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.

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1. Dirac delta function approximation

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}...
2. I Fermi's golden rule: why delta function instead of density states?

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...
3. Kramers-Kronig, parity and delta function

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...
4. A Time dependent problem with a delta function

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...
5. Mellin transform of Dirac delta function ##\delta(t-a)##

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}...

7. I What is Dirac delta function?

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...

10. Double Delta Potential: Solving Particle Scattering Problem

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]...
11. I Probability: why can we use the Dirac delta function for a conditional pdf?

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...
12. A sudden change in the depth of delta function potential well

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.
13. I Representations of a delta function

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.
14. Delta function for \nabla(log s), where s = \sqrt{x^2+y^2}

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...
15. A Dirac delta function confusion

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...
16. S-wave phase shift for quantum mechanical scattering

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...
17. A Evaluation of an improper integral leading to a delta function

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...

19. Scattering amplitude in scattering from a delta function

I tried to calculate the Fourier transform to get the amplitude, but I got lost
20. Engineering Dirac Delta Function in an Ordinary Differential Equation

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 =...
21. Solve $$\int_{∞}^{∞}dxf(x)\delta((x-x_1))$$: Dirac Delta Function

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...

40. I Understanding the Dirac Delta function

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##: $$\begin{split} \delta(x-y) f(x) &= f(y) \end{split}$$ And: \begin{split} \delta(x-x) f(x) &=...
41. 2D Integrating With Quadratic Arg. of Delta Function

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...
42. Cts approximation, delta function integration, stat mech

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...