Uncertain about what Heisenberg's Uncertainty Principle means? Worry no more - this video is here to help you :)
Let's start out this description with timestamps, because this video is super looong.
00:00 - Intro
00:42 - What is Heisenberg's Uncertainty Principle?
02:33 - Classical vs Quantum worlds, and what "uncertainty" even means
03:54 - A common description of, and misconceptions about the Uncertainty Principle
06:42 - Fourier transforms
12:22 - How Fourier transforms bring about Heisenberg's Uncertainty Principle
Most commonly, Heisenberg's Uncertainty Principle is used to describe a relationship between how much we know about two quantities: position and momentum. The principle tells us that there is a fundamental and universal limit to how much we can simultaneously know about both. In other words, the more confidently we know the position (of, let's say, a particle), the less confident we can be about its momentum. Now the word "uncertainty" just refers to the width of the probability distribution that describes a particle's position or momentum in the quantum world.
Before we find out where the Uncertainty Principle comes from, we will discuss a commonly used description that gets thrown around regarding the Principle. It's known as the Observer Effect, and it's the idea that the light we send into a particle in order to glean information about its position and momentum actually ends up changing the behaviour of the particle. High energy, small wavelength light tells us with more certainty the position of the particle, but changes its momentum by a lot so we are less certain about this. Low energy, large wavelength light doesn't give us much information about the position of the particle but does tell us more about its momentum. Annoyingly though, the observer effect is NOT the Uncertainty Principle. It's just a possible explanation for it, as suggested by Heisenberg himself.
So where can we look to find the origins of the Uncertainty Principle? Well a good place to start is to understand Fourier Transforms. Fourier transforms come about by first breaking down mathematical functions into sine wave building blocks - kind of like how we break vectors down into horizontal and vertical components. The sine wave building blocks are sine waves of different frequencies. We can then take the amplitude of each of the sine wave building blocks and plot that against the frequency of that sine wave to give us a new plot. This new plot is known as the Fourier Transform of the original function that we broke down.
The interesting thing to note is that if a function is super wide on its horizontal axis, then its Fourier transform is going to be super narrow, and vice versa. This is useful when we bring the whole thing back round to the Uncertainty Principle.
Now remember, in quantum mechanics we use "wave functions" to describe the probability distributions of position and momentum. But here's the clincher: The momentum wave function is the Fourier transform of the position wave function. This means that if we have a super wide position wave function (so it could be in a larger range of values, and thus we are more uncertain about it), then the momentum wave function is narrow (so it's in a smaller range of values and we are more certain about the momentum) and vice versa. This is where the Principle comes from - the more we know about position, the less we know about momentum, and vice versa once again.
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