How does this physically impossible function help us solve difficult problems in physics?
The Dirac Delta Function (named after Paul Dirac) or Unit Impulse function can be thought of as a spike. The value of the function is zero everywhere, except for at one particular value, where the function goes to infinity (i.e. its value is undefined). In this video, we look at how despite nothing in the real world behaving like this, we use the delta function to model various phenomena in theoretical physics.
Firstly, we define what the delta function looks like. Then we look at a couple of interesting mathematical properties of the function. One of these properties is that the integral of the delta function (which gives the area between the function and the horizontal axis) is equal to 1. This is a strange concept - how can the area under an infinitesimally thin, infinitely tall function be a finite value, and why is it specifically defined to be 1? Read up more about the function here: [ Ссылка ] and here [ Ссылка ]
We also see how the integral of a function corresponds to the area between that function and the horizontal axis, for those of us that are unfamiliar with this idea.
Secondly, we see that a delta function can be "moved" so the spike is at a different x position, in a similar way to how other functions are translated. If f(x) is centered on 0, then f(x-a) is centered on a.
This allows us to discover another property of the delta function - it can be used to pick out values of functions at specific points. For example, the integral of the product between a sine function and a delta function centered at a, is given by the value of the sine function at a. This is a remarkable property that allows us to encode many ideas in physics.
Firstly, we can treat particles as being point masses and charges. In reality their mass and charge are distributed over finite regions of space, but they're so small compared to us that we can very nearly pretend the masses and charges are concentrated at one infinitely small point. In this video we see how charged particles can be represented as point charges using the Dirac delta function. We take a function representing the charge (i.e. the magnitude of the charge on the particle), and multiply it by the delta function to give us the charge density. This way we can integrate the charge density to give us charge, while also encoding information about the position of the particle via the delta function.
Additionally, the delta function can be used to localize objects in time, as well as in space. This is often done when studying impulses (i.e. forces applied to objects for very short periods of time). An example is a footballer kicking a football - the force is exerted for a very short time. In that case, a delta function with time on the horizontal axis can be used to localize the force exerted on the ball to a particular instant in time!
So in summary, the Dirac Delta Function is a physically impossible but mathematically essential function (that's not really a function). it helps us greatly simplify many different ideas in theoretical physics.
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Timestamps:
0:00 - The Dirac Delta Function - What Does It Look Like?
1:48 - Mathematical Property: The Area Under the Delta Function is 1?!
3:24 - Translating the Function, and Using It to Pick out Function Values
4:26 - Uses in Theoretical Physics - Representing Point Charges
9:06 - Impulses - Localizing in Time Rather Than in Space
10:03 - Summarizing the Impossible (But Essential) Function
