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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24 views

What is a 'delta function optical pulse'?

What is a 'delta function optical pulse'? I suspect this means that the optical pulse can be modelled as having the properties of the Dirac delta function, but then what does it mean to say that an ...
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Why do we need large time assumption for energy conservation in electron transitions?

For electron absorption calculations (with an electric field perturbation $\Delta H = eE_0x cos(\omega t)$) we end up with an integral like: $$c_2(t) \propto \int \rho(\omega) \left( \frac{\sin(\frac{...
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Proof that infinite square well gets split into two by a Dirac delta function in the middle

I was reading these lecture notes https://faculty.biu.ac.il/~barkaie/TA6.pdf , and at the end, they claim that $\tan{kL}=0$ implying $kL=n\pi$, thus resulting in the same condition as before, means ...
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D'Alembertian of a delta-function of a space-time interval (i.e. on the light-cone)

How one differentiates a delta-function of a space-time interval? Namely, $$[\partial_t^2 - \partial_x^2 - \partial_y^2 - \partial_z^2] \, \delta(t^2-x^2-y^2-z^2) \, .$$ Somewhere I saw that the ...
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How to realize Heaviside $\theta(t-t')$ and Dirac $\delta(t-t')$ as matrices in numerics?

As is well known, single-particle Green's functions in the time domain might involve $$\theta(t-t')$$ for the retarded and advanced Green's functions. Sometimes, we also need $$\delta(t-t')$$ to ...
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Simplest possible solution to the Klein-Gordon field equation has a (KG) norm which is not constant in time

It is a fact that the Klein-Gordon inner product must be constant for all $t>0$, where the Klein-Gordon product is defined by $$ \langle f, g \rangle \ := \ i \int d^3x \; \left[ f^{\ast}(t,\mathbf{...
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Is there a way to see linear and surface charge density as a “special case” of volume charge density?

When deriving Gauss’s law in differential form (GLDF), $$\nabla \cdot \mathbf E = \frac{\rho}{\epsilon_0},$$ from Gauss’s law in integral form (GLIF) we get a tidier formula, which is however less ...
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How to solve double delta potential bound states by “brute force”

I just solved a problem in Griffiths' Intro to QM, where one had to find the bound states given the potential: $$V(x)=-\alpha [\delta (x-a)+\delta(x+a)]$$ In order to solve it, one had to exploit the ...
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Rigorous delta potential – a formulation using distributions?

It is common in QM, even in mathematical physics, to consider Hamiltonians with a Dirac delta in the potential: $$ \hat H = -\frac{\mathrm{d}^2}{\mathrm{d}x^2} + V(x) + \delta(x-a) \: . $$ The most ...
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The gradient of the d'Alembertian Green's function

So I have to prove that the d'alembertian of the associated green's function $G(t,t',\vec{r},\vec{r}')$ is equal to zero when given that $\vec{r}\neq\vec{r}'$ $$\left(\frac{1}{c^2}\partial^2 t-\Delta\...
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Are these bra-ket manipulations correct?

I'm reading an old paper ("Wigner's Function and Other Distributions in Mock Phase Spaces," Balazs and Jennings, Phys. Rep. 104(6), 1984), and came across the following statement (in which $\...
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Question regarding a special identity for $2\pi\delta(E-\epsilon_\alpha)$

I am reading Datta's book about Quantum Transport at the moment and I stumbled over an identity for the Dirac delta distribution, which is correct since it fullfils all the requirements for the Dirac ...
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Question on how to actually use momentum space Feynman rules in $\phi^4$-theory

The momentum space Feynman rules state that we "integrate over all undetermined momenta" and "impose momentum conservation at each vertex". This is given for example on page 95 of ...
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Alternative representations of the momentum operator in position space

The fundamental relation between the position and momentum basis in quantum mechanics is summed up in the canonical commutation relation: $[x,p]=i\hbar.I$ From here, one can get to the matrix elements ...
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Potential well with 2 delta potentials

I have a problem to derive transcendental equation for eigenenergies of bounded states for potential: $$ V(x)= \begin{cases} -\lambda\delta(x+a/4)-\lambda\delta(a-a/4), & |x|<a/2 \\ ...
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Can the vector potential be written as $\mathbf A = \nabla \chi$ for some singular function $\chi$?

Consider a magnetic field $\mathbf B= \Phi\delta(x) \delta(y) \hat z$. The corresponding vector potential becomes $\mathbf A = \frac{\Phi}{2\pi r} \hat\theta$ in the cylindrical coordinates. ...
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Dirac delta in 4 dimension - Continuity equation and energy momentum tensor

I had an exercise (Carroll Chapter 4, exercise 3) where I was asked to prove that writing the energy-momentum tensor for a single particle as a dust fluid, meaning $$ T^{\mu \nu}\left(y^{\sigma}\right)...
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Why wave function does not vanish at a Dirac delta potential?

I have studied that a wave function should vanish at the location of an infinite potential. Consider a direct Delta delta potential at $x=0$. Why does does function not become zero here at $x=0$?
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Influence of delta potential (particle in a box) [duplicate]

I have a question about the problem with a particle in a box and an additional delta potential: $V(x)=g\delta(x)+V_B(x)$ with $V_B(x)=0$ for $|x|\leq L$ and $V_B(x)=\infty$ elsewhere. From this I want ...
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Is there a sense in which one can interpret $\delta'(0) = 0$?

I wasn't sure whether to post this under physics or math (and landed on physics due to fear of being crucified for lack of rigor on math.stackexchange). In field theory, when we encounter divergences ...
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Plane wave basis in a box. Inconsistent terminology about Fourier series and transform

I will refer to the one-dimensional equivalent of the problem in the picture for simplicity. At the end of the picture, the author argues that the Fourier transform of the function $\psi(x)$ is the ...
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$\frac{0}{0}$ from Curvilinear Dirac Delta

The definition of the Dirac Delta in an arbitrary curvilinear coordinate: $$\delta(\vec{r})=\frac{\delta(x^1-x^1_0)\delta(x^2-x^2_0)\cdot \cdot\cdot \delta(x^N-x^N_0)}{h_1h_2\cdot\cdot\cdot h_N}$$ ...
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Is the Dirac $\delta$-function necessarily symmetric?

The Dirac $\delta$-function is defined as a distribution that satisfies these constraints: $$ \delta (x-x') = 0 \quad\text{if}\quad x \neq x' \quad\quad\text{and}\quad\quad \delta (x-x') = \infty \...
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2answers
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Dirac-delta-functions as eigenbasis of the position operator - pure nonsense? Or can more be said?

I remember overthinking equations like \begin{equation} \mathbf{1}=\int dx\ |x\rangle\langle x|\tag{1} \end{equation} and \begin{equation} X=\int dx\ |x\rangle\langle x|x\tag{2} \end{equation} when I ...
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1answer
38 views

What is the correct cosine-like integral representation of Dirac delta?

exploring the integral representations of the Dirac delta I found this in terms of an integral of cosine function (from wolfram's database, https://functions.wolfram.com/GeneralizedFunctions/...
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Dirac delta in spherical coordinates. What I'm doing wrong?

I must show that the integral $$\frac{1}{(2\pi)^{3}}\int_{\vec{k}}d^{3}k\frac{\cos(\vec{k}\cdot\vec{x})}{\left({\sqrt{|\vec{k}|^2+m^{2}}}\right)^{s}}=\delta^{3}(\vec{x})$$ when $s=0$ by using ...
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Why does this simple integral in electromagnetism involving the delta function seem to both vanish and not vanish depending on how you evaluate it?

This relates to David Tong's notes on electromagnetism, specifically page 18. We have as our charge density a delta function: $$\rho(\vec x)=Q\delta^{(3)}(\vec x).$$ We want to solve: $$\nabla^2\phi=-\...
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Quantum expressions involving Dirac delta function

I want to find the following quantum expression: $$ \langle x|PX|x'\rangle.$$ A. If I use $X|x'\rangle = x'|x'\rangle $, I will get: $$ \langle x|PX|x'\rangle = \langle x|Px'|x'\rangle = x'\langle x|...
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Understanding Maxwell's Equations in a Box

Maxwell's equations for $n$ charged particles each with charge $e_j$ are known to be (in cgs) $$\begin{align} \nabla\cdot\textbf{E}(t,x)&=4\pi\rho(t,x)\\ \nabla\cdot\textbf{B}(t,x)&=0\\ \...
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1answer
70 views

Approximation for the square of a Dirac delta function

I am working in the appendix to Section II.6 in Zee's QFT book, 2nd Ed. I am trying to compute the cross section for a meson to decay to two mesons as $\varphi\to\eta+\xi$ with three respective ...
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1answer
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Using Gauss' theorem [duplicate]

I am working through some questions and I am stuck on the working of this one: Where the working to the answer is here: Can someone explain the highlighted section as I can't see where it comes from?...
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Evaluating general solution of Dirac equation

In Ashok Das Lecture on QFT book, pg. 40, the solution of the Dirac equation for the general motion of a free particle with mass $m$ along an arbitrary direction is given by $$\psi (x)=\int d^4p \ a(...
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Completeness relation of spherical harmonics

In spherical coordinates, the resolution of the identity can be written as $$ 1=\int_0^{2\pi}d\phi\int_0^{\pi}\sin\theta\, d\theta\, |\theta,\phi\rangle\langle\theta,\phi| \equiv \int d\Omega |\Omega\...
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Bound States in Dirac Delta Potential [closed]

Consider a particle of mass m in the potential: $$V(x)=\left\{\begin{array}{rll}\infty & \text { for } & x \leq 0 \\ -V_{0} \delta(x-a) & \text { for } & x>0\end{array}\right.$$ ...
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What does happen if we use a delta function for density on the De Broglie–Bohm theory?

I was reading Pilot wave theory and De Broglie–Bohm theory pages on Wikipedia that I found how similar they are comparing to classical physics and I wondered what happens if we just replace the ...
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Dirac delta and quantum mechanics

I want to know why is this equality true: $$P(o)P(o') = \delta(o-o')P(o)$$ Where P is the projector operator I could see that: $$P(o)P(o') = |o\rangle\langle o|o'\rangle\langle o' | = |o\rangle\delta^{...
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Two-Body Decay Conservation of Energy

I was trying to derive transition rate for a two-body decay process. In one of the reference I'm following, it consider $a\rightarrow1+2$ decay, and said the daughter particles in center-of-mass ...
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1answer
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$N$-body phase space for Fermi golden rule

I was following along Mark Thomson's Modern Particle Physics, and stumble upe the derivation of d$n$ of Fermi golden rule on page 62: "... For the decay of a particle to a final state consisting ...
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Dirac-delta function in Derivation of Fermi's Golden Rule

I was following along Mark Thomson's Modern Particle Physics, and got stuck on this book's derivation of Fermi's Golden Rule (On page 53): "... If there are d$n$ accessible final states in the ...
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Dirac Notation and Coordinate transformation of a function

Lets say we have a 1 dimensional system with coordinate $x$ and the associated operator $\hat x$ with eigenstates $|x\rangle$. A function of $x$ is defined as $$ f(x) = \langle x |f\rangle \tag{1} $$ ...
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The Dirac comb with one $\delta$-function removed

Has anyone ever encountered the Dirac comb/Shah function with one removed $\delta$-function, $$ V(x)=\frac{\hbar^2\kappa}{m}\sum_{n\neq0}\delta(x-an), $$ in any literature? I want to find the solution ...
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How to handle a delta-function divergence from an infinite-dimensional trace of a quantum operator with a general continuous parameter?

Say I am working in the standard Schrödinger-style Hilbert space that corresponds to square-integrable functions on 3-space. Some normal operator $\hat{A}$ therefore has an eigenbasis indexed by 3 ...
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1answer
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How to use principal value in propagator definition?

The propagator for a scalar particle can be written as $$ \frac{1}{x + i\epsilon} = {\rm PV}\left( \frac{1}{x} \right) - i\pi\delta(x), \quad x = p^2 - m^2, \tag{1} $$ where $p, m$ are the momentum ...
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Hilbert Space and distribution theory in QFT

When developing Quantum Field Theory, we usually refer to the Minkowski coordinates which cover the whole space in the case it is flat. I don't know how to set distribution theory and Hilbert space ...
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Is continuity of the wavefunction “put in by hand” for the Dirac delta potentials?

In 1d, for $V(x) = g\delta(x)$, integrating the TISE yields (assuming that $\psi$ is bounded$^\dagger$, so as to suppress the term containing $E$) $$ -\frac{\hbar^2}{2m} \left( \psi'(\varepsilon) - \...
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What is $\langle x_1 |\hat V(\hat x)| x_2 \rangle$?

So I've become rusty in Quantum Mechanics. What is $\langle x_1 |V(\hat x)| x_2 \rangle$? Where $V$ is the potential and $|x \rangle$ is the postion eigenket? $$ \langle x_1 |\hat V(\hat x)| x_2 \...
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Do the free particle energy eigenfunctions satisfy the closure relation?

The energy eigenfunction of the free particle ($V(x)=0$) are given by $\psi_{E, \pm}=A(E)e^{\pm ik_Ex}$ where $A(E)=\left( m/\left(8\pi^2\hbar^2 E\right) \right)$ and $k_E=\sqrt{2mE}/\hbar$ for each $...
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Is $E=0$ included in the energy spectrum of the free particle in 1d?

In finding the eigenfunctions, $\psi_E$'s, of the free-particle Hamiltonian in 1d, $$ H = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2}, $$ with eigenvalues $E$'s, subject to the conditions that they are ...
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4answers
362 views

Schwarzschild solution, stress-energy side of Einstein's equation

The Schwarzschild solution in GR only has a singularity at the origin $r=0$: otherwise there is no matter content. The right-hand side of Einstein's equation is hence almost everywhere zero, but I ...
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119 views

Finding the Fourier Transform of a Plane Wave

Hello I am trying to find the fourier transform of a plane wave of the form $$\psi(x) = \frac {1}{\sqrt{2\pi \hbar}}\exp\left(\frac {i}{\hbar}p_0 x\right)$$ where $p_0$ is fixed and Real I've worked ...

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