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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What is dipolar charge distribution?
Electric dipole is a system of to opposite point charges separated by some distance.
$1.$ How can we have a continuous volume charge distribution from such a collection of point charges?
$2.$ Here ...
1
vote
1answer
76 views
Dirac delta function mathematical expression proof
In a discussion of the second order transitions in graphene this mathematical expression is used.
$$
\left|\frac{1}{\varepsilon + i \Gamma/2}\right|^2 = \frac{2\pi}{\Gamma}\delta(\epsilon)
$$
And I'm ...
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0answers
34 views
What is the symmetry behind this degeneracy?
I was working on a quantum mechanics problem, involving the perturbation of the 3D cubical potential well:
Suppose we perturb the infinite cubical well
\begin{equation}
V(x,y,z)=\begin{cases}
0, \...
0
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0answers
52 views
Potential with Diracs Delta [closed]
Could a potential with Diracs Delta exist so that the energy for a ground state would be positive? I'll appreciate any answear since this question is giving me a headache for 2 hours.
1
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1answer
63 views
Delta function eigenstate for non-zero potential
Consider the potential $V(x)=\frac{2}{x^2}$ and let $\frac{\hbar^2}{2m}=1$ for convenience. Now consider the function $\psi(x)=\delta(x)$.
According to Griffiths (electrodynamics book) problem 1.45(a),...
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1answer
58 views
Derivation of the QFT Propagator
I don't understand how we get from the RHS to the last line.
\begin{eqnarray}
\left[ \hat{H}_x - i \frac{\partial}{\partial t_x} \right] G^+(x,t_x,y,t_y) &=& -i \delta (t_x - t_y) \sum_n{\...
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0answers
23 views
Bosonic Pair Distribution Function
In Schwabls Book "Advanced Quantum Mechanics" in the chapter for Bosons he calculates the Bosonic pair distribution function for noninteracting bosons. He said the expectation value of \begin{align*}
\...
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1answer
35 views
Anomalous magnetic moment of the electron - integration problem
In Schwartz's QFT book (eqn 17.31), to find the anomalous magnetic moment of the electron from the form factors, near the end of the calculation the following integral needs to be evaluated:
$$ F_{2}(...
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1answer
39 views
Dirac delta normalization of electromagnetic fields
Usually, to quantize electromagnetic fields we use box normalization and therefore the normalization constant contains the dimensions of Volume V of the box.
But if we perform the Dirac-delta ...
1
vote
2answers
49 views
Different definitions of Functional Derivative
In studying QFT and General Relativity, I came across two different definitions of Functional Derivative, and I'd like to know if they are equivalent.
Firstly, in Wald's book General Relativity, as ...
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1answer
62 views
Explanation of the identities $\rho=\rho(\delta)$ ($\delta$ function) and $\rho=q\,\delta(\bar{r})$
Poisson's equation is $$\boldsymbol{\nabla}^{2}\varphi=-4\pi k_{e}\,\rho, \tag{*}$$ that in the case of a point charge $q$, already with spherical symmetry, has as solution
\begin{equation}
\varphi(...
2
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2answers
134 views
Curl and circulation of a vector field that is ill-defined at the origin: any interesting physical effects?
In the cylindrical polar $(\rho,\phi,z)$ coordinate, suppose the velocity field in a liquid is given by $$\vec{v}=\frac{K}{\rho}\hat{e}_{\phi}, \qquad K=\text{constant}.\tag{1}$$ It can be easily ...
3
votes
1answer
40 views
Divergence of $ \frac{1}{s}\hat{s}$ in cylindrical coordinates
In Griffiths' electrodynamics, the divergence of $\frac{1}{r^{2}}\hat{r}$ is evaluated in spherical coordinates to be $4\pi\delta(r)$. I encounter the same problem in case of $\frac{1}{s}\hat{s}$ ...
2
votes
2answers
154 views
A formula for delta function in quantum mechanics
I met a formula for delta function in a QM book ( not in English). The formula is used in the scattering theory. Its form is
$$\lim_{\alpha\rightarrow \infty}\exp[i\alpha x]=2i\delta(x), ~~(x\geq 0).$...
2
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0answers
74 views
Perturbation theory development for the ground state of the QM particle in the box with a centered dirac-delta spike
In the course of a discussion in the chat there emerged an interesting problem, namely a particle in an infinite well with an additional Dirac-delta function spike of scalable hight:
$$
H = -\frac{\...
2
votes
2answers
114 views
Wilson-Sommerfeld Quantization of Dirac delta in Infinite Square Well (ISW)
I am curious to find the energies of Dirac delta potential inside the ISW (walls at $x=0,L$)
$$
H = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V_0\delta(x-L/2)
$$
using Wilson-Sommerfeld ...
8
votes
1answer
180 views
If one puts a delta-function spike inside an infinite square well, is the resulting potential analytically solvable?
It was recently floated in chat that a particle in a box with a delta-function spike inside it, with hamiltonian
$$
H = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V_0\delta(x-a)
$$
and with ...
1
vote
1answer
64 views
Integrating Laplace's equation over a sphere
The Wikipedia page on Laplace's equation states that
if the Laplacian of $u$ is integrated over any volume that encloses the source point,
$$\iiint_V \nabla \cdot \nabla u \, d^3V =-1.$$
I can'...
0
votes
2answers
33 views
Is the DOS (density of states) wrong for degenerated case?
The density of states (DOS) is defined as $$\mathcal{N}\left(\lambda\right)=\sum_{n=1}^{M}\delta\left(\lambda-\lambda_{n}\right).$$ We can then get
$$\int d\lambda\mathcal{N}\left(\lambda\right)=M,$$ ...
1
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1answer
51 views
Mathematical expression of impulsive forces during a collision
I'm facing with problem regarding collision between bodies. Specifically, I need to understand the impulsive forces that arises during a collision. Sometime ago, I posted two questions (this and this) ...
3
votes
2answers
70 views
Kac-Moody algebra from WZW model via Poisson brackets
In 'Non-abelian Bosonization in Two Dimensions', Witten shows that the Poisson brackets of the currents that generate the $G\times G$ symmetry of the WZW model give rise to a Kac-Moody algebra upon ...
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2answers
54 views
$\delta$ potential has highest probability for highest potential
I can't understand this intuitively. Figure 2.9 in Griffith's QM says that the wavefunction at $x=0$ for a delta function potential is $\sqrt{\kappa}$, and to the right it decays like $\psi_+=\sqrt{\...
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0answers
47 views
Square root of Dirac delta Function? [duplicate]
I've puzzled by the appeareance and manipulation of a Square root of a Dirac Delta function.
The article is
https://www.researchgate.net/publication/...
1
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2answers
76 views
How can we argue $\nabla\cdot v = 0$ when ${\mathscr r} \ne 0$ on this vector function?
I am dealing with the vector field:
$$v = \dfrac{\hat{\mathscr r} }{{\mathscr r}^2}$$
And I am studying its divergence. If we compute it we get:
$$\nabla\cdot\left(\dfrac{\hat{\mathscr r} }{{\...
1
vote
1answer
59 views
A computation problem on reciprocal lattice
I am reading David Tong’s lecture notes on Application of Quantum Mechanics. My confusion is about the following paragraph:
Consider a function $f(\vec{x}) $, suppose $f(\vec{x})=f(\vec{x}+\vec{r})$...
0
votes
1answer
72 views
Trying to first understand position and momentum bases in Quantum Mechanics
In my lectures, I am told:
$$\langle x \mid \psi \rangle = \psi (x)$$
Which can only be valid if the overlap integral is:
$$\langle x \mid \psi \rangle = \int_{-\infty}^{\infty} \delta (x-x') \ \...
3
votes
4answers
283 views
Hamiltonian of quantum harmonic oscillator with $\psi(x)=\delta(x)$: comparison to classical mechanics
I was just reading the question Why can't $\psi(x)=\delta(x)$ in the case of a harmonic oscillator? The accepted answer says that $\psi(x)=\delta(x)$ is a mathematically valid state, though it's not ...
4
votes
1answer
162 views
Why can't $\psi(x) = \delta(x)$ in the case of Harmonic oscillator?
In the analysis of Harmonic Oscillator, it is claimed that $\langle\hat H\rangle$ cannot be zero, why is it so?
I mean $\hat H = \frac{ \hat p^2 }{2m } + \frac12 k \hat x^2$, and
$$\left<x^2\...
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0answers
85 views
A Naive Question about Delta Function and Wick Rotation
A delta function can be written as
$$\delta(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}dp\,e^{ipx}.$$
I have a very poor understanding of the Wick rotation technique used in quantum field theory. ...
1
vote
0answers
49 views
Delta function constraint as Lagrange multiplier in SYK model calculation?
In eq. (112) of these lecture notes the author is introducing a 1 into an integral in the following way
This looks like an integral representation of the delta function $\delta(x)\sim \int dy\, e^{i ...
1
vote
1answer
60 views
Greiner's Green's function for diffusion
I am reading Greiner's "Quantum Electrodynamics". In example 1.5 he derives the Green's function for diffusion. I am stuck on a step in the derivation.
He has the defining differential equation as
$$
...
5
votes
1answer
141 views
Infinite square well: wall with infinitesimal thickness
Given an infinite square well, it doesn't matter how thick the wall is, the particle is trapped inside the two walls. If we make the wall of arbitrarily small but finite thickness, the particle is ...
5
votes
1answer
152 views
Dirac Delta Function and Position [duplicate]
How does one prove that the Dirac Delta distribution is the eigenfunction of the position operator $\hat{x}$? In math, why does $\langle x’|x\rangle = \delta(x’-x)$?
-1
votes
1answer
64 views
Numerical approximation of the wavefunction in a delta-potential [closed]
I am trying to approximate the wavefunction of a particle in a delta potential $U(x) = -U_0 \delta(x)$ with $V_0 \gt 0$. I am using the following formula to calculate the wavefunction:
$\psi(x+\Delta ...
1
vote
1answer
62 views
Divergence of a displacement vector field multiplied by delta function
I'm trying to work out why
$$ \boldsymbol{\nabla\cdot[u}\,\delta^3(\mathbf{r})]=0,
$$
where $\boldsymbol{u}$ is the displacement field of a source of stress, $\boldsymbol{\nabla\cdot u}\ne 0$, and $\...
2
votes
1answer
53 views
Fermionic ghost path integral results in $\delta$ function?
This is related to a statement in pg 20 of hep-th/9408074 formula (2.39).
Suppose $$\mathcal{L}\sim\frac{i}{\lambda^{\prime}}\bar{\eta}^xg_{ij}U_x{}^i\psi^j+\cdots \tag{2.35}$$where $\bar{\eta}$ to ...
0
votes
1answer
62 views
Orthonormality and completeness in infinite dimensions: 2 different definitions [duplicate]
In finite dimensional vector spaces, orthonormality is defined as $\langle x_i|x_j \rangle=\delta_{ij}$ and the completeness relation is given simply by
$$I = \sum_i |x_i\rangle\langle x_i|.$$
To me, ...
1
vote
0answers
71 views
Orthonormality: from finite ($\delta_{ij}$) to infinite ($\delta(x-y)$) dimensional vector spaces [duplicate]
I've been reading Shankar's book on QM, but I'm unsatisfied with the section on "Generalization to Infinite Dimensions".
Given a finite dimensional vector space with a basis $\{x_i\}$, I understand ...
-1
votes
2answers
82 views
Dispersion Relations in Particle Physics [closed]
Please tell me how to get the identity(2) in this image
1
vote
0answers
48 views
Why Gauss divergence theorem isn't working? [duplicate]
$\vec{E}$ is electric field
$r$ is distance between source and field points
$\hat{r}$ is a unit vector from source point to field point
$x,y,z$ are Cartesian coordinates of field point
...
0
votes
1answer
56 views
Curl of magnetic field produced by current carrying wires with infinitesimal small area
Can Magnetic fields produced by thin current carrying wires with infinitesimal area have curl with a delta function in it ?? As area is Zero current density J definitely becomes infinite at where ...
4
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1answer
71 views
Equation for the field of a magnetic dipole
In my electrodynamics class, my professor derived the equation for the field of the magnetic dipole
$$\vec{B}(\vec{r})=\frac{\mu_0}{4\pi}\frac{1}{r^3}[3(\vec{m}\cdot\hat{r})\hat{r}-\vec{m}]+\frac{2\...
1
vote
2answers
81 views
Retarded potentials with a dirac delta fail to give Lienard-Wiechert
In the derivation of the Liénard-Wiechert potential the expression for the retarded potential is given
$$\varphi(\mathbf{r}, t) = \frac{1}{4\pi \epsilon_0}\int \frac{\rho(\mathbf{r}', t_r')}{|\mathbf{...
0
votes
2answers
74 views
How to plot the graph of this expression which involves Dirac delta function?
I was doing a problem on electrostatics which required finding the charge density from the given electric field and then plot a graph of the charge density. I was able to find the charge density which ...
0
votes
1answer
12 views
Synchrotron emissivity change of variables
I have an expression for an emissivity $j_\nu$
$$j_\nu =a_0 \left(\frac{p}{mc}\right)^2 B^2\; \delta\mathopen{}\left(\nu -a_1 \left(\frac{p}{mc}\right)^2B\right)$$
where $a_0$ and $a_1$ are ...
2
votes
0answers
57 views
Why does $\langle x' | x \rangle$ give the Dirac delta distribution? [duplicate]
I'm having difficulty understanding why the following is true:
$$ \int_\mathbb{R} \langle x' | x \rangle dx = \int_ \mathbb{R} \delta(x-x')dx$$
where $\delta(x)$ is the delta distribution. Are we ...
0
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1answer
38 views
Pöschl–Teller free wave solution normalization
I'm considering one-dimentional QM ($\hbar=1$, $m=1$) with the following potential
$$
V(x) = - \frac{1}{\cosh^2 x}\;.
$$
I know that free-wave solution are
$$
ψ_k(x) = e^{\pm i k x}(\tanh x \mp ik)\;.
...
1
vote
2answers
77 views
Can $E=\frac{q}{4\pi\epsilon_0 r^2}$ be directly derived from differential form of Maxwell equations?
The electric field of a point charge $q$ is well known to be
$$\mathbf E=\frac{q}{4\pi\epsilon_0 |\mathbf r|^3}\hat{\mathbf r}$$
This can be derived easily from integral form of Gauss’s law. Taking $...
0
votes
1answer
74 views
How to integrate by parts ghost fields in electrodynamics?
When applying Faddeev-Popov method to electrodynamics in the Lorenz gauge we obtain the ghost action
$$S=\int d^4xd^4y\bar\eta(x)\left(\partial^2\delta(x-y)\right)\eta(y),\tag{0}$$
where $\partial^2$ ...
3
votes
1answer
107 views
How to derive the $\frac{4\pi}{3}\vec{p}\delta^3(\vec{r})$ element for the dipole field, from its potential?
This might be a bit more general question about how to figure out what is the appropriate (delta) expression in singular points, but e.g. for the dipole, we can derive its potential by a taylor ...
