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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How can we evaluate the following integral using the tricks of delta functions? [migrated]
I am trying to teach myself the statistical field theory formulation of statistical mechanics. Not part of a class, just self study in my free time. I appreciate any help here.
I am starting with ...
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1answer
54 views
What is the relationship between wave function and classical distribution function?
The question is a bit weird since the wave function is quantum mechanical and the distribution function in phase space is really something classical.
But I would still like to know if I take the ...
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0answers
59 views
Green function derived from the idea of position representation in quantum mechanics
I am trying to study green's function by using a lecture by NPTEL its link is
https://www.youtube.com/watch?v=ZJ7v6VZQ32k&t=439s
I don't get this step by him at the minute of ten.
$$D_x G(x,x^{'...
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2answers
143 views
Mathematical rigorous definition for an electrical dipole
I've been reading Laurent Schwartz's Mathematics for the physical sciences, and in the chapter on distributions he makes many cool examples of ways to define in a mathematical rigorous way certain ...
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1answer
50 views
Position representation of an operator
$$\langle\ x\rvert M\lvert\ x'\rangle=M(x)\langle\ x\lvert\ x'\rangle=M(x)\delta(x-x')$$
I know this is true for if $M$ is a momentum operator or position operator, is this is true for a general ...
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1answer
49 views
Itzykson and Zuber: Reduced $T$-matrix
Could someone help me understand the reduced $T$-matrix mentioned in Itzykson and Zuber, eq.
$$\langle{f}| T|p_1p_2\rangle=(2\pi)^4\delta^4(P_f-p_1-p_2)\langle f|\mathcal{T}|p_1p_2\rangle. \tag{5-7}$...
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2answers
41 views
Is it true that any electrostatic charge density can be represented by Dirac Delta function? Give me a general example
My college professor said that any charge density can be replaced by Dirac Delta function but how I want to know.
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1answer
34 views
Finding normalization constant of a wave function with definite momentum
I try to read Sakurai's Modern Quantum Mechanics but I stuck at this point,
$$\delta(x^{'}-x^{''})=|N|^{2}\int dp^{'}\exp\Biggl({ip^{'}(x^{'}-x^{''})2\pi\over h}\Biggr)$$
This is an expression for ...
2
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0answers
15 views
Expected value of the current density operator
In Ullrich's TD-DFT book, the paramagnetic current-density operator is defined as
$$\hat{\mathbf{j}}(\mathbf{r})=\frac{1}{2i}\sum_{a=1}^{N}\left[\nabla_a\delta(\mathbf{r}-\mathbf{r}_a)+\delta(\mathbf{...
2
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1answer
70 views
Why do Candelas and Howard say that $\sum_{n=1}^\infty \cos\left( n \kappa \epsilon \right) \ = \ - \frac{1}{2}$?
In the paper Vacuum $\langle \phi^2 \rangle$ in Schwarzschild Spacetime by Candalas and Howard, they say that for each non-zero $\epsilon$ it is true that
$$
\sum_{n=1}^\infty \cos\left( n \kappa \...
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1answer
41 views
Problem in the continuum limit of a Kronecker delta
I am having troubles in understanding how to correctly perform the continuum limit of a double sum containing a Kronecker delta.
Imagine to integrate a function depending on $t$ and $t'$, both ...
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2answers
47 views
Factorising a 4D Dirac delta function in a rest frame
I'm working through a QFT problem and at one stage in the solutions we have this step:
$$\delta^{(4)}(p - q_1 - q_2) = \delta(E_1 +E_2 - M)\delta^{(3)}(\bf{q_1} - \bf{q_2}).$$
We are working in the ...
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1answer
45 views
Properties of Dirac delta function in Integral
I was reading commutation relation of canonical momentum in KG Field from Lectures of Quantum Field Theory by Ashok Das. In page 179, He has used Integration to derive the result where he expressed ...
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1answer
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What is the volume charge density (in spherical coordinates)?
What is the volume charge density (in spherical coordinates) of a uniform, in finitesimally thin spherical shell of radius $R$ and total charge $Q$, centered at the origin? Give your answer in terms ...
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1answer
53 views
$\delta^{(2)}$ convention
In this note: https://arxiv.org/abs/hep-th/0410165 at page 12 there is a delta-function constraint written as:
\begin{align}
\delta \left( ^ { U } M \right) = \prod _ { i < j } \delta ^ { ( 2 ) } \...
3
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1answer
59 views
A question about $\delta(x) \star \delta(p)$
Using the Moyal product between two delta functions in $(x,p)$-space one gets
$$ \delta(x) \star \delta(p) = \frac{1}{\pi} e^{2ixp}. $$
However, $\delta(-x)=\delta(x)$ and last time I checked $e^{...
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1answer
41 views
What is dipolar charge distribution?
An electric dipole is a system of two opposite point charges when their separation goes to zero and their charge goes to infinity in a way that the product of the charge and the separation remains ...
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1answer
78 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
44 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, \...
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1answer
64 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
64 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
26 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
38 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:
$$ F_{2}(...
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1answer
46 views
Dirac delta normalization of electromagnetic fields
Usually, to quantize electromagnetic fields we use box normalization and therefore the normalization constant contains the dimensions of Volume V of the box.
But if we perform the Dirac-delta ...
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2answers
54 views
Different definitions of Functional Derivative
In studying QFT and General Relativity, I came across two different definitions of Functional Derivative, and I'd like to know if they are equivalent.
Firstly, in Wald's book General Relativity, as ...
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1answer
69 views
Explanation of the identities $\rho=\rho(\delta)$ ($\delta$ function) and $\rho=q\,\delta(\bar{r})$
Poisson's equation is $$\boldsymbol{\nabla}^{2}\varphi=-4\pi k_{e}\,\rho, \tag{*}$$ that in the case of a point charge $q$, already with spherical symmetry, has as solution
\begin{equation}
\varphi(...
2
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2answers
142 views
Curl and circulation of a vector field that is ill-defined at the origin: any interesting physical effects?
In the cylindrical polar $(\rho,\phi,z)$ coordinate, suppose the velocity field in a liquid is given by $$\vec{v}=\frac{K}{\rho}\hat{e}_{\phi}, \qquad K=\text{constant}.\tag{1}$$ It can be easily ...
3
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1answer
42 views
Divergence of $ \frac{1}{s}\hat{s}$ in cylindrical coordinates
In Griffiths' electrodynamics, the divergence of $\frac{1}{r^{2}}\hat{r}$ is evaluated in spherical coordinates to be $4\pi\delta(r)$. I encounter the same problem in case of $\frac{1}{s}\hat{s}$ ...
2
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2answers
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A formula for delta function in quantum mechanics
I met a formula for delta function in a QM book ( not in English). The formula is used in the scattering theory. Its form is
$$\lim_{\alpha\rightarrow \infty}\exp[i\alpha x]=2i\delta(x), ~~(x\geq 0).$...
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0answers
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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:
$$
H = -\frac{\...
2
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2answers
128 views
Wilson-Sommerfeld Quantization of Dirac delta in Infinite Square Well (ISW)
I am curious to find the energies of Dirac delta potential inside the ISW (walls at $x=0,L$)
$$
H = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V_0\delta(x-L/2)
$$
using Wilson-Sommerfeld ...
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1answer
254 views
If one puts a delta-function spike inside an infinite square well, is the resulting potential analytically solvable?
It was recently floated in chat that a particle in a box with a delta-function spike inside it, with hamiltonian
$$
H = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V_0\delta(x-a)
$$
and with ...
1
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1answer
64 views
Integrating Laplace's equation over a sphere
The Wikipedia page on Laplace's equation states that
if the Laplacian of $u$ is integrated over any volume that encloses the source point,
$$\iiint_V \nabla \cdot \nabla u \, d^3V =-1.$$
I can'...
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2answers
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Is the DOS (density of states) wrong for degenerated case?
The density of states (DOS) is defined as $$\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,$$ ...
1
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1answer
58 views
Mathematical expression of impulsive forces during a collision
I'm facing with problem regarding collision between bodies. Specifically, I need to understand the impulsive forces that arises during a collision. Sometime ago, I posted two questions (this and this) ...
3
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2answers
85 views
Kac-Moody algebra from WZW model via Poisson brackets
In 'Non-abelian Bosonization in Two Dimensions', Witten shows that the Poisson brackets of the currents that generate the $G\times G$ symmetry of the WZW model give rise to a Kac-Moody algebra upon ...
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2answers
63 views
$\delta$ potential has highest probability for highest potential
I can't understand this intuitively. Figure 2.9 in Griffith's QM says that the wavefunction at $x=0$ for a delta function potential is $\sqrt{\kappa}$, and to the right it decays like $\psi_+=\sqrt{\...
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0answers
48 views
Square root of Dirac delta Function? [duplicate]
I've puzzled by the appeareance and manipulation of a Square root of a Dirac Delta function.
The article is
https://www.researchgate.net/publication/...
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2answers
79 views
How can we argue $\nabla\cdot v = 0$ when ${\mathscr r} \ne 0$ on this vector function?
I am dealing with the vector field:
$$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
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1answer
68 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 $f(\vec{x})=f(\vec{x}+\vec{r})$...
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1answer
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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:
$$\langle x \mid \psi \rangle = \int_{-\infty}^{\infty} \delta (x-x') \ \...
3
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4answers
297 views
Hamiltonian of quantum harmonic oscillator with $\psi(x)=\delta(x)$: comparison to classical mechanics
I was just reading the question Why can't $\psi(x)=\delta(x)$ in the case of a harmonic oscillator? The accepted answer says that $\psi(x)=\delta(x)$ is a mathematically valid state, though it's not ...
3
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1answer
171 views
Why can't $\psi(x) = \delta(x)$ in the case of Harmonic oscillator?
In the analysis of Harmonic Oscillator, it is claimed that $\langle\hat H\rangle$ cannot be zero, why is it so?
I mean $\hat H = \frac{ \hat p^2 }{2m } + \frac12 k \hat x^2$, and
$$\left<x^2\...
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0answers
105 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. ...
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0answers
82 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 ...
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1answer
68 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
$$
...
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1answer
175 views
Infinite square well: wall with infinitesimal thickness
Given an infinite square well, it doesn't matter how thick the wall is, the particle is trapped inside the two walls. If we make the wall of arbitrarily small but finite thickness, the particle is ...
4
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1answer
211 views
Dirac Delta Function and Position [duplicate]
How does one prove that the Dirac Delta distribution is the eigenfunction of the position operator $\hat{x}$? In math, why does $\langle x’|x\rangle = \delta(x’-x)$?
-1
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1answer
72 views
Numerical approximation of the wavefunction in a delta-potential [closed]
I am trying to approximate the wavefunction of a particle in a delta potential $U(x) = -U_0 \delta(x)$ with $V_0 \gt 0$. I am using the following formula to calculate the wavefunction:
$\psi(x+\Delta ...
1
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1answer
67 views
Divergence of a displacement vector field multiplied by delta function
I'm trying to work out why
$$ \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 $\...
