# 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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### Mean value with delta function [closed]

How do I compute this matrix element $$\langle 1|\delta(\hat x-b)|1\rangle$$ that models a 1-D harmonic oscillator? I have done the same for the ground state (by seting delta function as the Fourier ...
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### $D$-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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### Delta function potential, bound state energies

A lot of books find the bound state energies of the single delta function potential centred at $x=0$ by integrating the Schrodinger equation around $x=0$, using symmetric limits and making the width ...
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### The charge density $\rho_{\infty}$ of the sphere

I have two charged hemispheres (which are very close to each other, we consider them now as a sphere) with the charge density given as $\rho = \frac {Q}{\pi a^3 \frac {4}{3+n}} (\frac{r}{a})^n$, ...
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### Heat equation: Heat Kernel as $t\to0$

Consider heat flow on an infinite, 1D wire. The temperature T(x,t) obeys the diffusion equation, $$\frac{∂T}{∂t} = D \frac{∂^2T}{∂x^2}$$ with initial condition $T(x,0) = δ(x)$. The heat kernel is ...
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### 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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### 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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### Units of a dirac delta function in quantum mechanics

In quantum mechanics the eigenfunctions of the position are dirac delta functions, $A\delta(x-x_0)$, where $A$ is some constant. Eigenfunctions of the position are usually normalized with a "Delta ...
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### Understanding the Formalism of wave function collapse [duplicate]

For example, a perfect measurement ($\hat A=\hat x$) on position: $$\hat x\psi=x_0\psi$$ The eigenvalue $x_0$ is the result of measurement. The eigenfunction $\psi$ is the wavefunction after ...
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### 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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### Normalizable free particle wave on 1D

I am trying to solve a free 1D particle non-relativistic Schrodinger equation with initial wavefunction $\psi(x,0)=\delta(x)$, where $$\delta(x)=\lim_{a\to0}(a/2)|x|^{(a-1)}.$$ Here is my approach: ...
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### 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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### 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 ...
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### Trouble with position operator in quantum mechanics

I'm having some trouble with understanding the derivation of the action of the $X$ operator. It seems to be a result of the notation used and not a property of itself. The usual argument is to ...
For a free particle, we can derive the following well-known relation: $$\langle k|k'\rangle = \frac{1}{2\pi} \int_{-\infty}^{+\infty} e^{-i(k-k')x} \, dx = \delta(k-k'). \tag{1.10.33}$$ Reference: ...