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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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### Hilbert space of harmonic oscillator: Countable vs uncountable?

Hm, this just occurred to me while answering another question: If I write the Hamiltonian for a harmonic oscillator as $$H = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 x^2$$ then wouldn't one set of ...
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### Don't understand the integral over the square of the Dirac delta function

In Griffiths' Intro to QM [1] he gives the eigenfunctions of the Hermitian operator $\hat{x}=x$ as being $$g_{\lambda}\left(x\right)~=~B_{\lambda}\delta\left(x-\lambda\right)$$ (cf. last formula on ...
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### Divergence of a field and its interpretation

The divergence of an electric field due to a point charge (according to Coulomb's law) is zero. In literature the divergence of a field indicates presence/absence of a sink/source for the field. ...
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### What is the square root of the Dirac Delta Function?

What is the square root of the Dirac Delta Function? Is it defined for functional integrals? Can it be used to describe quantum wave functions? \begin{align} \int_{-\infty}^{\infty} f(x)\sqrt{\delta(...
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### Dirac Delta and Sloppy Notation

I am an undergraduate neuroscientist and recently I have been studying probability distributions in relation to information theory, and came across the definition of the Dirac Delta as a singular ...
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### The Dirac-delta function as an initial state for the quantum free particle

I want to ask if it is reasonable that I use the Dirac-Delta function as an initial state ($\Psi (x,0)$) for the free particle wavefunction and interpret it such that I say that the particle is ...
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### Bound states of the $V(x)=\pm \delta'^{(n)}(x)$ potential?

The $\delta(x)$ Dirac delta is not the only "point-supported" potential that we can integrate; in principle all their derivatives $\delta', \delta'', ...$ exist also, do they? If yes, can we look for ...
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### What really is a Dirac delta function?

Yesterday a friend asked me what a Dirac delta function really is. I tried to explain it but eventually confused myself. It seems that a Dirac delta is defined as a function that satisfies these ...
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### Group theory, character, representations, delta function, unit element

I am reading a physics paper (hep-th/9204083 v1 above eq (4.15)) where there is a nice formula that reads $$\delta(U-\mathbb{1}) = \sum\limits_{r} \dim(r)\, \mathrm{tr}_r(U)$$ Here $\:U\:$ is a ...
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### How do you do an integral involving the derivative of a delta function?

I got an integral in solving Schrodinger equation with delta function potential. It looks like $$\int \frac{y(x)}{x} \frac{\mathrm{d}\delta(x-x_0)}{\mathrm{d}x}$$ I'm trying to solve this by ...
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### Transition probability derivation: How to prove $\lim_{\alpha\rightarrow\infty} \frac{\sin^2\alpha x}{\alpha x^2} ~=~\pi\delta(x)$?

How to prove $$\lim_{\alpha\rightarrow\infty} \frac{\sin^2\alpha x}{\alpha x^2} ~=~\pi\delta(x)~?$$ I have encountered this limit while learning time dependent perturbation and transition ...
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### Maxwell Equations don't give unique Electric Field?

Consider the class of electric fields given by $$\mathbf{E}=\begin{cases} \ln (Cr)\hat{z} & 0\leq r < R\\ 0 & r> R \end{cases}$$ where $C$ is a constant and $r$ is the polar-distance ...
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### Mathematical interpretation of Poisson Brackets

Lets say we are working in a classical scalar field theory and we have two functional $F[\phi, \pi](x)$ and $G[\phi, \pi](x)$. In most of the references, starting with two functional the Poisson ...
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### Particle density operator in second quantization form

The particle-density operator is given by $n(\mathbf{x})=\sum_{\alpha}\delta^{(3)}(\mathbf{x}-\mathbf{x}_{\alpha})$, then how to derive its representation in terms of creation and annihilation ...
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### George Green's definition of Green's function

This is a curious question about the way George Green could have defined his Green's function. All the definitions I see have only Dirac-delta $\delta(x−x′)$ function as their source on the RHS. But ...
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### If one puts a delta-function spike inside an infinite square well, is the resulting potential analytically solvable?

It was recently floated in chat that a particle in a box with a delta-function spike inside it, with hamiltonian $$H = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V_0\delta(x-a)$$ and with ...
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### Does regularity of distributions have anything to do with definiteness of their product?

Recently I've gone through some literature concerning causal perturbation theory (CPT). As is well known, it deals with UV divergences in QFT by defining products of (operator-valued) distributions ...
The density of states (DOS) is generally defined as $D(E)=\frac{d\Omega(E)}{dE}$, where $\Omega(E)$ is the number of states. But why DOS can also be defined using delta function, as $$D(E)~=~\sum\... 1answer 363 views ### Simple question regarding the Green's function for the diffusion equation The differential operator for diffusion in three dimensions is given by \partial_t - k \nabla^2 where k is a constant. The Green's function is (according to Wikipedia)$$\theta(t)\left( \frac{1}{4\...
In Quantum Mechanics, when using Dirac's formalism one of its features is the expansion of state vectors into continuous basis of eigenvectors of unbounded self-adjoint operators. Let $\mathcal{H}$ be ...