot\nabla V\rangle$. Now I was wondering what would happen if V has te form: $V(\textbf{r}-\textbf{r}') = V_0\delta^{(D)...
6
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4answers
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Is the momentum operator diagonal in position representation?

The matrix elements of the momentum operator in position representation are: $$\langle x | \hat{p} | x' \rangle = -i \hbar \frac{\partial \delta(x-x')}{\partial x}$$ Does this imply that $\langle x |...
6
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4answers
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Delta Dirac Charge Density question

I have to write an expression for the charge density $\rho(\vec{r})$ of a point charge $q$ at $\vec{r}^{\prime}$, ensuring that the volume integral equals $q$. The only place any charge exists is at $...
6
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1answer
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Principal value of 1/x and few questions about complex analysis in Peskin's QFT textbook

When I learn QFT, I am bothered by many problems in complex analysis. 1) $$\frac{1}{x-x_0+i\epsilon}=P\frac{1}{x-x_0}-i\pi\delta(x-x_0)$$ I can't understand why $1/x$ can have a principal value ...
6
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1answer
227 views

Question on doing the integral for Fermi golden rule

Today in the lecture, my professor did something which confused me As an example, we consider the photoelectric effect, in which an electron bound in a Coulomb potential is ionized after ...
6
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1answer
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Can operators be argument of Dirac Delta function

In one part of Marc Bee's book on Quasielastic Neutron Scattering, he defines the pair correlation function $$ G(\textbf r,t) = \frac{1}{(2\pi)^3}\int I(\textbf Q,t)\text e^{-i\textbf Q.\textbf r}\ d^...
6
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1answer
591 views

George Green's definition of Green's function

This is a curious question about the way George Green could have defined his Green's function. All the definitions I see have only Dirac-delta $\delta(x−x′)$ function as their source on the RHS. But ...
6
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1answer
278 views

If one puts a delta-function spike inside an infinite square well, is the resulting potential analytically solvable?

It was recently floated in chat that a particle in a box with a delta-function spike inside it, with hamiltonian $$ H = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V_0\delta(x-a) $$ and with ...
6
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2answers
221 views

Does regularity of distributions have anything to do with definiteness of their product?

Recently I've gone through some literature concerning causal perturbation theory (CPT). As is well known, it deals with UV divergences in QFT by defining products of (operator-valued) distributions ...
6
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1answer
357 views

Simple question regarding the Green's function for the diffusion equation

The differential operator for diffusion in three dimensions is given by $\partial_t - k \nabla^2$ where $k$ is a constant. The Green's function is (according to Wikipedia) $$\theta(t)\left( \frac{1}{4\...
6
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2answers
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The definition of Density of States

The density of states (DOS) is generally defined as $D(E)=\frac{d\Omega(E)}{dE}$, where $\Omega(E)$ is the number of states. But why DOS can also be defined using delta function, as $$D(E)~=~\sum\...
6
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1answer
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How to make rigorous the idea of a continuous complete set?

In Quantum Mechanics, when using Dirac's formalism one of its features is the expansion of state vectors into continuous basis of eigenvectors of unbounded self-adjoint operators. Let $\mathcal{H}$ be ...
6
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3answers
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Schrodinger equation in term of Fokker-Planck equation

From Wikipedia on the Fokker-Planck equation: $$\tag{1}\frac{\partial }{\partial t}f\left( x^{\prime },t\right) ~=~\int_{-\infty}^\infty dx\left( \left[ D_{1}\left( x,t\right) \frac{\partial }{\...
6
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1answer
978 views

Using $\frac{1}{A+i\epsilon} = PV\frac{1}{A}-i\pi\delta(A)$ in Feynman Integrals

Are the following operations O.K.? This is related to the Feynman parameter trick. $$F:= \int_0^1 \mathrm{d}x\int_0^{1-x}\mathrm{d}y \frac{1}{f(x,y)+\mathrm{i}\epsilon}.$$ Now using $$\frac{1}{A+i\...