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.
372
questions
0
votes
1answer
40 views
How to get the imaginary part from the Källén-Lehmann propagator
During field theory course the Källén-Lehmann propagator was defined as follows:
$$D_F(p^2) = \frac{i}{p^2-m^2+i\epsilon} + \int^{\infty}_{4m^2}ds\rho(s)*\frac{i}{p^2-s+i\epsilon} \tag{1}$$
...
1
vote
0answers
29 views
Interpretation of induced force between two Dirac delta potential wells
This problem is based on MIT OCW course 8.04 problem set 6 question 5(e).
https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2013/assignments/MIT8_04S13_ps6.pdf
Consider two Dirac ...
0
votes
1answer
24 views
Deriving the form of radial distribution function in molecular dynamic simulation?
I am trying to derive the equation 8.16 in the attached image from the definition of the radial distribution function. However, I am unable to derive it with proper constants of $\frac{2}{N(N-1)}$
My ...
4
votes
4answers
149 views
Divergence of $\frac{ \hat {\bf r}}{r^2}$ , what is the 'paradox'?
I just started in Griffith's Introduction to electrodynamics and I stumbled upon the divergence of $\frac{ \hat r}{r^2}$ , now from the book, Griffiths says:
Now what is the paradox, exactly? ...
0
votes
1answer
34 views
Writing a two variable function$ f(x,t)$ in terms of Dirac-Delta $δ(x)$ function and a function $P(t)$?
How to write a two variable function $f(x,t)$ in terms of Dirac-Delta $\delta(x)$ function and a function $P(t)$?
For example;
I read something in a book. You can find the following picture.
But ...
1
vote
0answers
57 views
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 ...
0
votes
1answer
56 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 ...
0
votes
0answers
61 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.
$$D_x G(x,x^{'...
1
vote
2answers
144 views
Mathematical rigorous definition for an electrical dipole
I've been reading Laurent Schwartz's Mathematics for the physical sciences, and in the chapter on distributions he makes many cool examples of ways to define in a mathematical rigorous way certain ...
0
votes
1answer
51 views
Position representation of an operator
$$\langle\ x\rvert M\lvert\ x'\rangle=M(x)\langle\ x\lvert\ x'\rangle=M(x)\delta(x-x')$$
I know this is true for if $M$ is a momentum operator or position operator, is this is true for a general ...
1
vote
1answer
52 views
Itzykson and Zuber: Reduced $T$-matrix
Could someone help me understand the reduced $T$-matrix mentioned in Itzykson and Zuber, eq.
$$\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}$...
0
votes
2answers
44 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.
-1
votes
1answer
37 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
votes
0answers
19 views
Expected value of the current density operator
In Ullrich's TD-DFT book, the paramagnetic current-density operator is defined as
$$\hat{\mathbf{j}}(\mathbf{r})=\frac{1}{2i}\sum_{a=1}^{N}\left[\nabla_a\delta(\mathbf{r}-\mathbf{r}_a)+\delta(\mathbf{...
2
votes
1answer
70 views
Why do Candelas and Howard say that $\sum_{n=1}^\infty \cos\left( n \kappa \epsilon \right) \ = \ - \frac{1}{2}$?
In the paper Vacuum $\langle \phi^2 \rangle$ in Schwarzschild Spacetime by Candalas and Howard, they say that for each non-zero $\epsilon$ it is true that
$$
\sum_{n=1}^\infty \cos\left( n \kappa \...
1
vote
1answer
44 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 ...
1
vote
2answers
51 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 ...
0
votes
1answer
47 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 ...
0
votes
1answer
29 views
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 ...
4
votes
1answer
54 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
votes
1answer
65 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^{...
0
votes
1answer
45 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 ...
1
vote
1answer
79 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 ...
0
votes
0answers
47 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, \...
1
vote
1answer
67 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),...
0
votes
1answer
65 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{\...
0
votes
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*}
\...
0
votes
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:
$$ F_{2}(...
0
votes
1answer
49 views
Dirac delta normalization of electromagnetic fields
Usually, to quantize electromagnetic fields we use box normalization and therefore the normalization constant contains the dimensions of Volume V of the box.
But if we perform the Dirac-delta ...
1
vote
2answers
55 views
Different definitions of Functional Derivative
In studying QFT and General Relativity, I came across two different definitions of Functional Derivative, and I'd like to know if they are equivalent.
Firstly, in Wald's book General Relativity, as ...
0
votes
1answer
74 views
Explanation of the identities $\rho=\rho(\delta)$ ($\delta$ function) and $\rho=q\,\delta(\bar{r})$
Poisson's equation is $$\boldsymbol{\nabla}^{2}\varphi=-4\pi k_{e}\,\rho, \tag{*}$$ that in the case of a point charge $q$, already with spherical symmetry, has as solution
\begin{equation}
\varphi(...
2
votes
2answers
147 views
Curl and circulation of a vector field that is ill-defined at the origin: any interesting physical effects?
In the cylindrical polar $(\rho,\phi,z)$ coordinate, suppose the velocity field in a liquid is given by $$\vec{v}=\frac{K}{\rho}\hat{e}_{\phi}, \qquad K=\text{constant}.\tag{1}$$ It can be easily ...
3
votes
1answer
44 views
Divergence of $ \frac{1}{s}\hat{s}$ in cylindrical coordinates
In Griffiths' electrodynamics, the divergence of $\frac{1}{r^{2}}\hat{r}$ is evaluated in spherical coordinates to be $4\pi\delta(r)$. I encounter the same problem in case of $\frac{1}{s}\hat{s}$ ...
2
votes
2answers
169 views
A formula for delta function in quantum mechanics
I met a formula for delta function in a QM book ( not in English). The formula is used in the scattering theory. Its form is
$$\lim_{\alpha\rightarrow \infty}\exp[i\alpha x]=2i\delta(x), ~~(x\geq 0).$...
1
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0answers
99 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:
$$
H = -\frac{\...
2
votes
2answers
130 views
Wilson-Sommerfeld Quantization of Dirac delta in Infinite Square Well (ISW)
I am curious to find the energies of Dirac delta potential inside the ISW (walls at $x=0,L$)
$$
H = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V_0\delta(x-L/2)
$$
using Wilson-Sommerfeld ...
6
votes
1answer
276 views
If one puts a delta-function spike inside an infinite square well, is the resulting potential analytically solvable?
It was recently floated in chat that a particle in a box with a delta-function spike inside it, with hamiltonian
$$
H = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V_0\delta(x-a)
$$
and with ...
1
vote
1answer
66 views
Integrating Laplace's equation over a sphere
The Wikipedia page on Laplace's equation states that
if the Laplacian of $u$ is integrated over any volume that encloses the source point,
$$\iiint_V \nabla \cdot \nabla u \, d^3V =-1.$$
I can'...
0
votes
2answers
35 views
Is the DOS (density of states) wrong for degenerated case?
The density of states (DOS) is defined as $$\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,$$ ...
1
vote
1answer
59 views
Mathematical expression of impulsive forces during a collision
I'm facing with problem regarding collision between bodies. Specifically, I need to understand the impulsive forces that arises during a collision. Sometime ago, I posted two questions (this and this) ...
3
votes
2answers
90 views
Kac-Moody algebra from WZW model via Poisson brackets
In 'Non-abelian Bosonization in Two Dimensions', Witten shows that the Poisson brackets of the currents that generate the $G\times G$ symmetry of the WZW model give rise to a Kac-Moody algebra upon ...
0
votes
2answers
66 views
$\delta$ potential has highest probability for highest potential
I can't understand this intuitively. Figure 2.9 in Griffith's QM says that the wavefunction at $x=0$ for a delta function potential is $\sqrt{\kappa}$, and to the right it decays like $\psi_+=\sqrt{\...
0
votes
0answers
48 views
Square root of Dirac delta Function? [duplicate]
I've puzzled by the appeareance and manipulation of a Square root of a Dirac Delta function.
The article is
https://www.researchgate.net/publication/...
1
vote
2answers
79 views
How can we argue $\nabla\cdot v = 0$ when ${\mathscr r} \ne 0$ on this vector function?
I am dealing with the vector field:
$$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
vote
1answer
70 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
79 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:
$$\langle x \mid \psi \rangle = \int_{-\infty}^{\infty} \delta (x-x') \ \...
3
votes
4answers
304 views
Hamiltonian of quantum harmonic oscillator with $\psi(x)=\delta(x)$: comparison to classical mechanics
I was just reading the question Why can't $\psi(x)=\delta(x)$ in the case of a harmonic oscillator? The accepted answer says that $\psi(x)=\delta(x)$ is a mathematically valid state, though it's not ...
3
votes
1answer
174 views
Why can't $\psi(x) = \delta(x)$ in the case of Harmonic oscillator?
In the analysis of Harmonic Oscillator, it is claimed that $\langle\hat H\rangle$ cannot be zero, why is it so?
I mean $\hat H = \frac{ \hat p^2 }{2m } + \frac12 k \hat x^2$, and
$$\left<x^2\...
0
votes
0answers
111 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
88 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 ...
