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.
454
questions
3
votes
0answers
12 views
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 $|\psi\rangle$ ...
0
votes
1answer
9 views
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 ...
0
votes
3answers
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 ...
2
votes
0answers
33 views
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 ...
0
votes
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:
$$\...
0
votes
0answers
15 views
Calculation of a 2D scattering length with different masses along x and y - contact interaction
I want to calculate the 2D scattering length of two particles with equal (but anisotropic) masses interacting via a pseudo contact potential. In a relative coordinate system, the Schrödinger equation ...
1
vote
1answer
52 views
What is the meaning of Schrodinger equation solution for bound state of delta potential well?
Let's assume that we have delta potential well with $V = -\lambda\delta(x)$, where $\lambda >0$. Now if we solve Schrodinger equation, we get one eigenvalue $E_b=-\frac{m\lambda^2}{\hbar^2}$ with ...
2
votes
1answer
35 views
Potential energy of particle in delta function potential
What is the potential energy of a particle in the single bound state $\psi_b(x)=\frac{\sqrt{m\alpha}}{\hbar}e^{-\frac{m\alpha}{\hbar^2}|x|}$ of the Dirac-delta potential well
$$V(x) = -\alpha \delta(x)...
1
vote
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 $...
0
votes
1answer
40 views
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 $\delta(t) \delta(r)$:
$$(4\pi ct)^{-1} \delta(r-ct) \tag{1}...
2
votes
1answer
43 views
Issues with Feynman parameters
As a sanity check, I have tried to evaluate a Feynman parameter integral, and have been unable to reproduce the textbook result. I wish to verify the identity
$$\frac{1}{ABC} = \int\limits_0^1\int\...
0
votes
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.
$$ \{\...
1
vote
1answer
52 views
How to find the matrix elements of $ \hat{P}^2 $ in the $X$ basis?
In a resolution of a question in Shankar's book (https://www.physicspages.com/pdf/Shankar/Shankar%20Exercises%2005.01.02.pdf), the derivation of the matrix elements of $ P^2 $ is obtained as follows
$...
5
votes
1answer
46 views
Interpretation and applications of Sokhotski–Plemelj theorem in physics
Sokhotski–Plemelj theorem states:
$$ \tag{1}
\frac{1}{x + i0} = \text{P}\frac 1 x - i \pi\delta(x)
$$
I have seen this theorem being used in QFT and in non relativistic QM (collision theory, Green ...
0
votes
1answer
40 views
Quantum Tunneling in Dirac Delta potential
In quantum mechanics phenomenon of tunneling is well understood ; we know that there is some finite probability to find the particle in classically forbidden region but potential of this forbidden ...
0
votes
0answers
43 views
$\text{div} \ \vec{F} = \vec{0}$ for a conservative force? [duplicate]
I saw from "Advanced Engineering Mathematics, 10th Edition" by Kreyszig, p. 400, that the solution $V$ of the Laplace's equation,
$$\nabla^2 V = \frac{\partial^2V}{\partial x^2}+\frac{\partial^2V}{\...
0
votes
0answers
22 views
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 $$\vec \nabla \cdot \frac{\hat r}{r^2} = 4\pi \delta^3 \vec r \tag{1}$...
1
vote
1answer
46 views
Density of observable is expected value of Dirac delta
I am currently studying Statistical Mechanics and already have a background in probability and statistics. However, there are still things that remain unclear to me. So far I understand that time ...
0
votes
0answers
27 views
Gauss law in vector form [duplicate]
The electric field strength in a region is given by $\vec{E}=\dfrac {x\widehat{i} +y\widehat {j}}{x^{2}+y^{2}}$.
In order to calculate the net charge inside a sphere of radius $a$ centred at origin, ...
4
votes
2answers
142 views
Electric field of given charge density
Given the charge density
$$\rho(\vec{x})=\rho_0\delta(x_1)\delta(x_2),$$
where $\delta$ denotes the Dirac-distribution and $\vec{x}=(x_1,x_2,x_3)$, I am asked to calculate the electric field which
is ...
0
votes
0answers
21 views
Discontinuity of logarithmic terms
It is well known that the imaginary part (discontinuity) of a standard propagator is a $\delta$ function:
$$
\frac{1}{\omega^\prime-\omega-i\epsilon}-\frac{1}{\omega^\prime-\omega+i\epsilon}=2\pi i\...
1
vote
0answers
40 views
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 ...
1
vote
0answers
25 views
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:
$$\nabla^2G\left(\mathbf{r},\mathbf{r}'\right)=-4\pi\delta\left(...
-1
votes
2answers
72 views
Fourier transform of $f(x)=1/x^n$ at physicist level of rigour
Functions such as $f(x)=1/x^n$ where $n$ is a positive integer and $x$ is a real variable in $-\infty\leq x\leq \infty$, strictly do not have Fourier transforms. But when applied to physics, can we ...
2
votes
0answers
74 views
Charge density of a point charge [duplicate]
Is it right to say that the charge density of a point charge is infinity? if I were to take a uniformly charged ball with radius R and take the limit $R\to 0$, won't I get an infinite charge density ...
0
votes
0answers
13 views
Doubts about the Electric Field of Radiation Reaction and some integrals using derivatives of Dirac delta function
Summary
What's the correct result of the following integral and why?
$$
Y=\int_{0}^{t} \dot{x}_j(t') \delta^{(2)}[t'-t]\, dt'
$$
Here, $\dot{x}_j(t')$ is the observable for the position in the $j$ ...
0
votes
2answers
87 views
Why is integral of product of a test function and derivative of Dirac-delta function seems to diverge? [closed]
Suppose,we have to evaluate the integral $\int_{-\infty}^{\infty}f(x)\delta'(x)dx$
Traditionally to solve this,we integrate by parts so that the integral is equal to$-f'(0)$,which is finite if $0$ is ...
3
votes
2answers
92 views
Why does the range of this integral work out this way?
I have a bit of trouble in finding the same limits for the integral in Eq. (17.111) from Peskin & Schroeder. We have something like
$$ \int_0^1 dx' \int_0^1 dz f(x',z) \delta(x-zx').$$
Posing $y=...
1
vote
0answers
22 views
How to interpret the cross-spectral density of an incoherent field
I have been given a definition of the cross spectral density of a completely incoherent field:
$$W(x_1,x_2)=S_0\delta(x_1-x_2)$$
How do I interpret this? As I understand it, this means that there ...
2
votes
1answer
50 views
What paths are allowed in the Fourier form of the Dirac Delta distribution?
In this form of the Dirac Delta distribution
$$\delta(x) = \frac{1}{2 \pi i}\int_{- i \infty}^{i \infty}e^{-\omega x} d\omega$$
can $\omega(t)$ be evaluated over any path (that starts at $\omega(-\...
0
votes
2answers
88 views
Why does the S matrix always contain a factor of $(2\pi)^4?$
In quantum field theory, one usually defines the scattering amplitude as $$S-1=(2\pi)^4\delta(p_{out}-p_{in})M_{Scattering Amplitude}$$
Where S is the S matrix element for any scattering process. It's ...
-5
votes
1answer
42 views
Dirac delta function and Kronecker delta function
Can someone please tell me the difference between Kronecker delta function and Dirac delta function?
0
votes
2answers
98 views
Why is $\langle x|x'\rangle=\delta(x-x')?$ [duplicate]
Yes I have seen the explanation of why this is so in quantum mechanical textbooks. However, let's use the identity operator and do the following:
$$\langle x|x'\rangle =\langle x|I|x'\rangle =\int\...
0
votes
0answers
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 $$H\...
0
votes
1answer
91 views
Scattering theory with two Dirac potentials
i am trying to solve a toy model as a scattering problem containing two Dirac potentials $V(x) = u \delta(x) + u \delta(x-L)$ placed at $x=0$ and $x=L$. My main aim is to find resonance energy states ...
0
votes
1answer
53 views
Expectation Value Of a Charge density
so I just read
Consider a particle (without spin) and spatial coordinate vector
$\vec{q}$ and charge number Z. Classically, the >charge density of the particle is:
$$ \rho(\vec{r}) = Ze\...
4
votes
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 ...
2
votes
1answer
95 views
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
$$K (x^",t;x',t_0)=\sqrt {m\over {2\pi i\hbar (t-t_0)}} \exp \Biggl [{im (x^"...
1
vote
1answer
70 views
Different expression of $\Delta(x, m^2)=(2\pi)^{-3}\int e^{ip \cdot x}\theta(p^0)\delta(p^2+m^2)d^4p$
Let $$\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)^...
5
votes
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?
...
1
vote
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 ...
1
vote
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
$$
\Delta^{rs}_F(x - ...
1
vote
0answers
38 views
Delta function in Fermi's Golden rule [duplicate]
I am currently trying to understand the Fermi's golden rule.
We consider a system with Hamiltonian:
$$\hat H = \hat H_0 + \hat Ue^{i \omega t},$$
where the expectation value of $\hat U$ i much ...
0
votes
0answers
29 views
Books about distribution theory? [duplicate]
I need tips on (elementary) books about distribution theory, with applications to physics and engineering.
There is a one and a half page introduction (on page 910) to the Dirac delta function in ...
3
votes
1answer
81 views
QFT and measures on distributions
Recently I came across the following slogan: ,,constructing quantum field theory on a space $X$ means constructing a measure on the space of all (Schwarz?) distributions $\mathcal{S}'(X)$''. I would ...
5
votes
2answers
572 views
Green's function on torus
I have a question about the Green's function $G(z,w)$ on torus which takes the form (for example the first equation in the paper https://annals.math.princeton.edu/wp-content/uploads/annals-v172-n2-p03-...
0
votes
3answers
90 views
Verify that the electrostatic potential satisfies the Poisson equation [closed]
I'm reading Sect1.7 of Jackson's classical electrodynamics but I have trouble following his argument. Could someone help explain how exactly the Laplacian is evaluated in 1.30? Is it calculated with ...
2
votes
2answers
66 views
Product of projectors of a observable with continuous spectrum
Consider a hermitian quantum mechanical observable $\hat{N}$ with discrete non-degenerate eigenvalues $n_{i}$, and eigenstates $\left | N_{i} \right>$, thus
$$\hat{N}\left | N_{i} \right>=n_{i} ...
2
votes
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 ...
1
vote
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 ...
