What is Delta function: Definition and 377 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.
Hi,
I'm not sure if I have calculated task b correctly, and unfortunately I don't know what to do with task c?
I solved task b as follows
##\displaystyle{\lim_{\epsilon \to 0}} \int_{- \infty}^{\infty} g^{\epsilon}(x) \phi(x)dx=\displaystyle{\lim_{\epsilon \to 0}} \int_{\infty}^{\epsilon}...
Sakurai, in ##\S## 5.7.3 Constant Perturbation mentions that the transition rate can be written in both ways:
$$w_{i \to [n]} = \frac{2 \pi}{\hbar} |V_{ni}|^2 \rho(E_n)$$
and
$$w_{i \to n} = \frac{2 \pi}{\hbar} |V_{ni}|^2 \delta(E_n - E_i)$$
where it must be understood that this expression is...
Hi,
First of all, I'm not sure to understand what he Kramers-kronig do exactly. It is used to get the Real part of a function using the imaginary part?
Then, when asked to add a peak to the parity at ##\omega = -\omega_0##, is ##Im[\epsilon_r(\omega)] = \delta(\omega^2 - \omega_0 ^2)## correct...
Hello,
I try to solve a time dependent problem described by a Hamiltonian of the type $$ \mathcal{H}(t) = H_0 + V \delta(t) .$$
I started by trying to solve the Schrödinger equation with ##H_0 = p^2 / 2m##, but I'm getting a bit stuck.
I would like to know if you know of any books that deal...
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)...
Summary:: I have a problem with a particle, which gets scatterd at a double delta-potential
Hello, I am really stuck with the floowing problem:
A particle moves from the left along the x-axis and gets scatterd at a one-dimensional potential V(x)=a[dirac delta of x) +b [dirac delta of x-c]...
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...
is it correct that the continuum states will be free particle states? and the probability will be |< Ψf | ΨB>|^2 . Where Ψf is the wave function for free particle and ΨB is the wave function for the bound state when the depth is B.
There are many representations of the delta function. Is there a place/reference that lists AND proves them? I am interested in proofs that would satisfy a physicist not a mathematician.
My intuition for this problem is to use divergence theorem:
## \int_V \nabla^2 u dV = \int_S \nabla u \cdot \vec{n} dS##
But note that ##\vec{n}## is perpendicular to x-y plane, and makes ##\int_S \nabla \ln s \cdot \vec{n} dS = 0##
If we take laplacian in polar coordinate directly, then...
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...
a.) The potential is a delta function, so ##V \left( r \right) = \frac {\hbar^2} {2\mu} \gamma \delta \left(r-a \right)##, therefore ##V \left( r \right) = \frac {\hbar^2} {2\mu} \gamma ## at ##r=a##, and ##V \left( r \right) = 0## otherwise. I've tried a few different approaches:
1.) In...
Hi,
I have pasted two improper integrals. The text has evaluated these integrals and come up with answers. I wanted to know how these integrals have been evaluated and what is the process to do so.
Integral 1
Now the 1st integral is again integrated
Now the text accompanying the integration...
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
$$ δ(sin(x))=...
Form solid state physics, we know that the volume of k-space per allowed k-value is ##\triangle{\mathbf{k}}=\dfrac{8\pi^3}{V}##
##\sum_{\mathbf{k}}F(\mathbf{k})=\dfrac{V}{(2\pi)^3}\sum_{\mathbf{k}}F(\mathbf{k})\triangle{\mathbf{k}}##...
Hi!
I have a code that solve the poisson equation for FEM in temperature problems.
I tested the code for temperature problems and it works!
Now i have to solve an Electrostatic problem.
There is the mesh of my problem (img 01).
At the left side of the mesh we have V=0 (potencial).
There is a...
Moved from technical math section, so is missing the homework template.
How to solve this equation please?
I found charakteristic roots ia ##\pm \sqrt{-a^{-k^2}}##. Thank you
Moderator note: Edited the LaTeX above to show the exponent correctly.
Homework Statement
Consider a particle of mass m moving in a one-dimensional double well potential
$$V(x) = -g\delta(x-a)-g\delta(x+a), g > 0$$
This is an attractive potential with ##\delta##-function dips at x=##\pm a##.
In the limit of large ##\lambda##, find a approximate formula for the...
My function:
d2f/dx2 + cf = delta(x)
Condition: f is finite and f(50) = f(-50) = 0
Solution: f = C1exp(cx) + C2exp(-cx)
Due to condition,
f = C1exp(cx) for x<=0 and C2exp(-cx) for x>=0
f(50) = C2exp(-c*50) = 0 = > C2 = 0
Likewise, for C1
I don't know if I might have missed something...
In these notes, https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2016/lecture-notes/MIT8_04S16_LecNotes10.pdf, at the end of page 4, it is mentioned:
(3) V(x) contains delta functions. In this case ψ'' also contains delta functions: it is proportional to the product of a...
Hi. I was trying to test a code for the diffusion equation, using the analytical solution for infinite media, with a Dirac delta source term:
##q(\mathbf{r},t)=\delta (\mathbf{r}) \delta (t)##.
The code is not giving the analytical solution, and there might be several reasons why this is so...
Hi,
I am reading the Quantum Mechanics, 2nd edition by Bransden and Joachain. On page 777, the book gives an example of Dirac delta function.
$\delta_\epsilon (x) = \frac{\epsilon}{\pi(x^2 + \epsilon^2)}$
I am wondering how I can show $\lim_{x\to 0+} \int_{a}^{b} \delta_\epsilon (x) dx$...
Hi, if I have an interval on the x-axis, defined by the parameter L, can this, interval be transformed to a Dirac delta function instead, on the x-axis?
Thanks!
If I have a general (not a plain wave) state $$|\psi\rangle$$, then in position space :
$$\langle \psi|\psi\rangle = \int^{\infty}_{-\infty}\psi^*(x)\psi(x)dx$$
is the total probability (total absolute, assuming the wave function is normalized)
So if the above is correct, does that mean...
Homework Statement
Evaluate the integral:
$$\int_{-\infty}^{\infty} dx *\dfrac {\delta (x^2-2ax)} {x+b}$$
Homework Equations
$$ x^2-2ax=0 $$
The Attempt at a Solution
I know that the delta function can only be none zero when $$ x=2a$$ so then I have the following integral...
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...
Homework Statement
Prove the following
'()( − ) = −′()
∫-∞∞δ'(x)*f(x-a) = -f'(a)
Homework Equations
∫-∞∞δ'(x-a)*f(x) = f(a)
The Attempt at a Solution
[/B]
∫-∞ ∞δ'(x)*f(x-a) = ∫δ(x)*f(x-a)dx-∫f'(x-a)*δ(x)dx = f(-a) - f'(-a)
Went from 1st to second by integration by parts
Used...
Homework Statement
Ultimately, I would like a expression that is the result of an integral with the sin(nx)/x function, with extra terms from the expansion. This expression would then reconstruct the delta function behaviour as n goes to infty, with the extra terms decaying to zero. I...
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
I have a 2D integral that contains a delta function:
##\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\exp{-((x_2-x_1)^2)+(a x_2^2+b x_1^2-c x_2+d x_1+e))}\delta(p x_1^2-q x_2^2) dx_1 dx_2##,
where ##x_1## and ##x_2## are variables, and a,b,c,d,e,p and q are some real...
Homework Statement
Homework EquationsThe Attempt at a Solution
So cts approx holds because ##\frac{E}{\bar{h}\omega}>>1##
So
##\sum\limits^{\infty}_{n=0}\delta(E-(n+1/2)\bar{h} \omega) \approx \int\limits^{\infty}_{0} dx \delta(E-(x+1/2)\bar{h}\omega) ##
Now if I do a substitution...
Homework Statement
I just have a quick question about the delta function, I'm pretty confident in most other cases but in this simple one I'm not so sure.
$$\int_{-\infty}^{\infty} \phi (x)\delta (-x)dx$$
Homework EquationsThe Attempt at a Solution
[/B]
$$\int_{-\infty}^{\infty} \phi...
This is mostly a procedural question regarding how to evaluate a Bromwich integral in a case that should be simple.
I'm looking at determining the inverse Laplace transform of a simple exponential F(s)=exp(-as), a>0. It is known that in this case f(t) = delta(t-a). Using the Bromwich formula...
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...
I've come across the equation $$\int_0^1 dx \frac{dA(x)}{dx} + B = C = \text{finite}$$ in my readings on a certain topic in physics and, in both articles i have read, the following step is taken $$\int_0^1 dx \left( \frac{dA(x)}{dx} + (B-C)\delta(1-x) \right) = \text{finite}$$
For the...
I am quite new here, and was wondering if anybody knows how this 2D integral is evaluated.
$$ \int_{-\infty}^\infty \int_{-\infty}^\infty \delta(k_1 x-k_2y)\,dx\,dy$$Any help is greatly appreciated! Thanks!
Homework Statement
I have
##\int dx \int dy \delta (x^{2}+y^{2}-E) ## [1]
I have only seen expressions integrating over ##\delta## where the ##x## or the ##y## appear seperately as well as in the delta function and so you can just replace e.g ##y^2 = - x^{2} +E## then integrate over ##\int...
I want to calculate $$\langle x|XP|y \rangle$$ where X is the position operator and P the momentum operator, and the states are position eigenstates. But I get two different answers depending on if I insert a complete set of states.
First way:
$$\langle x|XP|y \rangle=x\langle x|P|y...