Understanding & Using the Delta Function in Physics

In summary: However, it does show up in a lot of places in physics, so if you're curious about it, it's worth looking into.
  • #1
theFuture
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I'm not sure what field this fits with, so I'll post here. I was introduced to the delta function in physics class. I understand what it means, but how do you use it?
 
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  • #2
Short answer - it is used under an integral sign, where the integral of a function turns out to be its value at a particular point. Beyond this, it all depends on context.
 
  • #3
That's what I understand. Where we used it was if, say you are walking at constant velocity, stop and turn around. Your v-t graph will be a line and the jump up to another line. That causes problems when you want to find the displacement if you cross the discontinuity. Can you think of any other physical examples?
 
  • #4
Originally posted by theFuture
Can you think of any other physical examples?
The "delta function" is not really a rigorously defined mathematical function, but nonetheless us physicists often use it as if it were.

The delta is one of those weird things like pi that just manages to show up in solutions all the time. You'll find deltas all over physics, particularly quantum mechanics.

To give a simple physical example, take a pure sine wave of one frequency -- say middle A, 440 Hz. If you plot the frequency (spectral) content of this signal, you'll see a delta function -- exactly one frequency is present, and all the signal's power is in it.

This same sort of thing happens in quantum mechanics -- if you have a particle with a precisely known momentum, its momentum is a delta function in momentum-space.

- Warren
 
  • #5
Actually, the delta function is defined rigorously. But in order to do it, it's not a function at all. It's actually a distribution. A distribution, from what I understand, is like a function except that it's input is not a number per se but instead another function. This is why you have to integrate it against something. Another way of thinking of the delta function is as a limit of a normalized gaussian function as the standard deviation approaches zero.

Suffice it to say, the delta function is a subtle creature and is difficult to understand.
 

1. What is the Delta Function in Physics?

The Delta Function, also known as the Dirac Delta Function, is a mathematical concept used in physics to represent a point mass or point charge. It is a function that is zero everywhere except at one specific point, where it is infinite. It is often used to simplify calculations in physics and is a fundamental tool in the study of quantum mechanics.

2. How is the Delta Function used in Physics?

The Delta Function is used in a variety of ways in physics. It is commonly used to represent a point particle, where the mass or charge is concentrated at one specific point. It is also used in probability and statistics, as it can be used to describe the probability of a particle being at a specific point in space. Additionally, it is used in signal processing and electrical engineering to describe the response of a system to an impulse or sudden change.

3. What are the properties of the Delta Function?

The Delta Function has several important properties that make it a useful tool in physics. These include:

  • It is zero everywhere except at one specific point.
  • The area under the Delta Function is always equal to one.
  • It is symmetric, meaning that the function is the same whether the point is on the left or right side of the origin.
  • It follows the sifting property, which states that the integral of the Delta Function multiplied by another function is equal to the value of the other function at the point where the Delta Function is located.

4. How is the Delta Function related to the Kronecker Delta?

The Delta Function is often confused with the Kronecker Delta, but they are two distinct concepts. While the Delta Function is a continuous function used in calculus and analysis, the Kronecker Delta is a discrete function used in algebra and combinatorics. The Kronecker Delta takes the value of one when its two indices are equal, and zero otherwise. However, both functions share similar properties, such as symmetry and the sifting property.

5. What are the limitations of using the Delta Function in Physics?

While the Delta Function is a powerful tool in physics, it does have its limitations. One limitation is that it is an idealization and does not exist in the physical world. It is a mathematical construct used to simplify calculations and models. Additionally, the Delta Function is undefined at the point where it is infinite, which can cause issues when trying to integrate over that point. Therefore, it is often used in conjunction with other mathematical techniques to overcome these limitations and provide more accurate results.

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