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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How to define the light “color” from a given spectral distribution?

The following question may be naive and incomplete in some way I don't know. I'm not a specialist on spectroscopy, colours and light curves, color spaces, etc. Suppose you have a simple power-law ...
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1answer
165 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 ...
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1answer
379 views

Understanding Heaviside and Dirac Delta for Quantum step function

Looking at the solution for from this site I'm a bit confused on how two quantities necessarily reduce. I'm given this wavefunction $$ \psi(x) = \begin{cases} Ax & 0<x<a/2 \\ ...
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Dirac delta function and correlation functions

I was reading this article, which introduces the Delta function as a general sequence of integrable functions, i.e. if $$\displaystyle\int_{-\infty}^{+\infty} g(x)~\mathrm dx = G,$$ where $g(x)$ could ...
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Are vacuum expectation values distributions?

In PCT, Spin, Statistics and all That (1964) on page 106 the vacuum expectation value is introduced in the following way. The question concerns the highlighted area, that is how can a tempered ...
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Green's function for the free particle Hamiltonian

For a free particle in 1D, the equation for its Green function (in atomic units) is $$\left(E+\frac{1}{2}\frac{d^2}{dx^2}\right)G(x,x',E) = \delta(x-x')$$ The textbook I'm following says the solution ...
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1answer
369 views

Confusing derivations of Planck's Law in different books

I was studying the derivation of Planck's Law. But I found confusing texts in different books. In the book "Quantum Physics of Atoms" by Eisberg and Resnick, it states that Planck considered the ...
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1answer
840 views

What are the dimensions for the Heaviside Step Function?

The Heaviside step function is defined as \begin{array}{ll} H(x) = 0 && \text{if }x<0 \\ H(x) = 1/2 && \text{if } x=0 \\ H(x) = 1 && \text{if } x>0 \end{array} ...
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Continuity equation involving vectors

A time dependent point charge $q\left ( t \right )$ at the origin $\rho\left ( \vec{r},t \right )=q\left ( t \right )\delta ^{3}\left ( \vec{r} \right ) $, is fed by a current $$\vec{J}\left ( \vec{r},...
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4answers
939 views

Why is the wave function inside a delta potential non-zero?

The wave function outside an infinite well is zero, owing to the fact that we assume particles to have finite energies. But in the case of a delta function potential $\delta(x-a)$, the wave function ...
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1answer
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Definitions of position operator in QM

We define the position operator $\hat{X}$ by $$\hat{X}|\psi\rangle := \bigg(\int dx |x \rangle x \langle x | \bigg) | \psi \rangle \tag{1}$$ for some state vector $| \psi \rangle \in \mathcal{H}$. ...
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Questions About Quantum Delta Function Potentials [closed]

I didn't think that it would be possible for a wave function to get through the delta function because there is no "leakage" of the wave function through an infinite potential barrier. I can ...
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1answer
541 views

How is the Dirac delta function used in classical mechanics? [closed]

If the contact force applied to a physical object (ex. empty bucket) is given by the Heaviside function: $$ F(t) = F_0~H(t)=\begin{cases} 0, t<0 \\ F_0, t \geq 0\\ \end{cases} $$ Then,...
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1answer
85 views

Practical way of expressing the $\delta$-function [closed]

I have got a problem in using the $\delta$-function. As we know, this function is often used to define a 'density'-related quantity. Such as the density of states or some correlation function. Take ...
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545 views

How can we justify identifying the Dirac delta function with the eigenfunction of position? [duplicate]

I can think of at least two different ways to understand eigenfunctions of operators in quantum mechanics. But neither one seems to provide a good explanation for why we take the position-basis ...
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Heisenberg EOM for $\langle x \rangle$ in momentum eigenstate - where is my error?

Equation of motion for expectation value of a quantum particle in a momentum eigenstate: $$\frac{d}{dt} \langle x \rangle = \frac{1}{i h} \langle [x,H] \rangle$$ and since it's in a momentum ...
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1answer
380 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}^{...
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683 views

Integration by parts with Dirac Delta function

I am having some hard time trying to understand the following "heuristic" integral, involving integration by parts with the Dirac's Delta. We start with the following relation $$ f(x) = \int_{-\infty}^...
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1answer
232 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 ...
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1answer
157 views

Inner product of standard-momentum one-particle states in Weinberg

My question has essentially already been addressed in Questions concerning some parts of the section on one-particle states in Weinberg's first volume on QFT (third question), but unfortunately ...
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1answer
102 views

How does a $\Theta$ function arise in this correlator?

I am currently reading the paper by Coleman on Symmetry breaking in 2d, which can be found here. On page 262 (4th page in the document), he is evaluating the following distribution: $$ F_{\mu}(k)=\...
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2answers
734 views

Bulk-to-Boundary propagator

How can I show that the bulk-to-boundary propagator $$ K(z,x;x')~=~\frac{z^{\Delta}}{[z^2+(x-x')^2]^{\Delta}} \tag{1} $$ goes as a delta function near the boundary $$ K(z,x;x')~\sim ~z^{d-\Delta}\...
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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 ...
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0answers
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Derivation of generation of time between two subsequent particle enters into computational domain

I have problem with understanding derivation of one equation in following problem. You have 1D computational domain (it is not 1D but because it is symmetrical and we are watching only radial ...
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1answer
284 views

Derivation of Biot Savart law for a curve

I'm ok with the following expression for Biot-Savart: $$ {\mathbf B}({\mathbf r}) = \frac{{\mu _0}}{{4\pi }}\int\frac{{{\mathbf J}({\mathbf r'}) \times {(\mathbf{r-r'})}}}{{|\mathbf{r-r'}|^3 }}dV'. $$ ...
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1answer
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Quantum Mechanics: how exactly does “delta function normalization” work for eigenfunctions in 1-d free space case?

The definition of "delta function normalization" says a basis of eigenfunctions of a particle in free space are orthonormal when $$\int_{-\infty}^{\infty}\phi_n^*(\vec{r})\phi_m(\vec{r})\mathrm{d}\vec{...
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868 views

How to calculate the second functional derivative of the action of a one-particle system?

Given the Lagrangian $$L(q,\dot{q})=m\dot{q}^2/2-V(q)$$ and the corresponding action $$S[q]\equiv\int_0^t dt' (m\dot{q}^2/2-V(q)),$$ I need to be able to evaluate the second functional derivative $\...
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1answer
2k views

Surface density charge, divergence of the electric field and gauss law

It´s known that the divergence of the electric field at a certain point is given by this formula: $$\nabla \cdot E=\dfrac{\rho (r)}{\epsilon_{0}}$$ Being $\rho (r)$ the volume charge density at that ...
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2answers
222 views

Commutation Relations in Second Quantization

I understand that if I have the field operators $\psi(r)$ and $\psi^\dagger(r)$, then I have the canonical commutation relation (in the boson case) $$[ \psi(r) , \psi^\dagger(r')]=\delta(r-r').$$ My ...
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1answer
255 views

Potential of an axisymmetric disc with constant rotation velocity

I am having trouble understanding why the form of the 3D potential for a disc with a constant rotation velocity for circular orbits of stars within the disc \begin{equation} v(R) = v_0, \tag{1} \end{...
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1answer
755 views

Book to study Dirac delta function from a physics point of view [duplicate]

I am a beginning physics graduate student. I am often bewildered by the strange properties of the Dirac delta function such as: $\delta (a x)= \frac{1}{a} \delta (x)$ The derivative of $\delta (x)$ ...
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43 views

Why is the inner product of position eigenstates not normalised? [duplicate]

I have read that $$<{\bf r}|{\bf r}'> = δ({\bf r}-{\bf r}').$$ I don't understand how this is correct, I want to say it is equal to 1 or 0, rather than an unnormalised delta function. Clearly ...
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247 views

graphical representation of Maxwell velocity distribution law

I have read Maxwell's distribution law it is the probabilistic representation of no. Of particles having velocity between $c$ to $c+DC$,through this representatation we can get the number of particle ...
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1answer
399 views

Imaginary Part of the Free Energy - Sohotski Plemenj theorem

I have posted this question already on Math Stack Exchange and I hope not to annoy the community if I post it here again, looking maybe for a better suited audience. I need to understand how the ...
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1answer
2k views

Why is the propagator the Green's function for Schrodinger equation? [duplicate]

Sakurai says that the propagator is simply the Green's function for the time-dependent wave equation satisfying $$\left [ -\frac{\hbar^2}{2m} \triangledown ''^2+V(\mathbf{x''})-ih\frac{\partial }{\...
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3answers
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On the completeness relation in Quantum Mechanics

Why does $$ \sum_n \Phi^{\ast}_n(x)\Phi_n(r)=\delta(x−r) $$ represents a completeness relation? Or, put differently, why does it imply completeness? Is there any way to see it intuitively? Maybe an ...
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242 views

Transition probability derivation

I have encountered this limit while learning time dependent perturbation and transition probability in Sakurai. How to show this limit? I tried to integrate around $x=0$ but didn't get anything useful?...
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1answer
743 views

Operators, Distributions and States in QFT

First of all, I will mention what I understand (pls. correct if wrong): States are vectors in the Hilbert space, to include continuous spectrum (and thus distributions), we expand this space to ...
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Representing propagators as Dirac delta functions [closed]

I have found online, in particular on the wolfram site, http://mathworld.wolfram.com/DeltaFunction.html, certain identities that allow one to represent a delta function as limits. Of particular ...
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777 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 ...
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energy distribution of electrons from the heated cathode in magnetic field

I have a very specific question which is troubling me. I use a heated disk cathode as an electron emitter. I know that the energy distribution of the electrons emitting from the cathode is $g(E)=\...
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3answers
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What does the Dirac delta function physically do while deriving Gauss Law form Coulomb's law?

While doing this derivation, the the source coordinates are mentioned as "$s$" and the coordinate of the point at which field is to be calculated is mentioned as "$r$". Kindly follow this Wikipedia ...
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1answer
405 views

Why is it true that Laplace's equation does not hold within the sphere in this case?

Find the average potential over a spherical surface of radius $R$ due to a point charge $q$ located inside. (In this case Laplace's equation does not hold within the sphere) This is a question from ...
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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\...
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1answer
431 views

In nonrelativistic Quantum Mechanics, is the expectation value of a sum of operators always equal to the sum of the expectation values?

Suppose that $\lvert \psi_n \rangle$ are the eigenvectors of a Hamiltonian, $\hat{H}$, which span some Hilbert space $\mathcal{H}$ and satisfy $$\hat{H}\lvert \psi_n \rangle = E_n \lvert \psi_n \...
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1answer
207 views

Heuristics behind Dirac delta function in Master equation for probability?

I'm reading this paper [Phys. Rev. Lett. 106, 160601 (2011)] and it studies simple diffusion where a particle stochastically resets to its initial position $x_0$ at a constant rate $r$. As you can see,...
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1answer
105 views

Integral Represation of Four Current with proper time

I have a question about the four current in covariant representation. the four current is defined as $$ J^{\alpha} = \binom{c\rho}{\vec{j}} $$ and i'm having a point charge, $$\rho(\vec{x},t)=e\...
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0answers
48 views

The momentum representation of $x$ and $ [x,p]$ [duplicate]

To deduce the momentum representation of $[x,p]$, we can see one paradom $$<p|[x,p]|p>=iℏ$$ $$<p|[x,p]|p>=<p|xp|p>−<p|px|p>=p<p|x|p>−p<p|x|p>=0$$ Why? If we ...
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1answer
563 views

The boundary condition for delta function

Beginning with the Schr\"odinger equation for $N$ particles in one dimension interacting via a $\delta$-function potential $$(-\sum_{1}^{N}\frac{\partial^2}{\partial x_i^2}+2c\sum_{<i,j>}\delta(...
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1answer
204 views

Dirac delta in field theory

We start with a function $${\Delta(x) = \displaystyle \int \dfrac{d^3k}{(2\pi)^3 2k^0}}\left( e^{ik^\mu x_\mu} - e^{-ik^\mu x_\mu} \right).$$ It is obvious to me that for $t = 0$ the above expression ...