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.
3
votes
2answers
54 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 ...
0
votes
2answers
45 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{\...
0
votes
0answers
18 views
Question about divergence and dirac delta function [closed]
How to derive 1.100? This is from Griffith's electrodynamics p.50.
0
votes
0answers
43 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
vote
2answers
71 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:
$$\nabla\cdot\left(\dfrac{\hat{\mathscr r} }{{\...
1
vote
1answer
49 views
A computation problem on reciprocal lattice
I am reading David Tong’s lecture notes on Application of Quantum Mechanics. My confusion is about the following paragraph:
Consider a function $f(\vec{x}) $, suppose $f(\vec{x})=f(\vec{x}+\vec{r})$...
0
votes
1answer
66 views
Trying to first understand position and momentum bases in Quantum Mechanics
In my lectures, I am told:
$$\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
263 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 ...
3
votes
1answer
153 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
$$\left<x^2\...
0
votes
0answers
74 views
A Naive Question about Delta Function and Wick Rotation
A delta function can be written as
$$\delta(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}dp\,e^{ipx}.$$
I have a very poor understanding of the Wick rotation technique used in quantum field theory. ...
1
vote
0answers
31 views
Delta function constraint as Lagrange multiplier in SYK model calculation?
In eq. (112) of these lecture notes the author is introducing a 1 into an integral in the following way
This looks like an integral representation of the delta function $\delta(x)\sim \int dy\, e^{i ...
0
votes
0answers
28 views
Need help about references for 2D delta “function” [migrated]
I am writing a paper about some numerical methods in the field of electrostatics and I remember from somewhere that the following equation is true: $$\left(
\frac{\partial^2}{\partial x^2}
+
\frac{\...
1
vote
1answer
51 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
$$
...
5
votes
1answer
117 views
Infinite square well: wall with infinitesimal thickness
Given an infinite square well, it doesn't matter how thick the wall is, the particle is trapped inside the two walls. If we make the wall of arbitrarily small but finite thickness, the particle is ...
5
votes
1answer
115 views
Dirac Delta Function and Position [duplicate]
How does one prove that the Dirac Delta distribution is the eigenfunction of the position operator $\hat{x}$? In math, why does $\langle x’|x\rangle = \delta(x’-x)$?
-1
votes
1answer
62 views
Numerical approximation of the wavefunction in a delta-potential [closed]
I am trying to approximate the wavefunction of a particle in a delta potential $U(x) = -U_0 \delta(x)$ with $V_0 \gt 0$. I am using the following formula to calculate the wavefunction:
$\psi(x+\Delta ...
1
vote
1answer
52 views
Divergence of a displacement vector field multiplied by delta function
I'm trying to work out why
$$ \boldsymbol{\nabla\cdot[u}\,\delta^3(\mathbf{r})]=0,
$$
where $\boldsymbol{u}$ is the displacement field of a source of stress, $\boldsymbol{\nabla\cdot u}\ne 0$, and $\...
2
votes
1answer
47 views
Fermionic ghost path integral results in $\delta$ function?
This is related to a statement in pg 20 of hep-th/9408074 formula (2.39).
Suppose $$\mathcal{L}\sim\frac{i}{\lambda^{\prime}}\bar{\eta}^xg_{ij}U_x{}^i\psi^j+\cdots \tag{2.35}$$where $\bar{\eta}$ to ...
0
votes
1answer
50 views
Orthonormality and completeness in infinite dimensions: 2 different definitions [duplicate]
In finite dimensional vector spaces, orthonormality is defined as $\langle x_i|x_j \rangle=\delta_{ij}$ and the completeness relation is given simply by
$$I = \sum_i |x_i\rangle\langle x_i|.$$
To me, ...
1
vote
0answers
67 views
Orthonormality: from finite ($\delta_{ij}$) to infinite ($\delta(x-y)$) dimensional vector spaces [duplicate]
I've been reading Shankar's book on QM, but I'm unsatisfied with the section on "Generalization to Infinite Dimensions".
Given a finite dimensional vector space with a basis $\{x_i\}$, I understand ...
-1
votes
2answers
73 views
Dispersion Relations in Particle Physics [closed]
Please tell me how to get the identity(2) in this image
1
vote
0answers
42 views
Why Gauss divergence theorem isn't working? [duplicate]
$\vec{E}$ is electric field
$r$ is distance between source and field points
$\hat{r}$ is a unit vector from source point to field point
$x,y,z$ are Cartesian coordinates of field point
...
0
votes
1answer
34 views
Curl of magnetic field produced by current carrying wires with infinitesimal small area
Can Magnetic fields produced by thin current carrying wires with infinitesimal area have curl with a delta function in it ?? As area is Zero current density J definitely becomes infinite at where ...
4
votes
1answer
65 views
Equation for the field of a magnetic dipole
In my electrodynamics class, my professor derived the equation for the field of the magnetic dipole
$$\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
73 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
$$\varphi(\mathbf{r}, t) = \frac{1}{4\pi \epsilon_0}\int \frac{\rho(\mathbf{r}', t_r')}{|\mathbf{...
0
votes
2answers
69 views
How to plot the graph of this expression which involves Dirac delta function?
I was doing a problem on electrostatics which required finding the charge density from the given electric field and then plot a graph of the charge density. I was able to find the charge density which ...
0
votes
1answer
11 views
Synchrotron emissivity change of variables
I have an expression for an emissivity $j_\nu$
$$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)$$
where $a_0$ and $a_1$ are ...
2
votes
0answers
55 views
Why does $\langle x' | x \rangle$ give the Dirac delta distribution? [duplicate]
I'm having difficulty understanding why the following is true:
$$ \int_\mathbb{R} \langle x' | x \rangle dx = \int_ \mathbb{R} \delta(x-x')dx$$
where $\delta(x)$ is the delta distribution. Are we ...
0
votes
1answer
37 views
Pöschl–Teller free wave solution normalization
I'm considering one-dimentional QM ($\hbar=1$, $m=1$) with the following potential
$$
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
62 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
70 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
102 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 ...
3
votes
1answer
111 views
How do you write the Wightman function $\langle\phi(t_1)\phi(t_2)\rangle$ for a massive scalar field in position space?
For a free real scalar field $\phi(t,\mathbf{x})$, we define the Wightman function as:
$$
W(t_1,t_2) \equiv \langle 0 | \phi(t_1,\mathbf{x}_1) \phi(t_2,\mathbf{x}_2) | 0 \rangle
$$
I'm suppressing the ...
0
votes
1answer
43 views
How can only 2 phases of a 3 phase power system be used to power a load?
When looking at 2 of the 3 phases on a graph, there's a point where they're both positive or both negative. How does one of the phases act as a return path?
0
votes
0answers
51 views
Fixing the Constant in the Schwarzschild Solution Using a Dirac Delta instead of the the Newtonian Limit
I have seen several derivations of the Schwarzschild solution of the Einstein equations, and they all invoke the Newtonian limit
$$
g_{00} \approx -1 + \frac{2GM}{r}
$$
for large $r$ to fix the ...
0
votes
1answer
43 views
How can one evaluate the expression: $\nabla_{i}\nabla_{j}\left(\frac{1}{r}\right)$, such that $i,j = x,y,z$?
I'm familiar with the Laplacian, but I'm unsure how to evaluate
$\nabla_{i}\nabla_{j}\left(\frac{1}{r}\right)$, such that $i,j = x,y,z$,
with this notation.
This is my attempt, assuming
$r=\left(...
1
vote
0answers
120 views
How can a Dirac delta function that does not occur under an integral be used to describe a transition rate?
In his excellent notes (found here), Mark Tuckerman shows that the transition rate of absorption between quantum states i and f, coupled by operator B, can be expressed as the fourier transform of the ...
1
vote
1answer
118 views
Under what conditions is a wavefunction $\psi(x)$ equal to the probability amplitudes $a(x)$?
For context, consider a general expansion of a wavefunction into continuous eigenstates of position, $\phi(x_m,x)$, multiplied by continuous probability amplitudes, $a(x_m)$
$$\begin{align}\psi(x) &...
0
votes
1answer
78 views
Delta function from poles of Green's function
In quantum mechanical scattering theory, we often use Green's functions which contain poles. For example, in Schroedinger quantum mechanics the free Green's function is given by
$$
G_0(\vec{p}) = \...
0
votes
0answers
42 views
Green's function regularization and delta distribution
I have a free Green's function which is proportional to a $2\times 2$ matrix:
$$
G_0 = \frac{1}{E^2-E_k^2}\begin{pmatrix} a & b \\ c & d \end{pmatrix}
$$
The total Green's function after ...
0
votes
1answer
97 views
To prove the Lorentz invariance of density distribution functions for massless particles in phase space
One defines the density distribution function of a collection of $N$ particles in phase space as follows, $$f(\vec{x},\vec{p},t)=\sum_{i=1}^N\delta^{(3)}(\vec{x}-\vec{x}_i)\delta^{(3)}(\vec{p}-\vec{p}...
0
votes
3answers
213 views
What is principal value in delta function integral? [closed]
The delta function may have different forms of definition. One related to Fourier transform is shown below,
$$\int_{-\infty}^{\infty}\!dt ~e^{i\omega t}~=~2\pi\delta(\omega).$$
then I wonder what if ...
1
vote
3answers
211 views
Is $\delta(r-ct)/4\pi r$, the 3D wave equation elementary solution, a transverse or longitudinal wave?
Background:
https://en.wikipedia.org/wiki/Longitudinal_wave
'Longitudinal waves are waves in which the displacement of the medium is in the same direction as, or the opposite direction to, the ...
2
votes
2answers
540 views
Derivative of delta function
I am reading and following along the appendices of "The Physical Principles Of The Quantum Theory", and trying to learn how he derives Schrödinger's Equation from his Matrix Mechanics, but I have run ...
2
votes
0answers
76 views
How can the propagator be written in the below integral form?
I’am finding it difficult to understand as to how the delta function is written as a product of many delta functions with the integral
2
votes
1answer
105 views
How to prove that divergence of the current density is equal to the minus time derivative of the charge density?
Namely, in Weinberg's book (Graviation and Cosmology...) on p. 40 after eq. 2.6.5 we see:
$$\begin{align}\nabla\cdot \vec J(\vec x,t) = \sum_n e_n \frac{\partial}{\partial x^i} \delta^3(\vec x-\vec ...
2
votes
0answers
57 views
Half of Dirac Delta within spectral integration
I was taking a course in Quantum Information along this last semester, and apart from some mathematical details it all made sense to me. One of these mathematical tricks that I haven't been able to ...
3
votes
1answer
157 views
Triple Delta Potential in Quantum Mechanics
I am facing a problem of Quantum Mechanics, and I gently need your help in continuing to solve it.
The problem is the old usual problem of a particle subject to a potential, which this time has the ...
-1
votes
1answer
40 views
Collision and impulsive forces: a formal approach
Consider two bodies $m$ and $M$. Suppose that $m$ is moving with constant velocity $v_0 > 0$ along a certain axis (e.g., it is moving on the right on the $x$-axis), and at a certain time, it ...
-1
votes
1answer
110 views
Wave-Function Normalization in Momentum Space Not Possible
Hello,
I just have a question about this passage; specifically, I do not understand why the result of the inner product (the integral of u_k* and u_k') being the delta function defies conventional ...
