What is Delta function: Definition and 378 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. K

    Computation about Gaussian and Dirac Delta Function

    I have a Gaussian distribution about t, say, N(t; μ, σ), and a a Dirac Delta Function δ(t). Then how can I compute: N(t; μ, σ) * δ(t > 0) Any clues? Or recommender some materials for me to read? Thanks!
  2. E

    Dirac Delta Function: Is delta(x-y) the Same as delta(y-x)?

    Sorry if the question seems naive but if we have the Dirac delta function delta(x-y) is it the same as delta(y-x)?? Or there are opposite in sign? And why ? Thank you for your time
  3. T

    Proper proof of a delta function

    Prove: tδ(t) = 0 The answer our TA has given isn't doing it for me: \int dt \delta(t)f(t) = (0)f(0) = 0 I'm wanting to write: t \frac{d}{dt}\int \delta(t) dt = t \frac{d}{dt}(1) = t * 0 = 0 Am I right here? This doesn't make use of a test function. I'm very sloppy with proofs! Thanks for...
  4. A

    How Does the Derivative of the Delta Function Affect y(t) in Signal Processing?

    I have an evil TA (who makes the assignments) who likes to give us torturously difficult assignments on stuff we haven't been taught (and in many cases don't even understand conceptually). Homework Statement The input signal, x(t) is a real-valued bandlimited signal with bandwidth W. Find...
  5. M

    What is the use of Dirac delta function in quantum mechanics?

    If you ask me define Dirac delta function, i can easily define it and prove its properties using laplacian or complex analysis method. But still i don't understand what is the use of DIRAC DELTA FUNCTION in quantum mechanics. As i have done some reading Quantum mechanics from Introduction to...
  6. U

    Delta function potential; Schrodinger Equation

    Homework Statement Consider the TISE for a particle of mass m moving along the x-axis and interacting an attractive delta function potential at origin: Part(a): What is the difference between a bound state particle and a free particle? Part(b): Show ##\psi _{(x)} = exp (-|k|x)## is a...
  7. P

    Double delta function and bound states.

    Homework Statement Given the delta function -α[δ(x+a) + δ(x-a)] where α and a are real positive constants. How many bound states does it possess? Find allowed energies for \frac{hbar2}{ma} and \frac{hbar2}{4ma} and sketch the wave functions. Homework Equations I know there are three parts of...
  8. C

    Lorentz transformation of delta function

    For two body decay, in CM frame, we know that the magnitude of the final particle momentum is a constant, which can be described by a delta function, ##\delta(|\vec{p^*}|-|\vec{p_0^*}|)##, ##|\vec{p_0^*}|## is a constant. When we go to lab frame (boost in z direction), what's the Lorentz...
  9. Ace10

    Infinite potential well with Delta function inside

    Hello guys, I need some serious help for the solution of a problem in Q.M, I'm not so sure if I deal with it properly.. Consider an infinite potential well with the traits: V(x):∞, for x>a and x<-a...
  10. L

    Inner product of dirac delta function

    Homework Statement Find the inner product of f(x) = σ(x-x0) and g(x) = cos(x) Homework Equations ∫f(x)*g(x)dx Limits of integration are -∞ to ∞ The Attempt at a Solution First of all, what is the complex conjugate of σ(x-x0)? Is it just σ(x-x0)? And I'm not sure how to...
  11. S

    Bound state of the delta function potential.

    What is the physical meaning to a bound state with negative energy? As I understand it, this is the case with the delta function potential, which admits only one bound state with a negative energy. If the potential function is identically zero throughout (except at the delta function peak)...
  12. L

    Showing the Representation of the Delta Function

    Homework Statement Show that ##\frac{1}{\pi}\lim_{\epsilon \to 0^+}\frac{\epsilon}{\epsilon^2+k^2}## is representation of delta function.Homework Equations ##\delta(x)=\frac{1}{2 \pi}\int^{\infty}_{-\infty}dke^{ikx}## The Attempt at a Solution...
  13. A

    Understanding the Delta Function: Integrating from -∞ to ∞

    My book loves to represent the delta function as: δ(r-r')=∫-∞∞exp(i(r-r')k)dk Now I can understand this formula if the integration was over the unit circle since. But this is an integration for which the antiderivative as no meaningful limit as x->±∞
  14. A

    Relation between residue integration and the Dirac Delta function

    Homework Statement OK so I'm doing a course on Signals and Systems and I'm taking inverse z transforms using residue integration. One particular formula in complex integration made me think a bit. \oint{\frac{f(z)}{z-z_0} dz} = 2\pi jf(z_0) This looks eerily similar to the definition...
  15. A

    Dirac Delta function and Divergence

    Homework Statement The Potential V(r) is given: A*e^(-lambda*r)/r, A and lambda are constants From this potential, I have to calculate: E(r), Rho(r) -- charge density, and Q -- total charge. Homework Equations The Attempt at a Solution I know that E(r) is simply minus...
  16. A

    Double delta function potential: two bound states vs one ?

    In the double delta function potential well, where one delta function ( -αδ(x) ) is at -a and one at +a, if the energy is less than zero, there can be either one or two bound states, depending on the magnitude of α...if α is large enough, there can be two bound states, but if α is small, there...
  17. skate_nerd

    Proving a property of the dirac delta function

    Homework Statement Prove this theorem regarding a property of the Dirac Delta Function: $$\int_{-\infty}^{\infty}f(x)\delta'(x-a)dx=-f'(a)$$ (by using integration by parts) Homework Equations We know that δ(x) can be defined as...
  18. F

    Residue of Dirac delta function?

    Does the Dirac delta have a residue? It seems like it might, but I don't know how to attack it, since I really know very little about distributions. For example, the Dirac delta does not have a Laurent-expansion, so how would you define its residue?
  19. Philosophaie

    Dirac Delta Function: What It Does & How to Evaluate It

    What does the Dirac Delta Function do? ##\delta^3(\vec{r})## How do you evaluate it? What are its values from -inf to +inf?
  20. T

    Delta function and derivative of function wrt itself

    Can someone explain the following profound truth: \frac {\partial f(x)}{\partial f(y)} =\delta(x-y)
  21. J

    Integral containing a delta function and a step function

    Homework Statement (a) Show that that δ(a-b)=∫δ(x-a)δ(x-b)dx (b) Show that ∂/∂x θ(x) = δ(x) where θ(x) is the heaviside step function (0 for x<0, 1 for x>0) (c) Show that ∫(-inf to inf) δ(x) f(θ(x))dx=∫(0 to 1) f(y)dy Homework Equations The definition of the delta function...
  22. Vahsek

    Dirac Delta Function: Definition & Mathematics

    It's been quite some time now since I decided to stop self-studying physics and to pay more attention to the math behind. I'm working towards gaining an understanding of 100% rigorous mathematics for now. One thing that has always bothered me is the Dirac delta function. What I want to know...
  23. R

    Does a particle have negative kinetic energy in delta function wall

    E is finite and V is infinite, So T should be negative infinite
  24. P

    Is δ(x+y)=δ(x-y) for Dirac Delta Function?

    Homework Statement Good day. May I know, for Dirac Delta Function, Is δ(x+y)=δ(x-y)? The Attempt at a Solution Since δ(x)=δ(-x), I would say δ(x+y)=δ(x-y). Am I correct?
  25. S

    The nature of the dirac delta function

    From what I can tell, it seems that 1/x + δ(x) = 1/x because if we think of both 1/x and the dirac delta function as the following peicewise functions: 1/x = 1/x for x < 0 1/x = undefined for x = 0 1/x = 1/x for x > 0 δ(x) = 0 for x < 0 δ(x) = undefined for x = 0 δ(x) = 0 for x > 0...
  26. H

    Delta function (hard question)

    Homework Statement Compute ∫^{∞}_{-∞}dx (x2+a2)-1δ(sin(2x)), without calculating the resulting sum. Homework Equations This is a very specific integral which ,has a delta function δ operating on sin function The Attempt at a Solution Does anyone know this integral ? I...
  27. M

    Understanding the Delta Function: Exploring the Role of k in Equations

    in this equation why does k take on both positive and negative values? isn't k a fixed constant that can only be positive or negative
  28. P

    The Double Dirac Delta Function Potential wave functions

    Homework Statement Consider the double Dirac delta function V(x) = -α(δ(x+a) + δ(x-a)). Using this potential, find the (normalized) wave functions, sketch them, and determine the # of bound states. Homework Equations Time-Independent Schrodinger's Equation: Eψ = (-h^2)/2m (∂^2/∂x^2)ψ +...
  29. Y

    Question about Dirac Delta function

    In page 555, Appendix B of Intro to electrodynamics by D Griffiths: \nabla\cdot \vec F=-\nabla^2U=-\frac{1}{4\pi}\int D\nabla^2\left(\frac{1}{\vec{\vartheta}}\right)d\tau'=\int D(\vec r')\delta^3(\vec r-\vec r')d\tau'=D(\vec r) where ##\;\vec{\vartheta}=\vec r-\vec r'##. Is it supposed to be...
  30. Y

    Question on Dirac Delta function

    I want to proof (1)##\delta(x)=\delta(-x)## and (2) ## \delta(kx)=\frac{1}{|k|}\delta(x)## (1) let ##u=-x\Rightarrow\;du=-dx## \int_{-\infty}^{\infty}f(x)\delta(x)dx=(0) but \int_{-\infty}^{\infty}f(x)\delta(-x)dx=-\int_{-\infty}^{\infty}f(-u)\delta(u)du=-f(0) I cannot proof (1) is equal as I...
  31. Y

    Question on Dirac Delta function in Griffiths

    My question is in Griffiths Introduction to Electrodynamics 3rd edition p48. It said Two expressions involving delta function ( say ##D_1(x)\; and \;D_2(x)##) are considered equal if: \int_{-\infty}^{\infty}f(x)D_1(x)dx=\int_{-\infty}^{\infty}f(x)D_2(x)dx\;6 for all( ordinary) functions f(x)...
  32. R

    Differentiating delta function composed with a function

    Dear all, I just wondered whether there was any standard identity to help me solve this equation: $$ \int \delta(f(x))^{\prime\prime}g(x) dx $$ Thanks in advance for your help.
  33. D

    Dirac Delta Function: Explanation & Usage

    I know this probably belongs in one of the math sections, but I did not quite know where to put it, so I put it in here since I am studying Electrodynamics from Griffiths, and in the first chapter he talks about Dirac Delta function. From what I've gathered, Dirac Delta function is 0 for...
  34. C

    Quick question about Kronecker delta function

    So in the infinite square well, the eigenfunctions are ## \psi_n (x) = \sqrt{\dfrac{2}{a}} \sin \left( \dfrac{n \pi}{a} x \right) ## Each state is orthogonal to each other, and so ## \displaystyle \int \psi_m (x) ^* \psi_n (x) dx = \delta_{mn} ## Does this also hold if they were cosines?
  35. Goddar

    Can I Solve This Complicated Equation with a Dirac Delta Function?

    Hi there, my version of Mathematica may be too old and I'm not finding this one by hand so any help would be appreciated: ψ''(z)=[k2/4 –M2 –kδ(z)]ψ(z), where δ(z) is the Dirac delta, k and M constants. i can solve the same equation without the M^2 term by exp(k|z|/2), but this one proves to...
  36. S

    Archived Calculus for delta function based on wave function

    Homework Statement Two-electron Wavefunction: ψ(r1,r2,r12) = exp(-Ar1-Br2-Cr12), r12 = |r1-r2| A, B, and C are coefficients Calculate <ψ|δ(r12)|ψ> and <ψ|δ(r1)|ψ> Homework Equations NO The Attempt at a Solution <ψ|δ(r12)|ψ> = ∫∫dv1dv2ψ2(r1,r2,r12)δ(r12)...
  37. B

    Double Delta function potential well

    Consider a one-dimensional system described by a particle of mass m in the presence of a pair of delta function wells of strength Wo > 0 located at x = L, i.e. V(x) = -Wo (x + L) - Wo(x - L) This is a rough but illuminating toy model of an electron in the presence of two positive. charges...
  38. F

    How Can Mathematical Modeling Describe Oscillations in a Weighted String System?

    A string of length L are connected in x = 0 and x = L. The point x = a, 0<a<L, are a point formed weight with mass m0 attached. Formulate the mathematical problem for small transversal oscillations . I got it to be: ∴ m0*utt(a,t) = s(a,t)(ux(a+,t)-ux(a-,t)) u(x,0) = g(x) ut(x,0) = h(x)...
  39. J

    Integral of delta function multiplied by a Heaviside step function

    Homework Statement consider two functions:ψ(x) which is eqaul to zero at a,that is ψ(a)=0 and f(x)=H(x-a)*β(x)+(1-H(x-a))*γ(x) where H(x-a) is the heaviside step function and β(x),γ(x) is the continuous function. it seems that the derivative of f(x) is not exist. the question is whether...
  40. L

    Calculate scattering amplitude by delta function potential

    Homework Statement I need to give scattering amplitude f(θ) in Born approximation to the first order in the case of delta function scattering potential δ(r). The problem is in spherical coordinate and I'll give major equation concerned.Homework Equations The equation for scattering amplitude is...
  41. J

    A question about Dirac delta function

    Hello, Is this correct: \int [f_j(x)\delta (x-x_i) f_k(x)\delta (x-x_i)]dx = f_j(x_i)f_k(x_i) If it is not, what must the left hand side look like in order to obtain the right handside, where the right hand side multiplies two constants? Thanks!
  42. J

    Simple equations in Dirac Delta function terms

    Hi there, I'm trying to comprehend Dirac Delta functions. Here's something to help me understand them; let's say I want to formulate Newton's second law F=MA (for point masses) in DDF form. Is this correct: F_i = \int [m_i\delta (x-x_i) a_i\delta (x-x_i)]dx Or is it this: F_i = [\int...
  43. B

    Integrating the Dirac Delta function

    Homework Statement I am trying to integrate the function \int _{-\infty }^{\infty }(t-1)\delta\left[\frac{2}{3}t-\frac{3}{2}\right]dt Homework Equations The Attempt at a Solution I think the answer should be \frac{5}{4} because \frac{2}{3}t-\frac{3}{2}=0 when t=9/4. then (9/4-1) = 5/4...
  44. G

    Dirac delta function how did they prove this?

    Hi all, I'm familiar with the fact that the dirac delta function (when defined within an integral is even) Meaning delta(x)= delta(-x) on the interval -a to b when integral signs are present I want to prove this this relationship but I don't know how to do it other than with a limit...
  45. F

    Understanding Dirac Delta Function: Time Derivative & Hankel Transformation

    Hi All, I have a problem in understanding the concept of dirac delta function. Let say I have a function, q(r,z,t) and its defined as q(r,z,t)= δ(t)Q(r,z), where δ(t) is dirac delta function and Q(r,z) is just the spatial distribution. My question are: 1. How can I find the time derivative...
  46. F

    Is there a coordinate independent Dirac delta function?

    I have been wondering exactly how one would express the Dirac delta in arbitrary spaces with curvature. And that leads me to ask if the Dirac delta function has a coordinate independent expression. Is there an intrinsic definition of a Dirac delta function free of coordinates and metrics? Or as...
  47. F

    Integrating with a dirac delta function

    Homework Statement I have to integrate: \int_0^x \delta(x-y)f(y)dy Homework Equations The Attempt at a Solution I know that the dirac delta function is zero everywhere except at 0 it is equal to infinity: \delta(0)=\infty I have to express the integral in terms of function...
  48. N

    Laplace transform of the dirac delta function

    Homework Statement L[t^{2} - t^{2}δ(t-1)] Homework Equations L[ t^{n}f(t)] = (-1^{n}) \frac{d^{n}}{ds^{n}} L[f(t)] L[δ-t] = e^-ts The Attempt at a Solution My teacher wrote \frac{2}{s^{3}} -e^{s} as the answer. I got \frac{2}{s^{3}} + \frac{e^-s}{s} + 2 \frac{e^-s}{s^2} + \frac{2e^-s}{s^3}
  49. D

    MHB What is the moral of conflicting definitions of the Heaviside function?

    Integrating the delta function: $$ \frac{4}{\pi^2}\int_0^{\pi}\int_0^{\pi}\delta(x - x_0,y - y_0)\sin nx\sin my dxdy $$ Would the solution be $\frac{4}{\pi^2}\sin nx_0\sin my_0$?
  50. D

    Laplace Transform of Delta Function

    Homework Statement Evaluate the Laplace transform: L{δ(t-∏)tan(t)} Homework Equations The Attempt at a Solution L{δ(t-∏)tan(t)} = ∫ δ(t-∏)tan(t) dt evaluated from 0 to ∞ =tan(∏)e-∏*s = 0 Could someone check my work on this one? I'm suspicious that my transform is just zero...
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