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

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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Why Does the Dirac delta Function Fix the Normalization of the Basis Vectors in Infinite Dimensions? [duplicate]

On page 60 of Shankar's intro to QM at the very bottom he says that the Dirac delta function fixes the normalization of the basis vectors with an infinite amount of dimensions. I don't understand why ...
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How to Solve this Integral

I am currently doing a problem in Quantum optics, specifically the problem of finding Wigner Function for Number states or Fock states. I am actually did the problem in a different way and found that ...
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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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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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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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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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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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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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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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What is the charge density for line and surface charges?

In electrostatics it is common to see line, surface and volumetric charges being described differently. A line distribution is a function defined on the line, a surface charge distribution is a ...
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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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49 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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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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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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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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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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107 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
51 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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38 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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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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263 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
165 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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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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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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85 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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60 views

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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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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116 views

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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50 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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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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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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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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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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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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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 ...
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Problem in understanding Feynman's explanation of the Dirac-Delta function

This is quoted from Feynman's Lectures' Normalization of the states in $x$: We return now to the discussion of the modifications of our basic equations which are required when we are dealing with ...
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Interpretation of the Dirac-measure property

First and foremost, apologies in advance for using an abuse of notation by placing the Dirac measure inside an integral. But given the circumstances, I have no choice. This is essentially a word by ...
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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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402 views

Normalized wave functions in position and momentum space

Using the following expression for the Dirac delta function: $$\delta(k-k')=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i(k-k')x}\mathrm{d}x$$ show that if $\Psi(x,t)$ is normalized at time $t=0$, ...
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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 ...
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238 views

Definition of a line charge with Dirac delta function [closed]

Is the following statement correct for a line charge distribution $λ(x)$? $$ρ(\mathbf r)=λ(x)δ(y)δ(z)$$ If yes - what does it say?
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State time evolution of a quantum harmonic oscillator with a Dirac-Delta function as an initial state [closed]

I have a question just like this Phys.SE problem here with a difference that our system is a harmonic oscillator (rather than a free particle). A particle with mass $m$ is connected to a string with ...
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Gauss' law in differential form for a point charge

I'm trying to understand how the integral form is derived from the differential form of Gauss' law. I have several issues: 1) The law states that $ \nabla\cdot E=\frac{1}{\epsilon 0}\rho$, but when ...
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102 views

How does one make sense of a delta function of a scalar field?

Disclaimer: Originally posted on math SE, but thought that it was better in physics SE, so deleted my post on math SE and posted here. In the classic review summary of stochastic quantization here, ...
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Why don't both equivalent forms of this delta function give the correct answer?

I am a bit confused on a basic problem involving a Dirac delta function being integrated over in a multiple integral. The original problem is to find the probability distribution in position-momentum ...
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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 ...
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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^...