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.
386
questions
0
votes
0answers
19 views
Coherent state basis of (relativistic) particle Fock space
For a neutral scalar bosonic particle of mass $m$, I consider a Fock space with an orthonormal basis of momenta eigenstates
\begin{equation}\label{Fock-p-states}
\left|p_1p_2\cdots p_n\right\rangle=\...
0
votes
2answers
59 views
Integrating TISE
Given the potential function:
$$
V(x) = \lambda \delta(x)
$$
I'm asked to integrate the TISE from $x = -\epsilon$ to $x = \epsilon$ for $\epsilon > 0$ and take the limit as $\epsilon \rightarrow ...
0
votes
1answer
28 views
Do Equal Time Commutation / Anticommutation relations automatically also apply for spatial derivatives?
The question is basically in the title.
My naive thought was that when a commutation relation holds for all field operators $\Psi(\vec{x})$ (by "all" I mean "at all positions $\vec{x}$") on a fixed ...
1
vote
0answers
69 views
How to prove scaled delta function relation mathematically? [migrated]
I am working through Shankar's Introduction to Quantum Mechanics. I have come across exercise 1.10.1, which asks the reader to show that:
$$\delta(ax)=\frac{\delta(x)}{|a|}.$$
I can understand it ...
2
votes
1answer
106 views
Limit of the $\sin^2$ function in the derivation of Fermi's golden rule
In the derivation of Fermi's golden rule one typically arrives at an expression of the form
$$
\frac{\sin^2(\omega t)}{\omega^2}
$$
which is then converted to
$$
\pi t\delta(\omega).
$$
I cannot ...
2
votes
1answer
32 views
Rationalizing the current operator
In eq. 1.191 of Mahan Many-Particle Physics, the current operator is defined as
$$j(r)=\frac{1}{2}\sum_ie_i[v_i\delta(r-r_i)+\delta(r-r_i)v_i].$$
I'm trying to make some sense of this definition. I ...
0
votes
0answers
29 views
Double Sommerfeld Expansion
Consider the Fermi-Dirac expansion for an arbitrary function $f(\epsilon)$:
$I(f)=\int_0^\infty d\epsilon\frac{f(\epsilon)}{e^{\beta(\epsilon-\mu)}+1}$
The large $\beta$ expansion of this quantity ...
1
vote
1answer
152 views
Why Dirac delta function is considered as wave function?
I have read in my textbook that dirac delta function is regarded as wave function. But isn't it violating the condition required for a wave function ie .
It becomes infinite $ δ(x-x_0)$ at $x=x_0$ ...
0
votes
0answers
56 views
Constraints in path integral and the Lagrange multiplier
I was reading some references on the slave-particle approach to the Kondo problem and Anderson model.
It is known that the slave-particle is introduced in the large Hubbard $U$ limit of the system so ...
0
votes
0answers
56 views
Colombeau Algebra for physics students
I have an undergraduate degree in physics, taken 2 years of calculus, and a rigorous course in linear algebra. I have not taken a math course in analysis, though have read a bit about it on my own.
...
0
votes
0answers
25 views
Confusion regarding the bound state of a Delta-function potential and Tunneling
I was reading (Griffith's QM book) about the Bound states for delta-function potential of the form $-\alpha \, \delta(x)$ where $\alpha > 0$.
I feel a bit conceptually unclear. Few doubts I have ...
1
vote
1answer
69 views
Covariant form of Green's function for wave equation
In J. D. Jackson's "Classical Electrodynamics", page 614 in the 3rd edition, he states that you can write the Green's functions for the wave equation in covariant form using the fact that
\begin{align*...
0
votes
1answer
66 views
${}$ Dirac delta potential
We know that the number of bound states for an attractive delta potential is one. If so what will the number of bound states for a particle in a repulsive delta potential?
If $V(x)= +a \cdot \delta(x)$...
2
votes
2answers
66 views
Deriving Current density of a Moving Point Charge Using the Continuity-Equation
Problem:
We know, a point charge at position $\mathbf{r}_q$ has the charge density
$$\rho_q(\mathbf{r})=q\delta(\mathbf{r}-\mathbf{r}_q) \tag{2}$$
if it moves with the velocity $\mathbf{v}$, we get ...
4
votes
1answer
76 views
How does a Lagrangian with delta potential transform to a Hamiltonian?
Suppose the Lagrangian was given as:
$$L = \frac{1}{2}\int_{-\infty}^{\infty} \underbrace{\left(\dot{A(}z)^2-A(z)^2\right)+\left(\dot{Q(}z)^2\delta(z)-Q(z)^2\delta(z)\right)+2\dot{Q(}z)\cdot A(z) \...
0
votes
1answer
58 views
How to get the imaginary part from the Källén-Lehmann propagator
During field theory course the Källén-Lehmann propagator was defined as follows:
$$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}$$
...
1
vote
0answers
35 views
Interpretation of induced force between two Dirac delta potential wells
This problem is based on MIT OCW course 8.04 problem set 6 question 5(e).
https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2013/assignments/MIT8_04S13_ps6.pdf
Consider two Dirac ...
0
votes
1answer
29 views
Deriving the form of radial distribution function in molecular dynamic simulation?
I am trying to derive the equation 8.16 in the attached image from the definition of the radial distribution function. However, I am unable to derive it with proper constants of $\frac{2}{N(N-1)}$
My ...
4
votes
4answers
247 views
Divergence of $\frac{ \hat {\bf r}}{r^2}$ , what is the 'paradox'?
I just started in Griffith's Introduction to electrodynamics and I stumbled upon the divergence of $\frac{ \hat r}{r^2}$ , now from the book, Griffiths says:
Now what is the paradox, exactly? ...
0
votes
1answer
39 views
Writing a two variable function$ f(x,t)$ in terms of Dirac-Delta $δ(x)$ function and a function $P(t)$?
How to write a two variable function $f(x,t)$ in terms of Dirac-Delta $\delta(x)$ function and a function $P(t)$?
For example;
I read something in a book. You can find the following picture.
But ...
0
votes
1answer
63 views
What is the relationship between wave function and classical distribution function?
The question is a bit weird since the wave function is quantum mechanical and the distribution function in phase space is really something classical.
But I would still like to know if I take the ...
0
votes
0answers
64 views
Green function derived from the idea of position representation in quantum mechanics
I am trying to study green's function by using a lecture by NPTEL its link is
https://www.youtube.com/watch?v=ZJ7v6VZQ32k&t=439s
I don't get this step by him at the minute of ten.
$$D_x G(x,x^{'...
1
vote
2answers
155 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
60 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
57 views
Itzykson and Zuber: Reduced $T$-matrix
Could someone help me understand the reduced $T$-matrix mentioned in Itzykson and Zuber, eq.
$$\langle{f}| T|p_1p_2\rangle=(2\pi)^4\delta^4(P_f-p_1-p_2)\langle f|\mathcal{T}|p_1p_2\rangle. \tag{5-7}$...
0
votes
2answers
66 views
Is it true that any electrostatic charge density can be represented by Dirac Delta function? Give me a general example
My college professor said that any charge density can be replaced by Dirac Delta function but how I want to know.
-1
votes
1answer
51 views
Finding normalization constant of a wave function with definite momentum
I try to read Sakurai's Modern Quantum Mechanics but I stuck at this point,
$$\delta(x^{'}-x^{''})=|N|^{2}\int dp^{'}\exp\Biggl({ip^{'}(x^{'}-x^{''})2\pi\over h}\Biggr)$$
This is an expression for ...
2
votes
0answers
26 views
Expected value of the current density operator
In Ullrich's TD-DFT book, the paramagnetic current-density operator is defined as
$$\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
74 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
$$
\sum_{n=1}^\infty \cos\left( n \kappa \...
1
vote
1answer
72 views
Problem in the continuum limit of a Kronecker delta
I am having troubles in understanding how to correctly perform the continuum limit of a double sum containing a Kronecker delta.
Imagine to integrate a function depending on $t$ and $t'$, both ...
1
vote
2answers
72 views
Factorising a 4D Dirac delta function in a rest frame
I'm working through a QFT problem and at one stage in the solutions we have this step:
$$\delta^{(4)}(p - q_1 - q_2) = \delta(E_1 +E_2 - M)\delta^{(3)}(\bf{q_1} - \bf{q_2}).$$
We are working in the ...
0
votes
1answer
52 views
Properties of Dirac delta function in Integral
I was reading commutation relation of canonical momentum in KG Field from Lectures of Quantum Field Theory by Ashok Das. In page 179, He has used Integration to derive the result where he expressed ...
0
votes
1answer
89 views
What is the volume charge density (in spherical coordinates)?
What is the volume charge density (in spherical coordinates) of a uniform, in finitesimally thin spherical shell of radius $R$ and total charge $Q$, centered at the origin? Give your answer in terms ...
4
votes
1answer
56 views
$\delta^{(2)}$ convention
In this note: https://arxiv.org/abs/hep-th/0410165 at page 12 there is a delta-function constraint written as:
\begin{align}
\delta \left( ^ { U } M \right) = \prod _ { i < j } \delta ^ { ( 2 ) } \...
3
votes
1answer
70 views
A question about $\delta(x) \star \delta(p)$
Using the Moyal product between two delta functions in $(x,p)$-space one gets
$$ \delta(x) \star \delta(p) = \frac{1}{\pi} e^{2ixp}. $$
However, $\delta(-x)=\delta(x)$ and last time I checked $e^{...
0
votes
1answer
55 views
What is dipolar charge distribution?
An electric dipole is a system of two opposite point charges when their separation goes to zero and their charge goes to infinity in a way that the product of the charge and the separation remains ...
1
vote
1answer
88 views
Dirac delta function mathematical expression proof
In a discussion of the second order transitions in graphene this mathematical expression is used.
$$
\left|\frac{1}{\varepsilon + i \Gamma/2}\right|^2 = \frac{2\pi}{\Gamma}\delta(\epsilon)
$$
And I'm ...
0
votes
0answers
58 views
What is the symmetry behind this degeneracy?
I was working on a quantum mechanics problem, involving the perturbation of the 3D cubical potential well:
Suppose we perturb the infinite cubical well
\begin{equation}
V(x,y,z)=\begin{cases}
0, \...
1
vote
1answer
74 views
Delta function eigenstate for non-zero potential
Consider the potential $V(x)=\frac{2}{x^2}$ and let $\frac{\hbar^2}{2m}=1$ for convenience. Now consider the function $\psi(x)=\delta(x)$.
According to Griffiths (electrodynamics book) problem 1.45(a),...
0
votes
1answer
81 views
Derivation of the QFT Propagator
I don't understand how we get from the RHS to the last line.
\begin{eqnarray}
\left[ \hat{H}_x - i \frac{\partial}{\partial t_x} \right] G^+(x,t_x,y,t_y) &=& -i \delta (t_x - t_y) \sum_n{\...
0
votes
0answers
29 views
Bosonic Pair Distribution Function
In Schwabls Book "Advanced Quantum Mechanics" in the chapter for Bosons he calculates the Bosonic pair distribution function for noninteracting bosons. He said the expectation value of \begin{align*}
\...
0
votes
1answer
42 views
Anomalous magnetic moment of the electron - integration problem
In Schwartz's QFT book (eqn 17.31), to find the anomalous magnetic moment of the electron from the form factors, near the end of the calculation the following integral needs to be evaluated:
$$ F_{2}(...
0
votes
1answer
59 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
150 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
83 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
159 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
74 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
191 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
$$\lim_{\alpha\rightarrow \infty}\exp[i\alpha x]=2i\delta(x), ~~(x\geq 0).$...
1
vote
0answers
118 views
Perturbation theory development for the ground state of the QM particle in the box with a centered dirac-delta spike
In the course of a discussion in the chat there emerged an interesting problem, namely a particle in an infinite well with an additional Dirac-delta function spike of scalable hight:
$$
H = -\frac{\...
2
votes
2answers
142 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 ...
