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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Mean value of observable in non-normalizable state

If $|\psi\rangle$ is a normalizable state in the Hilbert space of a quantum system and if ${\cal O}$ is some observable we can always evaluate the mean value of ${\cal O}$ on the state $|\psi\rangle$ ...
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Charge distribution and electric field on a conductive sheet

I have a sheet of paper, clad with a half-circle shape of conductive material. The half circle is not filled to the center. The inner radius is about 8cm, and the outer radius about 12cm, not that the ...
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63 views

Multiplying $X$ and $P$ operators in quantum mechanics using delta functions (on the $X$ basis)

Alright so I'm trying to figure out how to find the operator $XP$ in the $x$ basis, knowing that the elements of $X$ and $P$ are $x \delta(x-x')$ and $-ih \delta'(x-x')$ respectively. I know how to do ...
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Asymmetric delta potential well modelling wave function [closed]

I'm trying to model the wave function for an asymmetric delta function potential well, in which the left side of the well is at a potential of $0$, however the right side of the well has been moved ...
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2answers
75 views

What happens if I change the integration limits of the Fourier transform of $1$?

The Fourier transform of $1$ is the (one-dimensional) Dirac delta function: $$\delta(x) = \frac{1}{2\pi} \int_{-\infty}^\infty dp\ e^{-i p x}. \tag{1}$$ Now I would like to replace the RHS with: $$\...
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15 views

Calculation of a 2D scattering length with different masses along x and y - contact interaction

I want to calculate the 2D scattering length of two particles with equal (but anisotropic) masses interacting via a pseudo contact potential. In a relative coordinate system, the Schrödinger equation ...
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1answer
52 views

What is the meaning of Schrodinger equation solution for bound state of delta potential well?

Let's assume that we have delta potential well with $V = -\lambda\delta(x)$, where $\lambda >0$. Now if we solve Schrodinger equation, we get one eigenvalue $E_b=-\frac{m\lambda^2}{\hbar^2}$ with ...
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1answer
35 views

Potential energy of particle in delta function potential

What is the potential energy of a particle in the single bound state $\psi_b(x)=\frac{\sqrt{m\alpha}}{\hbar}e^{-\frac{m\alpha}{\hbar^2}|x|}$ of the Dirac-delta potential well $$V(x) = -\alpha \delta(x)...
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2answers
34 views

Solving 1d poisson equation with point source and periodic boundary condition

How can I solve poisson equation with two point charge sources in periodic 1D domain analytically. $$\dfrac{d^2\phi}{dx^2}=-\dfrac{q}{\epsilon_0}\left(\delta(x-x')-\delta(x+x')\right)\,,$$ where $...
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1answer
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Equations for 3D waves from an Impulsive Point Source

I think there should be two ways of writing the equation for the impulsive spherical wave from an impulsive point source at the origin, say $\delta(t) \delta(r)$: $$(4\pi ct)^{-1} \delta(r-ct) \tag{1}...
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1answer
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Issues with Feynman parameters

As a sanity check, I have tried to evaluate a Feynman parameter integral, and have been unable to reproduce the textbook result. I wish to verify the identity $$\frac{1}{ABC} = \int\limits_0^1\int\...
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1answer
39 views

Approximation from discrete Kronecker Delta to continuum Dirac Delta

I am working on second quantization of the Dirac field with discrete momentum I was asked to compute the creation/annihilation anticommutator by imposing the anticommutators on $\psi$ i.e. $$ \{\...
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1answer
52 views

How to find the matrix elements of $ \hat{P}^2 $ in the $X$ basis?

In a resolution of a question in Shankar's book (https://www.physicspages.com/pdf/Shankar/Shankar%20Exercises%2005.01.02.pdf), the derivation of the matrix elements of $ P^2 $ is obtained as follows $...
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1answer
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Interpretation and applications of Sokhotski–Plemelj theorem in physics

Sokhotski–Plemelj theorem states: $$ \tag{1} \frac{1}{x + i0} = \text{P}\frac 1 x - i \pi\delta(x) $$ I have seen this theorem being used in QFT and in non relativistic QM (collision theory, Green ...
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Quantum Tunneling in Dirac Delta potential

In quantum mechanics phenomenon of tunneling is well understood ; we know that there is some finite probability to find the particle in classically forbidden region but potential of this forbidden ...
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$\text{div} \ \vec{F} = \vec{0}$ for a conservative force? [duplicate]

I saw from "Advanced Engineering Mathematics, 10th Edition" by Kreyszig, p. 400, that the solution $V$ of the Laplace's equation, $$\nabla^2 V = \frac{\partial^2V}{\partial x^2}+\frac{\partial^2V}{\...
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Prove the given equation [duplicate]

While solving for the curl of the magnetic field($\vec \nabla\times \vec B = \mu_0 \vec J$), I got one formula which is written as $$\vec \nabla \cdot \frac{\hat r}{r^2} = 4\pi \delta^3 \vec r \tag{1}$...
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1answer
46 views

Density of observable is expected value of Dirac delta

I am currently studying Statistical Mechanics and already have a background in probability and statistics. However, there are still things that remain unclear to me. So far I understand that time ...
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27 views

Gauss law in vector form [duplicate]

The electric field strength in a region is given by $\vec{E}=\dfrac {x\widehat{i} +y\widehat {j}}{x^{2}+y^{2}}$. In order to calculate the net charge inside a sphere of radius $a$ centred at origin, ...
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2answers
142 views

Electric field of given charge density

Given the charge density $$\rho(\vec{x})=\rho_0\delta(x_1)\delta(x_2),$$ where $\delta$ denotes the Dirac-distribution and $\vec{x}=(x_1,x_2,x_3)$, I am asked to calculate the electric field which is ...
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Discontinuity of logarithmic terms

It is well known that the imaginary part (discontinuity) of a standard propagator is a $\delta$ function: $$ \frac{1}{\omega^\prime-\omega-i\epsilon}-\frac{1}{\omega^\prime-\omega+i\epsilon}=2\pi i\...
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Perturbation expansion in bound scattering states for double Dirac barriers [closed]

I was working my way through scattering theory notes by David Tong.In there,he discusses the analytical property of the $S$-matrix and uses it for the resonance states for the Double - Dirac potential ...
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Delta functions for surface of non trivial shapes

Say I'm trying to solve Laplace's equation on an object that isn't a sphere or a trivially characterized surface. I will want to solve: $$\nabla^2G\left(\mathbf{r},\mathbf{r}'\right)=-4\pi\delta\left(...
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2answers
72 views

Fourier transform of $f(x)=1/x^n$ at physicist level of rigour

Functions such as $f(x)=1/x^n$ where $n$ is a positive integer and $x$ is a real variable in $-\infty\leq x\leq \infty$, strictly do not have Fourier transforms. But when applied to physics, can we ...
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0answers
74 views

Charge density of a point charge [duplicate]

Is it right to say that the charge density of a point charge is infinity? if I were to take a uniformly charged ball with radius R and take the limit $R\to 0$, won't I get an infinite charge density ...
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Doubts about the Electric Field of Radiation Reaction and some integrals using derivatives of Dirac delta function

Summary What's the correct result of the following integral and why? $$ Y=\int_{0}^{t} \dot{x}_j(t') \delta^{(2)}[t'-t]\, dt' $$ Here, $\dot{x}_j(t')$ is the observable for the position in the $j$ ...
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2answers
87 views

Why is integral of product of a test function and derivative of Dirac-delta function seems to diverge? [closed]

Suppose,we have to evaluate the integral $\int_{-\infty}^{\infty}f(x)\delta'(x)dx$ Traditionally to solve this,we integrate by parts so that the integral is equal to$-f'(0)$,which is finite if $0$ is ...
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2answers
92 views

Why does the range of this integral work out this way?

I have a bit of trouble in finding the same limits for the integral in Eq. (17.111) from Peskin & Schroeder. We have something like $$ \int_0^1 dx' \int_0^1 dz f(x',z) \delta(x-zx').$$ Posing $y=...
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22 views

How to interpret the cross-spectral density of an incoherent field

I have been given a definition of the cross spectral density of a completely incoherent field: $$W(x_1,x_2)=S_0\delta(x_1-x_2)$$ How do I interpret this? As I understand it, this means that there ...
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1answer
50 views

What paths are allowed in the Fourier form of the Dirac Delta distribution?

In this form of the Dirac Delta distribution $$\delta(x) = \frac{1}{2 \pi i}\int_{- i \infty}^{i \infty}e^{-\omega x} d\omega$$ can $\omega(t)$ be evaluated over any path (that starts at $\omega(-\...
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2answers
88 views

Why does the S matrix always contain a factor of $(2\pi)^4?$

In quantum field theory, one usually defines the scattering amplitude as $$S-1=(2\pi)^4\delta(p_{out}-p_{in})M_{Scattering Amplitude}$$ Where S is the S matrix element for any scattering process. It's ...
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1answer
42 views

Dirac delta function and Kronecker delta function

Can someone please tell me the difference between Kronecker delta function and Dirac delta function?
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2answers
98 views

Why is $\langle x|x'\rangle=\delta(x-x')?$ [duplicate]

Yes I have seen the explanation of why this is so in quantum mechanical textbooks. However, let's use the identity operator and do the following: $$\langle x|x'\rangle =\langle x|I|x'\rangle =\int\...
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25 views

Greens function derivative [duplicate]

I´ve given a green's function defined as $$ G_N(\vec{x},\vec{x}') = \frac{1}{4\pi} \frac{1}{|\vec{x}-\vec{x}'|}+\frac{1}{4 \pi}\frac{1}{|\vec{x}-\vec{y}_s'|} $$ with $$y_s=(y_1,y_2,-y_3)$$ on $$H\...
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1answer
91 views

Scattering theory with two Dirac potentials

i am trying to solve a toy model as a scattering problem containing two Dirac potentials $V(x) = u \delta(x) + u \delta(x-L)$ placed at $x=0$ and $x=L$. My main aim is to find resonance energy states ...
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1answer
53 views

Expectation Value Of a Charge density

so I just read Consider a particle (without spin) and spatial coordinate vector $\vec{q}$ and charge number Z. Classically, the >charge density of the particle is: $$ \rho(\vec{r}) = Ze\...
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1answer
37 views

Action for extended objects

Take a spacetime $M$, with some $k$-manifold embedding $$X : \Sigma \to M$$ The image of $X$ represents some extended object (a $k$-brane as the string theory people say). If we only care about the ...
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1answer
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Proof for getting delta function on $t \to t_0 $ from the equation of the propagator for the free particle in 1 dimension

From Sakurai's quantum mechanics equation 2.5.16 give propagator for a free particle in 1 dimension. Equation 2.5.16 is $$K (x^",t;x',t_0)=\sqrt {m\over {2\pi i\hbar (t-t_0)}} \exp \Biggl [{im (x^"...
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1answer
70 views

Different expression of $\Delta(x, m^2)=(2\pi)^{-3}\int e^{ip \cdot x}\theta(p^0)\delta(p^2+m^2)d^4p$

Let $$\Delta(x, m^2)=(2\pi)^{-3}\int e^{ip \cdot x}\theta(p^0)\delta(p^2+m^2)d^4p.$$ Here $\theta$ is the step function at $0$. I would like to show that this is the same as $$\Delta(x, m^2)=(2\pi)^...
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2answers
181 views

What is a delta potential?

I know how a delta potential is described mathematically but how can a delta potential be a 'well'? Does it have particles outside the 'well' and 'bind' it or does it somehow have particle inside it? ...
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1answer
68 views

Dirac delta, Fourier transform & exponentials

Consider the following equation/identity: $$ \int d^3x e^{i(\vec{p}+\vec{q})\cdot\vec{x}}=(2\pi)^3\delta^{(3)}(\vec{p}+\vec{q}). $$ I am trying to calculate some commuters I'm encountering in my ...
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0answers
52 views

Propagators and Green functions for general fields

In my QFT class we have defined the Feynman propagator of a field $\phi^r$ (where $r$ could be a vector or spinor index, or even a multiindex if $\phi$ is a tensor field etc.) as $$ \Delta^{rs}_F(x - ...
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38 views

Delta function in Fermi's Golden rule [duplicate]

I am currently trying to understand the Fermi's golden rule. We consider a system with Hamiltonian: $$\hat H = \hat H_0 + \hat Ue^{i \omega t},$$ where the expectation value of $\hat U$ i much ...
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0answers
29 views

Books about distribution theory? [duplicate]

I need tips on (elementary) books about distribution theory, with applications to physics and engineering. There is a one and a half page introduction (on page 910) to the Dirac delta function in ...
3
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1answer
81 views

QFT and measures on distributions

Recently I came across the following slogan: ,,constructing quantum field theory on a space $X$ means constructing a measure on the space of all (Schwarz?) distributions $\mathcal{S}'(X)$''. I would ...
5
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2answers
572 views

Green's function on torus

I have a question about the Green's function $G(z,w)$ on torus which takes the form (for example the first equation in the paper https://annals.math.princeton.edu/wp-content/uploads/annals-v172-n2-p03-...
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3answers
90 views

Verify that the electrostatic potential satisfies the Poisson equation [closed]

I'm reading Sect1.7 of Jackson's classical electrodynamics but I have trouble following his argument. Could someone help explain how exactly the Laplacian is evaluated in 1.30? Is it calculated with ...
2
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2answers
66 views

Product of projectors of a observable with continuous spectrum

Consider a hermitian quantum mechanical observable $\hat{N}$ with discrete non-degenerate eigenvalues $n_{i}$, and eigenstates $\left | N_{i} \right>$, thus $$\hat{N}\left | N_{i} \right>=n_{i} ...
2
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1answer
40 views

$r$-representation of Operator

I am watching this video https://www.youtube.com/watch?v=sYgX5pdncG8 at 14:30, it has $\langle r|H|r'\rangle = H(r) \delta(r-r') $ Can you help me to understand why it is so? I thought it should ...
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1answer
41 views

How can I find the normaliztion constant of probabilty amplitude for small space-time path?

For free particle with $V=0$ case we get $$<x_n,t_n;x_{n-1},t_{n-1}>=\frac{1}{w(\Delta t)}\exp \left [\frac{im(x_n-x_{n-1})^2}{2\hbar\Delta t}\right ] \tag{6.42}$$ given in eqn. 6.42 of ...

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