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.
772
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How to understand the scattering cross section of a delta potential field is not zero? [closed]
how to calculate the scattering cross section of a delta field?
and how to understand why it is not zero?
because a hard-sphere potential’s scattering cross section of a delta field equal to $$\pi r^2$...
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4
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1
answer
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Modeling a pure dipole as a function similar to a Dirac delta function
I am taking an undergraduate course in E&M following Griffiths.
I was wondering if there is a good way to embed the information of a dipole into the charge distribution (and if it would be of any ...
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0
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43
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Discretize a point on the surrounding grid and maintain Gaussian distribution
I now have a point located at x, and the value at that point is q(x). I want to discretize the point to the surrounding grid points and maintain a Gaussian distribution. A one-dimensional grid is fine....
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4
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If quantum fields are operator valued distributions, why aren't they always smeared?
I don't completely understand the distributional character of a quantum field because I never see them "smeared" in basic textbooks. As I understand it, if $\chi : \mathcal{F} \rightarrow \...
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Derivation of some property of the Dirac delta function [duplicate]
I have some question about the property of the delta function:
$$
\int g(x)\delta(f(x))dx=\frac{g(0)}{|f'(0)|}.
$$
I know how to derive this identity, but what I am not so sure is why there is the ...
- 49
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1
answer
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Calculation of the Effective action - Lewis H. Ryder
I have been studying the book on Quantum Field Theory by Lewis H. Ryder and I am finding a Gaussian integration a little bit confusing. In the book, the transition amplitude (Eq. $(5.15)$) is given as ...
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2
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Preserving the hermiticity of differential operators when acting on bra
In position basis, $\psi(x_o)=\langle x_o|\psi\rangle$ where $|\psi\rangle=\int dx|x\rangle\psi(x)$
So, we have defined $\langle x_o|x\rangle=\delta(x-x_o)$
Thus, $\langle x_o|\psi\rangle=\langle x_o|\...
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1
answer
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Integral of derivative of delta function gives strange answer [closed]
So I've been doing some QM and I keep coming across the following type of integrals:
$$
\int f(x) \frac{\partial}{\partial x} \delta(x-x') dx.
$$
I know that I should integrate by parts but then I ...
1
vote
0
answers
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Proof that $-\partial^2 G(x, y) = \delta(x-y)$ for free field propagator
I recently realized that there is a slightly pedantic issue when one normally proves that the equations of motion acting on the free field propagator gives a delta function which I have become ...
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2
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62
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Cancelling one-loop divergences in non-linear sigma model expansion term
In the appendix A of this paper by Braaten et al., the authors try to compute the divergences of two integrals that come from an expansion of an action $I$ in $\langle e^{iI} \rangle$, via dimensional ...
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0
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Why does the surface integral over the $B$-field in a Stokesian loop tend to zero as the surface tends to zero (boundary conditions)?
I am confused by the standard argument used for deriving the boundary conditions at the interface of two media as told by Jackson, e.g. see here.
My question concerns the fact that Jackson says that ...
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1
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How to compute the Feynman propagator for the Proca field?
I was repeating each step of the exercise 6.4 of the Greiner's book "Field quantization" when I discovered that there is a passage which I can't reproduce, the calculations are lengthy and ...
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Is this a valid alternative definition of the delta function?
The delta function can be defined as:
$$
\delta(x) = \int_{-\infty}^{\infty} e^{-2\pi i k x} \, dk
$$
Loosely speaking, I can understand this because unless $x=0$, the complex exponential oscillates ...
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Doublet impulse force harmonic oscillator
I initially asked this on a math forum, but I see now that physics was a better choice. I'm considering an at-rest simple harmonic oscillator (m,k) and want to model the force by a doublet (derivative ...
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2
answers
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How to calculate the rotation at a singularity?
An electrodynamics lecture asks me to prove that
$$
\nabla \times \left( \frac{\vec{M} \times \vec{x}}{ |\vec{x}|^3} \right) = \frac{8 \pi}{3} \vec{M} \delta^3(\vec{x})- \frac{\vec{M}}{|\vec{x}|^3}+ \...
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How does transition rate behave under $T \rightarrow \infty$ limit
I am supposed to learn Fermi's Golden Rule, and the book I am using for that is Modern Particle Physics by Mark Thomson. On page 52, he goes :
The transition rate $d\Gamma_{fi} = \frac{1}{T}|{T_{fi}^...
2
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2
answers
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What went wrong in the following calculation of $\langle p'|[x,p]|p'\rangle$? [duplicate]
We know that $$[x,p]=i\hbar. $$
Consider now the diagonal element in the momentum representation, $$\langle p'|[x,p]|p'\rangle=i\hbar\langle p'|p'\rangle=i\hbar\delta(0).$$
But the LHS = $$\langle p'|...
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The unitarity of the $\delta(x)$ potential
One of the common potentials to solve in quantum mechanics is when $$H=\frac{p^{2}}{2m}+\delta(x).$$ Is this Hamiltonian considered to produce unitary evolution?
In particular, I'm not sure what is ...
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1
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In formalizing QFT, are mathematical issues of canonical quantization approach and that of path integral approach related?
In QFT, many mathematical issues arise. Setting aside renormalization, these deal with rigorous constructions of objects underlying QFT:
i) In the canonical quantization approach, the main issue comes ...
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0
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55
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Product of delta functions in fermion self-energy at finite temperature
In the calculation of the fermion self-energy at finite temperature, there seems to be a term containing the product of two delta functions which when combined equal zero, however I fail to see why ...
3
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203
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Problem obtaining Klein-Gordon equation solutions
I am having some problems and also some questions regarding how can one get the general solution to the Klein-Gordon equation, which usually appears in the literature as
$$
\phi(t,\mathbf{x})=\int\...
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votes
2
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889
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Distributions "more singular than a Dirac delta" must have negativity
I am looking at properties of the Glauber P-functions, which are distributions (in the sense of a dirac delta) on the complex plane, normalized so that $\int_{\mathbb{C}} d^2 \alpha P(\alpha) = 1$. On ...
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votes
3
answers
228
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Schrödinger equation of the double delta potential in momentum space
This may be a silly question but it has gotten me stumped. I am currently trying to see if I can get anywhere by putting a simple symmetric double delta potential
\begin{align}
V(x) = -\gamma (\...
-1
votes
3
answers
115
views
How is the resolution of the identity carried out in the eigenbasis of the position operator?
I'm currently reading Robert Littlejohn's Physics 221A (Graduate Quantum Mechanics) Notes, and I encountered this in "Notes 1"
I'm a bit confused about how he "substituted this back ...
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0
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Scattering from a dirac delta potential in momentum space [duplicate]
I was curious about solving the reflection and transmission amplitudes for the dirac delta potential barrier using fourier transform.
I'm able to take the fourier transform but I'm unable to interpret ...
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0
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74
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What is $\left<x\right|\hat x\hat p-\hat p\hat x\left|x\right>$? [duplicate]
This is simple question, but I don't know how to do it.
$\left<x\right|\hat x\hat p-\hat p\hat x\left|x\right>=?$,
I can solve it in two ways.
One of them is
$$\left<x\right|\hat x\hat p-\hat ...
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0
votes
2
answers
111
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Matrix elements of the commutator of the delta function with $p^2$ [closed]
Suppose we have potential $V(x) = \delta(x)$. I want to evaluate $\langle x' \vert [p^2, \delta(x)] \vert x'' \rangle$. Unprimed quantities are operators while primed quantities are eigenvalues/...
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1
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What would the measurement device show me if I measured the dye concentration at the pipe location where all of the mass is concentrated at?
I was studying the methods for solving the diffusion equation. Here I found an example problem with the Dirac delta distribution initial condition and zero boundary conditions (example $9.5.1$):
$$ {\...
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votes
1
answer
86
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$n$-dimensional Gauss Law and Dirac Delta [duplicate]
I was searching for a fundamental proof of Gauss Law using the divergence of the electric field. In three dimensions the divergence of $\hat{r}/r^2$ evaluates to 0. So a book I was reading said that ...
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Reference request - classical field theory and mathematics
I am looking for references (books, lecture notes etc) on mathematical classical field theory. By that, I mean classical field theory under a rigorous point of view. However, I am more interested in ...
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0
answers
43
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Is nature discretizable? [duplicate]
From mathematical perspective, can we describe all the realistic quantum mechanical phenomena at any given moment by functions alone, or is it correct that distributional behavior can also be observed ...
- 1,697
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0
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Dirac delta in Fourier space for finite volume
In some class notes about cosmology I have found the following claim. The author starts by stating that the Dirac delta is given by:
$$\delta^{(D)}(\vec{x}+\vec{x}')=\int\dfrac{d^3q}{(2\pi)^3}e^{i(\...
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1
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"Special Kind of Infinity" Showing Up In the Product of a Bra and Ket Vector Under and Integral
As a summer "project" I have been reading through Paul Dirac's Principles of Quantum Mechanics to try and acquaint myself with Quantum Theory. In a section on observables, Dirac poses a ...
0
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1
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43
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Wave propagation in free space in the limit $d\to 0$
In Goodman's Introduction to Fourier Analysis, the evolution of the field amplitude $U(x_0)$ of a wave propagating in free space is described by the Fresnel approximated operator $$\mathcal R[d]\{U(...
3
votes
1
answer
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How to normalize the states in the continuous limit?
In quantum field theories we can perform the continuous limit, where we take the limit $V\rightarrow\infty$ for the volume system. In quantum optics, we can start by absorbing a factor $\left(\frac{L}{...
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vote
0
answers
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Where did the the $\delta(0)$ go? [closed]
I am reading my notes for a lecture on quantum field theory. For some example of a 1d system with spatial coordinate $x$ and for a wave with frequency $\omega$, it starts with defining field operators ...
- 855
3
votes
0
answers
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Trouble showing that the "naive" dilaton vertex operator is not primary
Context:
On page 167 of David Tong's string theory notes, he writes that
"a naive construction of the [dilaton's] vertex operator is not primary".
I assume that by this he means an ...
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votes
1
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Question about integral containing derivative of Dirac delta distribution
The result of the integral of the dirac delta δ(x-a) times a function f should be f(a) right? Then why isn't the integral just the final result directly without doing all of this, where did the ...
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Dirac equation solution for a point particle
I want to solve a Dirac equation with a special condition for the probabillity
$\bar{\psi}\gamma^0\psi=\delta\left(\vec{r}\right)$
Since Dirac spinor is a $4\times 1$ matrix, I am not sure what ...
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1
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54
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Is there a closed-form solution to the free evolution of a delta-function via de-Broglie waves?
One of the most fascinating parts of de Broglie's thesis is that, if you have a "hyperplane" wave in spacetime, with a frequency of $\omega_0$ and no spatial dependence in its "rest&...
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1
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Bosonic representation of delta function for Grassmann-even quantity
Suppose I have 2 Grassmann scalars $\theta$ and $\bar{\theta}$ and form the bosonic quantity $X = \bar{\theta}\theta$. Is there a purely bosonic representation of the delta function $\delta(X - \...
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0
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Generalised hydrodynamics and the Dirac delta potential
Generalised Hydrodynamics is a theory of hydrodynamics for Quantum integrable systems. Those system are integrable in the sense that one can find an infinite number of conserved charges i.e. ...
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1
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Is the $i\varepsilon$ prescription in the Feynman propagator just as "outrageous" as $1+2+3+... = -1/12$?
When the calculation of the Feynman propagator is introduced in QFT (or QM), the inclusion of the $i\varepsilon$ term is often described as a minor technical detail that is there just to "make ...
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0
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Time evolution of Gaussian wave packet in free space with initial condition Dirac delta function [duplicate]
I learn something about the Time evolution of Gaussian wave packet in free space.
if the initial condition is a Dirac delta function at $t=0$, then the wave function is
$$
\psi (x,t)=\frac{1}{\sqrt{2\...
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3
votes
1
answer
149
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Dirac delta of matrix argument - Matrix model path integral vs Hilbert space
Assume a Hamiltonian $H$ with $N$ orthonormal eigenstates $\{\vert n\rangle\}$ of energies $\epsilon_n$. One can define a density of states,
\begin{align}
\rho(E)&=\mathrm{tr}\,\hat{\delta}(E-\hat{...
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0
votes
2
answers
80
views
How to calculate the a delta function times derivative of a function? [closed]
It is clear to me that when we are talking about delta function, we should always understand it as a distribution, for example,:
\begin{equation}
\int \delta(x-x') f(x) \, \mathrm{d}x = f(x')
\end{...
1
vote
1
answer
75
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Fourier Transformation if we only have relative coordinates
Let’s say we want to do a Fourier Transformation (FT) of a function $f(t-t‘,r-r‘)$ i.e. the function to be Fourier transformed only depends on relative coordinates. This is for example the case if we ...
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0
votes
0
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91
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References on the difference between Dirac's and Von Neumann's approach to Quantum Mechanics (rigged Hilbert Space vs Hilbert Space only)
I have not found any clear and comparative explanation between the Dirac and von Neumann versions of Quantum Mechanics (rigged Hilbert Space vs Hilbert Space only). I have found some short articles. ...
1
vote
0
answers
51
views
What is the relationship between $\partial_x^2\frac1r$ and $\delta^3(r)$? [closed]
We have the equation
\begin{equation}
\nabla^2\frac1r=-4\pi\delta^3(r).
\end{equation}
I first encountered this equation in electrodynamics. So what is $\partial_x^2\frac1r$ then? It looks like the ...
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0
votes
2
answers
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Dirac Delta Function by M.S. Howe
I'm reading 'Hydrodynamics and Sound' by M.S. Howe. I want to understand the point source like the figure. I could understand $\nabla^2\phi=0$, if $r>0$. But I couldn't understand $r\rightarrow 0$ ...
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