Understanding Dirac Delta Squares: Clarifying Doubts

In summary, the conversation discusses the meaning of Dirac delta square \delta(x-x_1)\delta(x-x_2) and its integration, which is equal to \delta(x_1-x_2). The expression only makes sense under an integral sign and has a value of one or zero. More information about the Dirac delta function can be found on Wikipedia.
  • #1
hermitian
6
0
hi,

may someone help me to clarify my doubts...

in my work, i encounter diracdelta square [tex]\delta(x-x_1)\delta(x-x_2)[/tex] i am not sure what it means... it seems if i integrate it

[tex]\int dx \;\delta(x-x_1)\delta(x-x_2) = \delta(x_1-x_2)[/tex] is either zero of infinity.

is this correct?

thanks
 
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  • #2
Technically, saying that it has "value zero or infinity" doesn't make sense. Any Dirac delta only makes sense under an integral sign (although in physics, we tend to think of it as being an "infinite spike with a finite area").

It is correct that
[tex]
\int dx \;\delta(x-x_1)\delta(x-x_2) = \delta(x_1-x_2)
[/tex]

So, again, this expression again only makes sense inside an integral, like
[tex]\int dx_1 \int dx \; \delta(x - x_1) \delta(x - x_2) = \int dx_1 \; \delta(x_1 - x_2)[/tex]
which is one or zero (depending on whether or not x2 lies in the integration interval of the x1 integral).
 
  • #3

FAQ: Understanding Dirac Delta Squares: Clarifying Doubts

What is the Dirac Delta function?

The Dirac Delta function, also known as the impulse function, is a mathematical function that is used to represent a point mass or point charge in the field of physics and engineering. It is defined as 0 everywhere except at the origin, where it is infinite. It is often used to simplify calculations in certain areas of math and physics.

What are Dirac Delta squares?

Dirac Delta squares refer to the squared version of the Dirac Delta function. It is defined as 0 everywhere except at the origin, where it is infinite. Unlike the Dirac Delta function, it has a finite integral over the entire real line. It is often used in quantum mechanics and signal analysis.

How can Dirac Delta squares be interpreted graphically?

Graphically, Dirac Delta squares can be thought of as a series of infinitely narrow and infinitely tall rectangles, centered at the origin. The area of each rectangle represents the magnitude of the function at that point. As the width of the rectangles approaches zero, the function becomes more and more concentrated at the origin.

What are some properties of Dirac Delta squares?

Some properties of Dirac Delta squares include:

  • They are even functions, meaning they are symmetric about the y-axis.
  • The integral of the function over any interval that includes the origin is equal to 1.
  • They satisfy the sifting property: ∫f(x)δ(x)dx = f(0).
  • They can be scaled and shifted by constants without affecting their properties.

How are Dirac Delta squares used in real-world applications?

Dirac Delta squares have various applications in the fields of physics, engineering, and mathematics. Some common uses include modeling point charges in electromagnetism, representing impulses in control systems, and simplifying calculations in quantum mechanics. They are also used in signal processing to analyze and manipulate signals in the time and frequency domains.

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