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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12
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246 views

Derivatives of distributions in general relativity

I am having some trouble when trying to reproduce some calculations involving the description of distributions (mostly used in spacetime junction conditions). I am trying to reproduce the ...
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186 views

$D$-dimensional Schrodinger's equation with a Dirac delta potential

I know that for $D\geq 2$, there is no bound state for a Dirac potential $V=-\alpha \delta(\textbf{x})$ unless we use an ultraviolet cutoff $k_{max}=1/a$. I showed this by solving the Schrodinger's ...
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496 views

Free path distribution

I'm studying statistical mechanics, and I'm trying to resolve some problem known from my thermodynamics course. So I want to calculate mean free path for particles with a concentration $n$ and ...
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278 views

Laplacian and Dirac Delta function

Although I find it mathematically dubious, we said that $$\Delta \frac{1}{r} ~=~ -4\pi \delta^3({\bf r}).$$ Now, I was wondering is there a similar relation to the delta function if we look at $$\...
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1answer
127 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=\...
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219 views

Sommerfeld expansion for temperature dependent spectral function

Consider the integral \begin{align} I(\beta)=\int_{-\infty}^{\infty}\frac{\text{d}\epsilon}{\pi}\frac{\rho(\epsilon,\beta)}{1+e^{\beta \epsilon}} \end{align} where $\rho(\epsilon,\beta)$ is a generic ...
2
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1answer
107 views

Delta Function of a Curve

I need to evaluate the integral \begin{equation} \int_0^1\mathrm dt\,f\left(t\right)\delta^{\left(3\right)}\left(\vec r\left(t\right)-\vec r_0\right) \end{equation} where there is only one $0\leq t_0\...
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43 views

Expression for the (1+1)-dimensional retarded Dirac propagator in position space

Where an expression for the (1+1)-dimensional retarded Dirac propagator in position space can be found, especially including the generalized funcion supported on the light-cone? In particular, is it ...
2
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1answer
102 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{...
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2answers
217 views

Resonance propagator properties

(This is part c & d of problem 7.9 from Schwartz' book on QFT). Show that a propagator only has an imaginary part if it goes on-shell. Explicitly, show that $$Im(M)=-\pi\delta(p^2-m^2)$$ when $$...
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101 views

How can the propagator be written in the below integral form?

I’am finding it difficult to understand as to how the delta function is written as a product of many delta functions with the integral
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115 views

Half of Dirac Delta within spectral integration

I was taking a course in Quantum Information along this last semester, and apart from some mathematical details it all made sense to me. One of these mathematical tricks that I haven't been able to ...
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183 views

Classical field theory: marriage of differential geometry to functional analysis?

I have always wondered (and thoroughly - but perhaps not enough - searched for literature) how the purely geometrical formulation of classical field theory (take for example a hardcore text on this ...
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1answer
58 views

Why is it physically resonable to consider a function as a regular distribution?

In "Mathematical Methods in Physics" by Blanchard and Brüning the introduction of regular distributions and then general distributions is motivated by the following idea: When we think about the ...
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116 views

Under what circumstances is the functional derivative (of an action functional) an actual function?

In general, for a functional $F[\phi]$, the functional derivative is $$\frac{\delta F[\phi]}{\delta \phi} [f(x)] = \lim_{\varepsilon \to 0} \frac{F[\phi + \varepsilon f ] - F[\phi]}{\varepsilon}$$ ...
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76 views

Are vacuum expectation values distributions?

In PCT, Spin, Statistics and all That (1964) on page 106 the vacuum expectation value is introduced in the following way. The question concerns the highlighted area, that is how can a tempered ...
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824 views

Integration by parts with Dirac Delta function

I am having some hard time trying to understand the following "heuristic" integral, involving integration by parts with the Dirac's Delta. We start with the following relation $$ f(x) = \int_{-\infty}^...
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1k views

How to calculate the second functional derivative of the action of a one-particle system?

Given the Lagrangian $$L(q,\dot{q})=m\dot{q}^2/2-V(q)$$ and the corresponding action $$S[q]\equiv\int_0^t dt' (m\dot{q}^2/2-V(q)),$$ I need to be able to evaluate the second functional derivative $\...
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79 views

energy distribution of electrons from the heated cathode in magnetic field

I have a very specific question which is troubling me. I use a heated disk cathode as an electron emitter. I know that the energy distribution of the electrons emitting from the cathode is $g(E)=\...
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101 views

Poisson-like green functions

How can I verify that equation $$\nabla ^2 f (r) = - \frac{e}{4 \pi \epsilon ^2} \delta (r-\epsilon)$$ in 3D has a solution of the form $$f (r) = a - \frac{e}{4 \pi r} \theta (r-\epsilon) -\frac{e}{...
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285 views

Calculation of the Poisson bracket of a (Classical) Yang-Mills generator

This question might be too technical or minute, but I believe someone can give me the right advise. What I want to calculate is a Poisson bracket algebra of classical YM gauge generators, \begin{...
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255 views

Finding the moments of the Boltzmann/Gibbs Distribution

I am trying to compute the moments of the Boltzmann distribution using a moment generating function, by taking the Fourier transform of the distribution and then taking derivatives to find the ...
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17 views

How to interpret the cross-spectral density of an incoherent field

I have been given a definition of the cross spectral density of a completely incoherent field: $$W(x_1,x_2)=S_0\delta(x_1-x_2)$$ How do I interpret this? As I understand it, this means that there ...
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50 views

Propagators and Green functions for general fields

In my QFT class we have defined the Feynman propagator of a field $\phi^r$ (where $r$ could be a vector or spinor index, or even a multiindex if $\phi$ is a tensor field etc.) as $$ \Delta^{rs}_F(x - ...
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21 views

Delta Potential Boundary Conditions on the wavefunction

I'm reading over how the delta function potential problems are solved and I can't really understand the origin of these boundary conditions: $(1) \,\,\psi \,\,$ is always continuous $(2) \,\, \dfrac{...
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60 views

Interpretation of induced force between two Dirac delta potential wells

My question 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 ...
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2answers
277 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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160 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{\...
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149 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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269 views

How can a Dirac delta function that does not occur under an integral be used to describe a transition rate?

In his excellent notes (found here), Mark Tuckerman shows that the transition rate of absorption between quantum states i and f, coupled by operator B, can be expressed as the fourier transform of the ...
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103 views

is it true that $\lim\limits_{\epsilon \to 0^{+}} \ln( x \pm i \epsilon ) = \mathscr{P}\ln|x| \pm i \pi \Theta(-x)$?

Is the following statement true? $$ \lim\limits_{\epsilon \to 0^{+}} \ln( x \pm i \epsilon ) = \mathscr{P}\ln|x| \pm i \pi \Theta(-x) $$ where $\mathscr{P}$ is the Cauchy principal value. The above ...
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583 views

Relationship between Green's function and impulse response

In my field, electrical engineering, we frequently study linear time-invariant systems of the following form: $$ a_n\frac{d^ny}{dt^n} + a_{n-1}\frac{d^{n-1}y}{dt^{n-1}} + \ldots + a_1\frac{dy}{dt} + ...
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344 views

Density of a thin disk by delta Dirac function

How to write a volume (charge for example) density using delta Dirac function in spherical coordinates in this case: There are thin charged disk (radius $a$) with surface charge density $\sigma$. The ...
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136 views

Can Maxwell's Equations in differential form be viewed as equalities of measures?

What I have in mind is the following - Suppose we choose to model the universe as a 3 dimensional flat Euclidean space $\mathbb{R}^3$ equipped with the standard topology and the Borel-sigma algebra. ...
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179 views

Counting experiment and standard error

I have a counting experiment: Let's say I have N identical bees. I take one of them, expose it to $\gamma$-radiation and look if it has died or survives. If it survives, I count it as 1 count. If it ...
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109 views

Second-order functional dervative of the Yang-Mills action by DeWitt

DeWitt calculated a second order functional dervative of a Yang-Mills action in Table I p. 1201 in his paper PR 162 (1967) 1195. The action is the usual one, $$S=-(1/4)\int d^4x F_{a\mu\nu}F^{a\mu\nu}...
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2k views

Green's function for the free particle Hamiltonian

For a free particle in 1D, the equation for its Green function (in atomic units) is $$\left(E+\frac{1}{2}\frac{d^2}{dx^2}\right)G(x,x',E) = \delta(x-x')$$ The textbook I'm following says the solution ...
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29 views

Derivation of generation of time between two subsequent particle enters into computational domain

I have problem with understanding derivation of one equation in following problem. You have 1D computational domain (it is not 1D but because it is symmetrical and we are watching only radial ...
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248 views

Notation - d.o.f.'s for Grassmann delta functions in a SUSY field theory amplitude

I was reading the following paper http://arxiv.org/pdf/1306.2962v1.pdf as I stumbled upon an issue concerning counting and assigning the Grassmann degrees of freedom that appear in grassmann delta ...
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2k views

Delta correlated white noise

I am studying Brownian motion, specifically Langevin equation. This equation includes a force expressed by a white noise, say $\xi(t)$. One of the hypothesis is that it is $\delta$-correlated (since ...
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201 views

Particle in a higher-dimensional box with an attractive delta potential

Suppose you have a particle in the box $[0,L]^d$, with an attractive Dirac delta potential $-\delta_{\vec w}(x)$ at $\vec w$. How do you solve the Schroedinger equation for this system? In the case $...
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59 views

Fourier transform and commutation of integral and laplacian

While determining the Breit-Fermi potential, I carry out a Fourier transform by using the following identity: $$\int\frac{d^3q}{(2\pi)^3}e^{-i\vec{q}\cdot\vec{r}}\frac{1}{|\vec{q}|^2}=\frac{1}{4\pi r}$...
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1answer
51 views

Scattering cross section from sum of delta functions in 3D

we had the following question in our exam: I know basic scattering concepts like partial waves, born approximation etc. and the solution of common potentials like coulomb or hard spheres but have ...
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42 views

Integration by parts with Dirac delta function in deriving the Lienard-Wiechert fields

Lienard-Wiechert fields can be derived by directly differentiating the Lienard-Wiechert potentials. But for convenience many textbook authors choose to differentiate under the integration sign of the ...
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54 views

Proof of commutator in Tong's notes on QFT

I am following David Tong's notes on QFT: http://www.damtp.cam.ac.uk/user/tong/qft.html . In equation 2.21, he tries to prove $$[\phi(\vec{x}),\pi(\vec{y})] = i\delta^{(3)}(\vec{x}-\vec{y}).$$ Here, $\...
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1answer
23 views

quantum work distribution from sudden quench

I'm trying to calculate the quantum work distribution P(W) from a sudden quench, which the expression is $P(W) = \sum_{n,m}p_n^0p_{m|n}^\tau \delta[W - (\epsilon_m^\tau - \epsilon_n^0)],$ where $p_n^...
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64 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 ...
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119 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 ...
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62 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. ...
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30 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 ...