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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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Convolution And deconvolution of functions

It is apparent that any attempt to measure the value of a physical quantity is limited, to some extent, by the finite resolution of the measuring apparatus used. On the one hand, the physical quantity ...
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Resolution and delta function

Any attempt to measure the value of a physical quantity is limited, by the finite resolution of the measuring apparatus used. on the one hand the physical quantity we wish to measure will be in ...
Hello's user avatar
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Does the exponential representation of Dirac delta function depend on choice of Fourier convention?

Is it always true that $$\delta(\omega) = \frac{1}{2\pi}\int_{-\infty}^{+\infty} e^{i \omega t} dt , $$ regardless of your Fourier convention? For example, if I choose to use the Fourier convention ...
photonica's user avatar
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Why do delta wells and delta barriers behave equivalently, but normal wells and normal barriers don't?

In Griffith's Intro to QM, it is shown that for $V(x)=-\alpha \delta(x)$ regardless of whether $\alpha$ is positive (a "delta well") or negative (a "delta barrier"), the ...
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Is $\sin(\infty) = 0$? [migrated]

My theoretical QM prof writes $$ \delta(x-a) = \frac{1}{\pi} \lim_{l \to \infty} \left[ \frac{ \text{sin}(l (x-a))}{x-a} \right] $$ which for $x \neq a$ means $$ 0 = \frac{1}{\pi} \lim_{l \to \infty} \...
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Why is Dirac Delta Function term in QFT's momentum operator not infinite?

In Weigand's Quantum Field Theory notes, he has the following equations for the spatial components of the momentum operator. $$ \begin{align} P^i &= \int \frac{d^3p}{(2\pi)^3} \dot{\phi(\vec{x})}\...
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Lorentz-invariant phase space integral

Consider the following Lorentz invariant integral associated to a $2\to 2$ scattering: \begin{equation*} I = \int \frac{d^3\mathbf{p_3}}{(2\pi)^3 2E_3} \int \frac{d^3\mathbf{p_4}}{(2\pi)^3 2E_4} \...
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Schrödinger equation, 2D delta function potential, and confusion

Apropos of nothing in particular, I thought I would play around with the Schrödinger equation in 2D with a delta function potential. To keep things simple I thought I would concentrate on the bound ...
bob.sacamento's user avatar
2 votes
2 answers
103 views

Closed form expression of 2D CFT integral

I am currently working on a 2d CFT and am wanting to compute a complex plane integral, making sure I take into consideration potential contact terms as well. The integral in question is $$ \int d^2z \...
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White noise fluctuation amplitude

I'm trying to understand better noise processes, and have a very basic question. Suppose I have a stochastic process characterised by white noise, namely $$ \langle X(t) \rangle = \overline{X} \,;\...
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A simple question in quanum mechanics on position and momenum eigenstates

The eigenfunctions (eigenstates) for the momentum of a particle are given by the plane waves $$\phi(x,t) = \sin(kx - \omega t)$$ If we sum a large number of these waves in a range from $0$ to $k_m$, ...
Anky Physics's user avatar
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Diverging Scattering Amplitudes and Transmission/Reflection Coefficients

I am currently studying scattering theory from Sakurai and Griffiths and I have noticed that for the 1D Dirac potential, the transmission and reflection coefficients diverge when the energy ...
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Manipulation of functions inside a Dirac Delta function [closed]

It is not clear to me how this derivation proceeds through the steps. Could someone help me understand how to arrive at this result or point me towards a resource that explains these algebraic ...
Jasper amirante's user avatar
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2 answers
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Grassmann variables and orthogonality of coherent fermionic states

Let a coherent fermionic state $$ \left|\phi\right> := \left|0\right> + \left|1\right> \phi,\tag{0} $$ where $\phi$ is a Grassmann number (i.e. it anticommutes with other Grassmann numbers). ...
Gabriel Ybarra Marcaida's user avatar
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2 answers
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How does the Green's function related the wavefunctions at different space-time points in Schrödinger's equation?

I have been trying to study Quantum Field Theory and have come across Green's Functions for the first time. While referring to Tom Lancaster's book Quantum Field Theory for the Gifted Amateur, the ...
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How to understand this Dirac delta function?

I am reading this paper about quantization of the electromagnetic field, and there is a point where the author imposes the fundamental commutation relation between the vector potential and its ...
Claudio Saspinski's user avatar
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How do we know Schwinger functions exist?

Let $\mathcal{D}'(\mathbb{R}^n)$ denote the dual of $C^\infty_C(\mathbb{R}^n)$, that is distributions on the set of infinitely differentiable functions with compact support. If $d\mu$ is a probability ...
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Continuum bases: why do we use dirac delta function? [duplicate]

In discrete bais, we can express a vector as $$ |\psi\rangle=\sum_{i} c_i|e_i\rangle $$ with orthonormality $$ \langle e_i|e_j\rangle=\delta_{ij}.$$ $\delta_{ij}$ is usual kronecker delta. If we ...
B. Silasan's user avatar
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Questions about derivation of Fano resonance

In Fano's original paper about Fano resonance [https://doi.org/10.1103/PhysRev.124.1866], starting from equation (3b) $$V_{E'}a+E'b_{E'}=Eb_{E'}\tag{3b}$$ one gets an expression for $b_{E'}$ $$b_{E'}=\...
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Dirac Delta applied to the gradient of a function

The supplementary section of a paper I am reading uses the "substitution" property of the Dirac delta function : $$\mathbf{v}\left(\mathbf{x}_j\right)\delta\left(\mathbf{x}-\mathbf{x}_j\...
haricash's user avatar
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1 answer
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How to standardize the energy of a Dirac delta function relative to sample rate (width) and amplitude?

Background I was instructed that a Dirac delta function (impulse from $0$ to $A$ then back to $0$ at short duration) has a white noise audio frequency type excitation distribution here ie. It should ...
mike's user avatar
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2 answers
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Neutral Hydrogen Atom Time-Averaged Potential

I'm self-studying the 3rd edition of Jackson's Classical Electrodynamics and I have a question about a problem. The fifth problem of Chapter 1 asks the reader to determine the volume charge ...
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2 answers
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Green Function for Poisson Equation Derivation

I've read the Green function derivation for Poisson Equation (electrostatics) in this document. There are some points which are not clear for me. On page 10, the document starts with the Poisson ...
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Coulomb potential from QFT in the external field approximation [closed]

In eq. (13.6.8) at page 558 of the first volume of the Quantum Theory of Fields by Weinberg, the following identity is given: \begin{align} \left[\frac{1}{(q_1\cdot p +i \varepsilon)((q_1+q_2)\cdot p +...
Tanatofobico's user avatar
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2 answers
152 views

Position operator action on a wavefunction [closed]

In a 1 dimensional infinite potential well with width $a$, the ground state wave-function is given by $$\psi(x) = \sqrt{\frac{2}{a}}\sin(\frac{\pi}{a}x)$$ The action of the position operator in the ...
Anky Physics's user avatar
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2 answers
175 views

Solving Wave Equation in 1+1D via Fourier Transforms with Dirac Delta function initial condition

I'm trying to use the Fourier transform method to solve the following PDE: This is a an infinite string with a pulse for it's initial condition. (At $t=0$, the string is stricken sharply so that the ...
MoreDust's user avatar
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1 answer
57 views

A tricky derivation accompanied by delta function

I have been reading a book on Thermal Field theory by Michel Le Bellac During the reading I have come into a seemingly trivial but indeed tricky derivation. On page 26(2.47), we are supposed too prove ...
quantumology's user avatar
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1 answer
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Trouble Understanding Computation In Nucleon Scattering Example in David Tong Lecture Notes

I am struggling to understand the following computation from page 59 of Tong's QFT notes http://www.damtp.cam.ac.uk/user/tong/qft.html The expression $$ (-ig)^{2} \int \frac{d^{4}k}{(2\pi)^{4}} \frac{...
user480172's user avatar
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2 answers
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Is the entropy for the classical microcanonical ensemble defined... up to an infinite number?

In the microcanonical ensemble all states $(p, q) \in \Gamma$ (where $\Gamma$ is the phase space of a system with $3N$ coordinates) with the same energy have the same probability density. I would ...
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Expectation value in terms of Dirac function in Quantum Mechanics

Recently, I started with principles of Quantum Mechanics by R. Shankar, and I got stuck at a part in my calculation of the Expectation value of Position and Momentum operator in the case where the ...
Charu _Bamble's user avatar
2 votes
0 answers
99 views

Why is the general solution of the Klein-Gordon equation a distribution? [closed]

I posted this question on math SE but I thought it might also be appropriate to post it here. Consider the Klein-Gordon equation: $$(\square + m^2)\phi = (\partial_t^2 - \Delta + m^2)\phi = 0 \tag{1}$$...
CBBAM's user avatar
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4 votes
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 ...
Pallav Pant's user avatar
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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....
Zhao Dazhuang's user avatar
9 votes
4 answers
3k views

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 ...
Tomer's user avatar
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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 ...
Jack's user avatar
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2 answers
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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|\...
Iti's user avatar
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-1 votes
1 answer
106 views

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 ...
Gytis Vejelis's user avatar
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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 ...
pseudo-goldstone's user avatar
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67 views

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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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 ...
F L's user avatar
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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 ...
zzz's user avatar
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2 votes
2 answers
147 views

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}+ \...
F L's user avatar
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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}^...
Mahammad Yusifov's user avatar
2 votes
2 answers
88 views

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'|...
Enigma's user avatar
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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 ...
Yair's user avatar
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6 votes
1 answer
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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 ...
Sam Park's user avatar
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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 ...
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