Understanding & Using the Delta Function in Physics

[theFuture]

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?

 

[mathman]

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.

 

[theFuture]

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?

 

[chroot]

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

 

[farmerwa]

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.