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Results tagged with dirac-delta-distributions
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user 227119
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.
16
votes
Physical meaning of the Jacobian in relation to Dirac delta function
We can handle easily integrals where the vector variable of integration, let $\:\mathbf{u}\:$, is the argument of the $\:\delta-$function, for example
\begin{align}
\iiint\d …
- 16.4k
3
votes
Accepted
Why does it follow that the Dirac delta function is a scalar "because determinant of the Lor...
So we want to prove that the expression $\,\delta^4\left[\boldsymbol{x-x}\left(\tau\right)\right]\,$ is a Lorentz invariant scalar, that is
\begin{align}
& \delta^4\left[\boldsymbol{x'}\boldsymbol{- …
- 16.4k
5
votes
Derivative of delta function
The Dirac $\;\delta\;$ function is even. This is more clear looking it as limit of proper functions.
While its 1rst derivative is an odd function.
- 16.4k
1
vote
Accepted
How to evaluate the coordinate transformation of Dirac delta function?
Of course we can evaluate the given integral without transforming the Dirac $\delta-$functions since :
\begin{equation}
\int\limits_{\boldsymbol{-\infty}}^{\boldsymbol{+\infty}}\!\!\delta\left(x_{_{\r …
- 16.4k
3
votes
Accepted
Greiner's Green's function for diffusion
Let a real function $\;f(x)\;$ of the real variable $\;x\in\mathbb{R}\;$ for which
\begin{align}
f(x)\boldsymbol{=}0 \quad & \text{for any} \quad x\boldsymbol{\ne} x_{0} \quad \textbf{and}
\tag{01a}\l …
- 16.4k
2
votes
The gradient of the d'Alembertian Green's function
We need frequently to differentiate expressions of Dirac $\delta$-function when the argument of the latter is a function $f\left(z\right)$ of the variable $z$ with respect to which we want to differen …
- 16.4k
3
votes
Transition probability derivation: How to prove $\lim_{\alpha\rightarrow\infty} \frac{\sin^2...
Consider a real function $\;f(x)\;$ of the real variable $\;x\in\mathbb{R}\;$ for which
\begin{align}
f(x)\boldsymbol{=}0 \quad & \text{for any} \quad x\boldsymbol{\ne} x_{0} \quad \textbf{and}
\tag{0 …
- 16.4k
4
votes
Accepted
Why is the propagator the Green's function for Schrodinger equation?
Hint : Check if this "modified" Schrodinger equation is satisfied by the "modified" propagator
\begin{equation}
\widetilde{K}(\mathbf{x''},t \; \boldsymbol{;} \;\mathbf{x'},t_{0})=\theta(t-t_{0})\;K …
- 16.4k
6
votes
Accepted
Normalization of eigenfunction to Dirac-delta function
For a function $f(x)$ the Fourier transform is defined as:
\begin{equation}
\overset{\boldsymbol{\sim}}{f}\left(k\right)=\dfrac{1}{\sqrt{2\pi}}\int\limits_{\boldsymbol{-}\infty}^{\boldsymbol{+}\infty} …
- 16.4k
1
vote
Accepted
Heaviside function in the form of an integral
$\newcommand{\bl}[1]{\boldsymbol{#1}}
\newcommand{\e}{\bl=}
\newcommand{\p}{\bl+}
\newcommand{\m}{\bl-}
\newcommand{\mr}[1]{\mathrm {#1}}
\newcommand{\gr}{\bl>}
\newcommand{\les}{\bl<}
\newcommand{\g …
- 16.4k
13
votes
Divergence of $\frac{ \hat {\bf r}}{r^2} \equiv \frac{{\bf r}}{r^3}$, what is the 'paradox'?
You must use the Dirac $\:\delta-$function and its properties.
The point charge $\:q\:$ being at rest at $\:\mathbf{r}_{0}\:$ we have
\begin{equation}
\mathbf{E}\left(\mathbf{r},t\right)\boldsymbol{= …
- 16.4k