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

In summary, the Dirac Delta Function is a mathematical tool used to model the behavior of point-like particles in three dimensions. It is a function of the vector r and can be evaluated by taking the product of three 1-dimensional delta functions. It is commonly introduced in Quantum Mechanics or Electromagnetism courses and can be found in textbooks such as Griffiths' on these subjects.
  • #1
Philosophaie
462
0
What does the Dirac Delta Function do?

##\delta^3(\vec{r})##

How do you evaluate it?

What are its values from -inf to +inf?
 
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  • #2
Did google yield no useful results?
 
  • #3
It did not explain the vector r
 
  • #4
The vector just means that it is a function of x, y, and z.

Beyond that, I would say that you should try a bit harder to learn about the delta function without just asking us to explain it. It is typically introduced in either a Quantum mechanics course or an Electromagnetism course. Griffiths text on both subjects gives a non rigorous intro to the delta function.
 
  • #5
So are you familiar with the 1-dimensional Dirac delta function? If so, the 3-dimensional delta function is just the product of three 1-dimensional delta functions, one for each component of the vector ##\vec{r}##:

##\delta^3(\vec{r}) = \delta(r_x)\delta(r_y)\delta(r_z)##
 

1. What is the Dirac Delta Function?

The Dirac Delta Function, also known as the unit impulse function, is a mathematical function that is defined to be zero everywhere except at the origin, where it is infinitely large. It is often represented by the symbol δ(x) and is used to model point-like objects or phenomena in physics and engineering.

2. How is the Dirac Delta Function evaluated?

The Dirac Delta Function is evaluated using the following properties:

  • δ(x) = 0 for all x ≠ 0
  • ∫δ(x)dx = 1
  • ∫δ(x-a)f(x)dx = f(a) for all a
  • δ(ax) = 1/|a|δ(x) for all a ≠ 0

Using these properties, the Dirac Delta Function can be evaluated at any point or interval.

3. What is the physical significance of the Dirac Delta Function?

The Dirac Delta Function is often used to represent point sources or impulses in physical systems. It is also used in the study of distributions, which are generalized functions that are not defined in the traditional sense. In physics, the Dirac Delta Function is used to model point particles, such as electrons, and point forces, such as those in electromagnetism.

4. Can the Dirac Delta Function be integrated?

No, the Dirac Delta Function cannot be integrated in the traditional sense. It is not a function in the usual sense but rather a distribution. However, it can be integrated in the context of distributions using the properties mentioned above.

5. How is the Dirac Delta Function used in engineering and mathematics?

The Dirac Delta Function is used in engineering to model point-like objects or phenomena, such as a point force or a point load. In mathematics, it is used to solve differential equations, evaluate integrals, and define other distributions. It is also used in signal processing and control systems to analyze and design systems with impulse responses.

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