The Impossible Function
In mathematics (and most of theoretical physics), the Dirac delta function, δ(x), is a generalized function over real numbers. Its value is zero everywhere except at zero, and its integral from the infinites is equal to one.
Formulated by Paul Dirac, as the name suggests, the graph of the Dirac delta is usually thought of as following the whole x-axis and the positive y-axis. In other words, for every value of non-zero x, the function takes a value of zero. But at zero, the value of the function is incredibly large. It happens to infinite. This may sound a little hand-wavy but, in principle, the delta function is a “spike” that is infinitesimally thin and infinitely large.
Needless to say, that’s a pretty strange graph. What do we mean by “infinitesimally thin” and “infinitely large”? All it really means is that the function spikes only at one value — zero. For any other x value, no matter how close to zero it may be, the delta function is always zero.
Of course, there are a ton of mathematical intricacies here. You may feel inclined to ask “is it a true mathematical function?” Or even “how do we deal with a function that doesn’t change continuously”? But we won’t get into too much of that detail here. I hope some of this becomes apparent as you progress
Before we delve into the physics of it, another property worth noting about the delta function is that its integral is exactly 1. For any general function, we find the area under the curve when we integrate it. The area is the integral of the function with defined limits.
Now, if you’re unfamiliar with this, the idea is that it’s not geometrically easy to find this area. We resort to mathematical formulations instead. It’s the same as splitting the area under our curve into infinitely many rectangles and adding all their areas up. Something like this:
Now, the delta function is specially defined so that its integral is equal to one. Over all space, you find that the area bound by the delta function and the x-axis is one.
This is incredibly strange to think about considering there is no area under the graph anywhere except for where x = 0. The width of this function is zero and the height is infinite and yet somehow the area under this function happens to be finite: 1. If you want to know more about this, feel free to google it. This is merely a property of the Dirac delta; we’d digress if we got into the math of it all. For what it’s worth, take me on blind faith with this one.
Now there’s another intricacy here. The delta function does not need to spike specifically at x = 0. We can move this function around to spike at exactly what we’d like. If you’re familiar with translating functions, you’d know that if you took some dependent variable, x, and subtracted some quantity from it, let’s say a, then what you’re doing is actually shifting the entire graph rightwards by a units.
We could do the same with the delta function if we wanted to.
The reason this is important is that we can now take another function, let’s say the sine function, and multiply it by δ(x-a). Then, if we were to integrate this, we’d end up with the value of the function — sin(x) in this case — at the point x = a.
In other words, when used in this way, the delta function can be used to “pick out” values of any function. But this is how far the math goes.
The Physics
The delta function is more than just mere mathematical intricacies that make our lives easier. Of course, with this comes a quite important question: “how does an infinitely narrow and infinitely high function help us describe anything real?”.
Well, even though these sorts of infinities don’t show up in our lives, theoretical physics is full of them. To keep things simple, we often treat particles as point masses. We assume that the mass of small particles — like electrons — is concentrated at a point. By definition, a point has no size. It is infinitesimally small.
Consequently, we often treat charged particles as “point” charges. The source is a mere point in space. Call it a pixel if you’d like. We say that their charge is localized to that point. This makes the math a whole lot easier as opposed to considering a distribution of charges over finite space. We just say it’s shoved into a very, very, very narrow area. But where does the Dirac delta come into this?
We can start by thinking not about the charge of the particle but rather the charge density of the system we’re considering. Charge density is simply a quantity that represents the amount of charge we find per unit volume. To put it in a slightly more “correct” way, it’s actually the rate of change of charge with respect to volume.
Rearranging this to find the charge of the particle, we find that q is simply the integral of density, rho, with respect to volume:
This is essentially the opposite of taking the derivative. Multiply both sides by dV and take the integral on both sides. There, you have it. Now, the physical interpretation of this is that the charge of the particle can be found by knowing how much charge there is for every tiny bit of volume. We get the charge density and multiply it by the tiny bit of volume to find the tiny bit of charge in that volume. It’s a little weird to explain this in words so we’ll just resort to math instead, dq = ρdV.
We add up all tiny little dqs to give us q. But if you recall, for a point mass, we assume that the charge is not distributed at all. Instead, it’s all found in one single point. Let’s say, x = b.
To keep things simple, we’ll seek refuge in a one-dimensional system. In principle, you can apply the same idea to both the y and z coordinates too. Here’s what we do: we say that the charge density is equal to the charge localized at the point multiplied by the delta function of x-b:
Why is this useful? Well, check this out. Substituting it back into our equation where q is the integral of rho dV, we get this:
Where q1is simply the magnitude of our point charge. Once again, this is not quite correct because we’re ignoring y and z but let’s stick to this. Earlier, we said that the delta function picks out a value of another function. Well, in this case, q1 is that function. Mathematically, if we were to encode q1, then it would simply be a constant function. But the delta function helps us pick it out at the point in space where the charged particle is located. Once again, I’d like to clarify that the delta function wouldn’t be appropriate in real life — with its infinities. The charge distribution would be spread out and not just concentrated on one point in space.
It’s just that when we deal with charged particles, they’re so incredibly small compared to our usual units of measurement, that the particle may as well be treated as though it were infinitely small.
The Dirac delta, however, goes beyond this too. It can be used to find the value of a wave function at a particular point. The delta function is expedient in quantum mechanics. The wave function of a particle gives the probability amplitude of finding a particle within a given region of space. Wave functions are assumed to be elements of the Hilbert space. When the Dirac delta is applied to the wavefunction, it spits out probabilistic values. More importantly, however, it also allows for the existence of the delta potential: a potential well mathematically described by the Dirac delta. This can be used to simulate situations where a particle is free to move in two regions of space with a barrier between the two regions. For example, an electron can move almost freely in a conducting material, but if two conducting surfaces are put close together, the interface between them acts as a barrier for the electron that can be approximated by a delta potential.
The Dirac delta is physically impossible but mathematically essential. For a more detailed discussion, check this out by Joseph Mellor:
With that, I will end this here. Thank you for reading!
