What is Dirac delta function: Definition and 197 Discussions
In mathematics, the Dirac delta function (δ function) is a generalized function or distribution, a function on the space of test functions. It was introduced by physicist Paul Dirac. It is called a function, although it is not a function R → C.
It is used to model the density of an idealized point mass or point charge as a function equal to zero everywhere except for zero and whose integral over the entire real line is equal to one. No function has these properties, such that the computations made by theoretical physicists appeared to mathematicians as nonsense until the introduction of distributions by Laurent Schwartz to formalize and validate the computations. As a distribution, the Dirac delta function is a linear functional that maps every function to its value at zero. The Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1, is a discrete analog of the Dirac delta function.
In engineering and signal processing, the delta function, also known as the unit impulse symbol, may be regarded through its Laplace transform, as coming from the boundary values of a complex analytic function of a complex variable. The convolution of a (theoretical) signal with a Dirac delta can be thought of as a stimulation that includes all frequencies. This leads to a resonance with the signal, making the theoretical signal "real" (i.e. causal). The formal rules obeyed by this function are part of the operational calculus, a standard tool kit of physics and engineering. In many applications, the Dirac delta is regarded as a kind of limit (a weak limit) of a sequence of functions having a tall spike at the origin (in theory of distributions, this is a true limit). The approximating functions of the sequence are thus "approximate" or "nascent" delta functions.
I need help to understand how equation (27) in this paper has been derived.
The definition of P(k) (I discarded in the question ##\eta## or the integration with respect for it) is given by (26) and the definition of h(k) and G(k) are given by Eq. (25) and Eq. (24) respectively.
In my...
Hi,
I am not sure if I have solved the following task correctly
I did the plotting in mathematica and got the following
Would the plots be correct? What is meant by check for normalization, is the following meant?
For the approximation for ##\epsilon > 0##, does it mean that for the...
I have read lots but still, there're some really unproductive explanations of dirac delta function. So hopefully, you can explain it by following my arguments and not formal definition because I've read it all.
It's shown to be as ##\delta (x) = 0## when ##x \neq 0## and ##\delta (x) =...
In 3 d spherical coordinates we know that
$$\triangledown \cdot \frac{\hat{\textbf{r}}}{r^2}=4π\delta^3(\textbf{r})$$
Integration over all## R^3## is 4π
So when we remove the third dimensions and enter 2d polar coordinates then
$$\triangledown \cdot...
My goal is to develop the equation 21. You should asume that \delta(r_2-r_1)^2 =0. These is named renormalization. Then my question is , do my computes are correct with previous condition ?
Hi,
I found Laplace transform of this Dirac delta function which is ##F(s) = e^{-st}## since ##\int_{\infty}^{-\infty} f(t) \delta (t-a) dt = f(a)##
and that ##\delta(x) = 0## if ##x \neq 0##
Then the Mellin transform
##f(t) = \frac{1}{2 \pi i} \int_{\gamma - i \omega}^{\gamma +i \omega}...
I came across it in the derivation of Gauss' law of electric flux from Coulomb's law. I did some research on it, but the wikipedia page about it was slightly confusing. All I know about it is that it models an instantaneous surge by a distribution. However I am still perplexed by this concept...
Hey guys! Sorry if this is a stupid question but I'm having some trouble to express this charge distribution as dirac delta functions.
I know that the charge distribution of a circular disc in the ##x-y##-plane with radius ##a## and charge ##q## is given by $$\rho(r,\theta)=qC_a...
hi, there. I am doing some frequency analysis. Suppose I have a function defined in frequency space $$N(k)=\frac {-1} {|k|} e^{-c|k|}$$ where ##c## is some very large positive number, and another function in frequency space ##P(k)##. Now I need integrate them as $$ \int \frac {dk}{2 \pi} N(k)...
Hi,
I have a quick question about something which I have read regarding the use of dirac delta functions to represent conditional pdfs. I have heard the word 'mask' thrown around, but I am not sure whether that is related or not.
The source I am reading from states:
p(x) = \lim_{\sigma \to...
My first attempt revolved mostly around the solution method shown in this "site" or PowerPoint: http://physics.gmu.edu/~joe/PHYS685/Topic4.pdf .
However, after studying the content and writing down my answer for the monopole moment as equal to ##\sqrt{\frac{1}{4 \pi}} \rho##, I found out the...
As a part of a bigger problem, I was trying to evaluate the D'Alambertian of ##\epsilon(t)\delta(t^2-x^2-y^2-z^2)##, where ##\epsilon(t)## is a sign function. This term appears in covariant commutator function, so I was checking whether I can prove it solves Klein-Gordon equation. Since there's...
1.) Laplace transform of differential equation, where L is the Laplace transform of y:
s2L - sy(0) - y'(0) + 9L = -3e-πs/2
= s2L - s+ 9L = -3e-πs/2
2.) Solve for L
L = (-3e-πs/2 + s) / (s2 + 9)
3.) Solve for y by performing the inverse Laplace on L
Decompose L into 2 parts:
L =...
If the question was
$$ \int_{∞}^{∞}dxf(x)δ((x - x_1)) = ? $$ The answer would be ##f(x_1)##
So the delta function has two roots, I searched the web and some books but I am not sure what approach should I use here. I guess there's sometihng happens when ##x_1 = -x_2##.
So I am not sure what...
I have to integrate this expression so I started to solve the delta part from the fact that when n=0 it results equals to 1.
And the graph is continuous in segments I thought as the sumation of integers
$$ \int_{-(n+1/2)π}^{(n+1/2)π} δ(sin(x)) \, dx $$
From the fact that actually
$$ δ(sin(x))=...
Form solid state physics, we know that the volume of k-space per allowed k-value is ##\triangle{\mathbf{k}}=\dfrac{8\pi^3}{V}##
##\sum_{\mathbf{k}}F(\mathbf{k})=\dfrac{V}{(2\pi)^3}\sum_{\mathbf{k}}F(\mathbf{k})\triangle{\mathbf{k}}##...
Homework Statement
Consider a particle of mass m moving in a one-dimensional double well potential
$$V(x) = -g\delta(x-a)-g\delta(x+a), g > 0$$
This is an attractive potential with ##\delta##-function dips at x=##\pm a##.
In the limit of large ##\lambda##, find a approximate formula for the...
I've been thinking about this problem and would like some clarification regarding the value of the divergence at a theoretical point charge.
My logic so far:
Because the integral over all space(in spherical coordinates) around the point charge is finite(4pi), then the divergence at r=0 must be...
Hi. I was trying to test a code for the diffusion equation, using the analytical solution for infinite media, with a Dirac delta source term:
##q(\mathbf{r},t)=\delta (\mathbf{r}) \delta (t)##.
The code is not giving the analytical solution, and there might be several reasons why this is so...
Hi,
I am reading the Quantum Mechanics, 2nd edition by Bransden and Joachain. On page 777, the book gives an example of Dirac delta function.
$\delta_\epsilon (x) = \frac{\epsilon}{\pi(x^2 + \epsilon^2)}$
I am wondering how I can show $\lim_{x\to 0+} \int_{a}^{b} \delta_\epsilon (x) dx$...
Hi, if I have an interval on the x-axis, defined by the parameter L, can this, interval be transformed to a Dirac delta function instead, on the x-axis?
Thanks!
If I have a general (not a plain wave) state $$|\psi\rangle$$, then in position space :
$$\langle \psi|\psi\rangle = \int^{\infty}_{-\infty}\psi^*(x)\psi(x)dx$$
is the total probability (total absolute, assuming the wave function is normalized)
So if the above is correct, does that mean...
Homework Statement
Show that this is a valid representation of the Dirac Delta function, where ε is positive and real:
\delta(x) = \frac{1}{\pi}\lim_{ε \rightarrow 0}\frac{ε}{x^2+ε^2}
Homework Equations
https://en.wikipedia.org/wiki/Dirac_delta_function
The Attempt at a Solution
I just...
Is the square of a Dirac delta function, ##(\delta(x))^2##, still a Dirac delta function, ##\delta(x)##?
A Dirac delta function peaks at one value of ##x##, say 0. If it is squared, it still peaks at the same value, so it seems like the squared Dirac delta function is still a Dirac delta...
I just want to make sure that I am understanding the Dirac Delta function properly. Is the following correct?:
For two variables ##x## and ##y##:
\begin{equation}
\begin{split}
\delta(x-y) f(x) &= f(y)
\end{split}
\end{equation}
And:
\begin{equation}
\begin{split}
\delta(x-x) f(x) &=...
Homework Statement
\begin{equation}
\int_V (r^2 - \vec{2r} \cdot \vec{r}') \ \delta^3(\vec{r} - \vec{r}') d\tau
\end{equation}
where:
\begin{equation}
\vec{r}' = 3\hat{x} + 2\hat{y} + \hat{z}
\end{equation}
Where d $\tau$ is the volume element, and V is a solid sphere with radius 4, centered...
Homework Statement
I have the second order diff eq:
Solving by Laplace transform gets me to:
I could use the inverse laplace transform that takes me back to e^{at}cos(bt) with b=0, but that only solves for the homogeneous (complementary) part of the equation, it won't reproduce the dirac...
Homework Statement
hi
i have to find the result of
##\int_{0}^{\pi}[\delta(cos\theta-1)+ \delta(cos\theta+1)]sin\theta d\theta##
Homework Equations
i know from Dirac Delta Function that
##\int \delta(x-a)dx=1##
if the region includes x=a and zero otherwise
The Attempt at a Solution
first i...
Hello,
I'm stuck with this exercise, so I hope anyone can help me.
It is to prove, that the density of states of an unknown, quantum mechanical Hamiltonian ##\mathcal{H}##, which is defined by
$$\Omega(E)=\mathrm{Tr}\left[\delta(E1\!\!1-\boldsymbol{H})\right]$$
is also representable as...
Homework Statement
Differential equation: ##Ay''+By'+Cy=f(t)## with ##y_{0}=y'_{0}=0##
Write the solution as a convolution (##a \neq b##). Let ##f(t)= n## for ##t_{0} < t < t_{0}+\frac{1}{n}##. Find y and then let ##n \rightarrow \infty##.
Then solve the differential equation with...
Homework Statement
Find the Fourier spectrum of the following equation
Homework Equations
##F(\omega)=\pi[\delta(\omega - \omega _0)+\delta(\omega +\omega_0)]##
The Attempt at a Solution
Would the Fourier spectrum just be two spikes at ##+\omega _0## and ##-\omega _0## which go up to infinity?
Given the definition:
δ(x) = 0 for all x ≠ 0
∞ for x = 0
∫-∞∞δ(x)dx = 1
I don't understand how the integral can equal unity. The integral from -∞ to zero is zero, and the integral from 0 to ∞...
Homework Statement
I am trying to determine whether
$$f(x)g(x')\delta (x-x') = f(x)g(x)\delta (x-x') = f(x')g(x')\delta(x-x')$$
where \delta(x-x') is the Dirac delta function and f,g are some arbitrary (reasonably nice?) functions.
Homework Equations
The defining equation of a delta function...
Homework Statement
Given the following wave function valid over -a \le x \le a and which is 0 elsewhere,
\psi(x) = 1/\sqrt{2a}
Find the uncertainty in \left<\left(\Delta p\right)^2\right> momentum, and the uncertainty product \left<\left(\Delta x\right)^2\right>\left<\left(\Delta...
Homework Statement
For each of these sketch and provide a formula for the function (i.e. in terms of ##u(t)##, ##\delta(t)##) and its derivative and anti-derivative. Denote the ##\delta## function with a vertical arrow of length 1.
(a) ##f(t)=\frac{|t|}{t}##
(b) ##f(t)=u(t) exp(-t)##...
Homework Statement
Find the solution to:
$$\frac{d^2}{dt^2} x + \omega^2 x = \delta (t)$$
Given the initial condition that ##x=0## for ##t<0##. First find the general solution to ##t>0## and ##t<0##.
Homework Equations
The Attempt at a Solution
This looks like a non-homogeneous second...
Homework Statement
Find the scalar product of diracs delta function ##\delta(\bar{x})## and the bessel function ##J_0## in polar coordinates. I need to do this since I want the orthogonal projection of some function onto the Bessel function and this is a key step towards that solution. I only...
The following integral arises in the calculation of the new density of a non-uniform elastic medium under stress:
∫dx ρ(r,θ)δ(x+u(x)-x')
where ρ is a known mass density and u = ru_r+θu_θ a known vector function of spherical coordinates (r,θ) (no azimuthal dependence). How should the Dirac...
I am currently reading Modern Electrodynamics by Andrew Zangwill and came across a section listing some delta function identities (Section 1.5.5 page 15 equation 1.122 for those interested), and there is one identity that really confused me. He states:
\begin{align*}
\frac{\partial}{\partial...
Homework Statement
Compute the average value of the function:
f(x) = δ(x-1)*16x2sin(πx/2)*eiπx/(1+x)(2-x)
over the interval x ∈ [0, 8]. Note that δ(x) is the Dirac δ-function, and exp(iπ) = −1.
Homework Equations
∫ dx δ(x-y) f(x) = f(y)
The Attempt at a Solution
Average of f(x) = 1/8 ∫from...
Hi - firstly should I be concerned that the dirac function is NOT periodic?
Either way the problem says expand $\delta(x-t)$ as a Fourier series...
I tried $\delta(x-t) = 1, x=t; \delta(x-t) =0, x \ne t , -\pi \le t \le \pi$ ... ('1' still delivers the value of a multiplied function at t)...
Ok so for equations of spherical wave in fluid the point source is modeled as a body force term which is given by time dependent 3 dimensional dirac delta function f=f(t)δ(x-y) x and y are vectors.
so we reach an equation with ∫f(t)δ(x-y)dV(x) over the volume V. In the textbook it then says that...