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.
Homework Statement
\int_{-\infty}^t (cos \tau)\delta(\tau) d\tau
Evaluate the integral. I'm supposed to evaluate this for all t I believe, so I'm concerned with t<0, t=0, t>0.
Homework Equations
\int_{-\infty}^{\infty} f(x)\delta(x) dx = f(0)
The Attempt at a Solution...
Homework Statement
Need to integrate using the dirac delta substitution:
\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\!x^2\cos(xy)\sqrt{1-k^2\sin^2(y)}\, dx\, dy
Homework Equations
\cos(xy) = \frac{1}{2}\left(e^{ixy} + e^{-ixy}\right)
\delta\left[g(t)\right] =...
Hi guys.
I play now a bit with EM fields and I have encountered some problems connected with Dirac delta. By coincidence I visited this forum and I thought I could find some help in here.
The problem is that in order to get a potential in some point from a single charge you need to just...
Previously I posted a question on the Dirac delta function and was informed it was not a true function, but rather a distribution. However, I have to admit I still did not understand why its integral (neg inf to pos inf) is unity. I've thought about this and came up with the following...
Homework Statement
How do you show that int[delta(t)]dt from negative infinity to infinity is 1?
Homework Equations
Dirac delta function defined as infinity if t = 0, 0 otherwise
The Attempt at a Solution
My teacher said that it has to do with m->infinity for the following...
1. The problem statement
Show that:
\int_{-\infty}^{\infty} f(x) \delta^{(n)}(x-a) dx = (-1)^n f^{(n)}(a)
The Attempt at a Solution
I am trying to understand how to prove:
\int_{-\infty}^{\infty} f(x) \delta '(x) dx =- f'(x)
I know that we need to use integration by parts, but I'm...
Griffiths' section 1.5.3 states that the divergence of the vector function r/r^2 = 4*Pi*δ^3(r). Can someone show me how this is derived and what it means physically? Thanks in advance.
Homework Statement
Starting with the definition of the Dirac delta function, show that \delta( \sqrt{x}) um... i have looked in my book and looked online for a problem like this and i really have no clue where to start. the only time i have used the dirac delta function is in an integral...
Hi
I have been trying to learn dirac delta function. but its kind of confusing. I come across 2 contrasting definitions for it. The first one states that the function delta(x-xo) is infinite at x=x0 while the other states that delta(x-x0) tends to infinite as x tends to x0. Now both of them...
Hey there!
I'm faced with this problem:
http://img7.imageshack.us/img7/4381/25686658nz9.png
It's a 1D nonhomogeneous wave equation with a "right hand side" equaling to the dirac delta function in x * a sinusoidal function in t. I have to find its general solution with the constraints...
By definition of the Dirac delta function, we have:
\int f(x) \delta(x-a) dx=f(a)
This is fair enough. But in ym notes there is a step that goes like the following:
\mathbf{\nabla} \wedge \mathbf{B}(\mathbf{r})=-\frac{\mu_0}{4 \pi} \int_V dV'...
I'm posting this here because I'm asking about the mathematical properties of the Dirac delta function, delta(x) which is zero for all non-zero real values of x and infinite when x is zero. The integral (-inf to +inf) of this function is said to be 1. How is this derived?
Homework Statement
Hi there, i am trying to do a proof that H'(t)= δ(t)
Homework Equations
We have been given the following:
F is a smooth function such that lim (t-->±∞)F(t)=0
Therefore the integral between ±∞ of [H(t)F(t)]'=[H(t)F(t)]∞-∞=0
I understand it up until this point...
Hi.
Recently day, I tried to solve quantum mechanics problem in liboff fourth version to prepare
graduate school.
But what make me be confused a lot is Dirac Delta Function.
One of my confusing on Dirac Delta is what i wrote below.
-One of the formula describing Dira Delta...
hello every body
i am a new M.S student
and i can't understand the Dirac delta function can anyone simply describe it to me in order to simplify it.
thank you
Find a distribution F in R^2 that satisfies (Dx) F(x,t) = t*Delta(x)
It is apperantly not t*H(x) as in R.
* is multiplication, D is dirac delta, H is Heavyside , (Dx) is derivation with respect to x (in the sense of distributions)
Sorry for not using Latex.
Indeed I am trying to...
Derivative Using Dirac Delta Function
Homework Statement
Let \theta(x) be the step function:
\theta(x) be equivalent to
1, if x > 0
0, if x \leq 0
Show that \frac{d \theta }{dx} = \delta(x)
Homework Equations
In the previous portion I was able to prove
x \frac{d}{dx}...
Show that
\stackrel{lim}{\alpha \rightarrow \infty} \int^{\infty}_{-\infty}g(x)\sqrt{\frac{\alpha}{\pi}}e^{-\alpha x^2} dx = g(0)
where g(x) is continuous.
To use the continuity of g(x) I started from
\left|g(x)-g(0)\right|<\epsilon
and tried to put it in into the integral...
Given:
f(x)=\delta(x-a)
Other than the standard definitions where f(x) equals zero everywhere except at a, where it's infinity, and that:
\int_{-\infty}^{\infty} g(x)\delta(x-a)\,dx=g(a)
Is there some kind of other definition involving exponentials, like:
\int...
Homework Statement
Three-dimensional particle is placed in a Dirac delta potential:
V = -aV_{0}\delta(r-a)
Find energy states and eigenfunctions for the angular quantum number l = 0.[/
Homework Equations
The Attempt at a Solution
It's not clear to me what boundary...
Three-dimensional particle is placed in a Dirac delta potential:
V = -aV_{0}\delta(r-a)
Find energy states and eigenfunctions for the angular quantum number l = 0.
Hi, I need your help with a very standard proof, I'll be happy if you give me some detailed outline - the necessary steps I must follow with some extra clues so that I'm not lost the moment I start - and I'll hopefully finish it myself. I am disappointed that I can't proof this all by myself...
Homework Statement
Find the bound state energy for a particle in a Dirac delta function potential.
Homework Equations
\newcommand{\pd}[3]{ \frac{ \partial^{#3}{#1} }{ \partial {#2}^{#3} } } - \frac{\hbar^2}{2 m} \ \pd{\psi}{x}{2} - \alpha \delta (x) \psi (x) = E\psi (x)
where \alpha >...
[SOLVED] Dirac delta function
Homework Statement
Prove that \delta(cx)=\frac{1}{|c|}\delta(x)
Homework Equations
The Attempt at a Solution
For any function f(x),
\int_{-\infty}^{\infty}f(x)\delta(cx) dx = \frac{1}{c}\int_{-\infty}^{\infty}f(t/c)\delta(t) dt
where I have...
This is probably a silly question to some, but I've been struggling to understand how the delta function behaves when given a complex argument, that is \delta(z), z \in C. I guess the basic definition is the same that the integral over all space is 1, but I'm looking for a more detailed guide on...
[SOLVED] Dirac delta function and Heaviside step function
In Levine's Quantum Chemistry textbook the Heaviside step function is defined as:
H(x-a)=1,x>a
H(x-a)=0,x<a
H(x-a)=\frac{1}{2},x=a
Dirac delta function is:
\delta (x-a)=dH(x-a) / dx
Now, the integral:
\int...
Trying to solve the ODE
mx''(t) + bx'(t) + kx(t) = F(t)
with m measured in Kg, b in Kg/s and Kg/s^2, F(t) in Kgm/s^2 and x(t) in m with initial conditions x(0) = 0 and x'(0) = 0, i got the following Green's function
G(t,t') = \frac{1}{m\omega} e^{-\omega_1(t-t')}\sinh\left[\omega(t-t')\right]...
OK, so my basic understanding of Dirac Delta Function is that it shows the probability of finding a point at (p,q) at time t. Dirac Delta is 0 everywhere except for (p_{0},q_{0}).
So my question comes
Is it possible that a point enters the (p_{0},q_{0}) and stays there (for some period of...
Does anyone know what the Dirac delta function would look like in a space with curvature and torsion? The Dirac delta function is a type of distribution. But that distribution might look differently in curved spacetime than in flat spacetime. I wonder what it would look like in curved spacetime...
Alright...so I've got a question about the convolution of a dirac delta function (or unit step). So, I know what my final answer is supposed to be but I cannot understand how to solve the last portion of it which involves the convolution of a dirac/unit step function. It looks like this:
10 *...
Alright, I'm in my first QM course right now, and one of the topics we've looked at is solving the one-dimensional time-independent Schrodinger equation for various potentials, such as the harmonic oscillators, infinite and finite square wells, free particles, and last, but not least, the dirac...
Homework Statement
The function \delta(cosx) can be written as a sum of Dirac delta functions:
\delta(cosx)=\sum_{n} a_{n}\delta(x-x_{n})
Find the range for n and the values for a_{n} and x_{n}
The Attempt at a Solution
Well, taking the integral of \delta(cosx), we only get spikes when...
The Dirac delta function, \delta (x) has the property that:
(1) \int_{-\infty}^{+\infty} f(x) \delta (x) dx = f(0)
Will this same effect happen for the following bounds on the integral:
(2) \int_{0}^{+\infty} f(x) \delta (x) dx = f(0)...
I have a line charge of length L and charge density /lambda on the Z-axis. I need to express the charge density in terms of the Dirac Delta function of theta and phi. How would I go about doing this?
I am trying to evaluate the following integral.
\int_{-\infty}^{\infty}{\delta(2t-3)\sin(\pi t) dt}
where delta represents the Dirac delta function.
I am told that the answer is -1. However, when I evaluate it in MATLAB and Maple 11, I get an answer of -1/2. What is the correct way...
Homework Statement
I'm trying to prove that \delta'(y)=-\delta'(-y).
Homework Equations
The Attempt at a Solution
I'm having trouble getting the LHS and the RHS to agree. I've used a test function f(y) and I am integrating by parts.
For the LHS, I have...
Homework Statement
Hi there, I'm stuck at a problem where I have (sorry i don't know how to use mathtype so I'll try my best at making this clear) the integral of a dirac delta function squared:
int[delta(x*-x)^2] between minus infinity and infinity (x*=constant)
I know that the function...
So what we have so far is that any and all subsets are implied by a set. If there exist a set, then all the subsets within it are implied to exist also. This includes the elements of a set. The elements of a set are implied by the existence of a set.
One of the most natural things to do with...
Dirac Delta Function:
If, at time t =a, the upper end of an undamped spring-mass system is jerked upward suddenly and returned to its original position, the equation that models the situation is mx'' + kx = kH delta(t-a); x(0) = x(sub zero), x'(0) = x(sub 1), where m is the mass, k is the...
1. The ProblemHomework Statement
4 Parts to the Assignment. Finding the Displacement of a beam assuming w to be constant.
1. Cantilever beam, free at one end. Length =l, Force P applied concentrated at a point distance rl from the clamped end. Boundary Conditions y(0)=0, y'(0)=0, y"(l)=0, and...
[b]1. Homework Statement
\int x[delta(x)-delta(x/3+4)] dx
Homework Equations
so I'm supposed to use this principle:
\int f(x)delta(x-xo)dx=f(xo)
The Attempt at a Solution
So it seems simple but I just want to make sure that I'm applying the above principle correctly.
I...
Homework Statement
SO I'm given a dirac delta function, also known as a unit impulse function.
d(t-t'_=(1/P) sum of e^[in(t-t')], for n from negative to positive infinity.
I need to graph this.
Homework Equations
I understand that at t', there is a force made upon the system which...
Q: Integral of Delta(x-b)dx and the lower limit is (-) infinity and upper is a
Please help me in steps tried my best to solve.Note this is not homework I was doing the book problems or my practice
Thanks
Homework Statement
How would one show that dirac delta is the limit of the normal distribution?
http://en.wikipedia.org/wiki/Dirac_delta
using the definition \delta(k) = 1/(2\pi)\int_{-\infty}^{\infty}e^{ikx}dx
Homework Equations
The Attempt at a Solution
OK, I'm currently reading Hughes' Finite Element Method book, and I'm stuck on a chapter the goal of which is to prove that the Galerkin solution to a boundary value problem is exact at the nodes.
So, the author first speaks about the Dirac delta function: "Let \delta_{y}(x) = \delta(x-y)...
I have seen two formulations of the dirac delta function with the Fourier transform. The one on wikipedia is
\int_{-\infty}^\infty 1 \cdot e^{-i 2\pi f t}\,dt = \delta(f)
and the one in my textbook (Robinett) is
1/2\pi \int_{-\infty}^\infty 1 \cdot e^{-i f t}\,dt = \delta(f)
I...
I do not know if it is true but is this identity true
\frac{\delta _{n}^{x} }{h} \rightarrow \delta (x-n)
as h tends to 0 ?, the first is Kronecker delta the second Dirac delta.
i suspect that the above it is true but can not give a proof
Dirac developed his delta function in the context of QM. But there are various functions under the integral that give the delta function. My question is does one Dirac delta function equal any other? Are all ways of getting the Dirac delta function equivalent? Thanks.