Name: The more general uncertainty principle, regarding Fourier transforms
Uploaded: 2025-05-09T10:48:27.000Z
Duration: 19 min 21 s
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You've probably heard of the Heisenberg Uncertainty Principle from quantum mechanics, that the more you know about a particle's position, the less certain you can be of its momentum and vice versa.

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My goal here is for you to come away from this video feeling like this is utterly reasonable.

It'll take some time, but I think you'll agree that digging deep is well worth it.

You see, the Uncertainty Principle is actually one specific example of a much more general trade-off that shows up in a lot of everyday, totally non-quantum circumstances involving waves.

The plan here is to see what this means in the context of sound waves, which should feel reasonable, and then Doppler radar, which should again feel reasonable and a little bit closer to the quantum case, and then for particles, which if you're willing to accept one or two premises of quantum mechanics, hopefully feels just as reasonable as the first two.

The core idea here has to do with the interplay between frequency and duration.

And I bet you already have an intuitive idea of this principle before we even get into the math or the quantum.

If you were to pull up behind a car at a red light, and your turn signals were flashing together for a few seconds, you might kind of think they have the same frequency, but at that point, for all you know, they could fall out of sync as more time passes, revealing that they actually had different frequencies.

So an observation over a short period of time gave you low confidence over what their frequencies are.

But if you were to sit at that red light for a full minute, and the signals continued to click in sync, you would be a lot more confident that the frequencies are actually the same.

So that certainty about the frequency information required an observation spread out over time.

And this trade-off right here, between how short your observation can be, and how confident you can feel about the frequency, is an example of the general uncertainty principle.

The shorter it lasts in time, the less certain you can be about what its exact frequency is.

In the extreme, I could ask you what the pitch of a clap or a shockwave is, and even someone with perfect pitch would be unable to answer.

And on the flip side, a more definite frequency requires a longer duration signal.

Or rather than talking about definiteness or certainty, it would be a little more accurate here to say that the short signal correlates highly with a wider range of frequency, and that the signal correlating strongly with only a narrow range of frequencies must last for a longer time.

Here, that's the kind of phrase that's made a little clearer when we bring in the actual math.

So let's turn now to talking about the Fourier transform, which is the relevant construct for analyzing frequencies.

The last video I put out was a visual intuition for this transform, and yes, it probably would be helpful if you've seen it, but I'm going to go ahead and give a quick recap here just to remind ourselves of how it went.

Let's say you have a signal, and it plays 5 beats per second over the course of 2 seconds.

The Fourier transform gives a way to view any signal not in terms of the intensity at each point in time, but instead in terms of the strength of various frequencies within it.

The main idea was to take this signal and to kind of wind it around a circle.

As in, imagine some rotating vector whose length is determined by the height of the graph at each point in time.

Right now, this little vector is rotating at 0.3 cycles per second, that's the frequency with which we're winding the graph around the circle.

And for most frequencies, the signal is kind of just averaged out over the circle.

This was the fun part of last video, don't you think?

Just seeing the different patterns that come up as you wind a pure cosine around a circle like this.

But the key point is what happens when that winding frequency matches the signal frequency, in this case 5 cycles per second.

As our little vector is rotating around and it draws, all of the peaks line up on one side and all of the valleys on another side, so the whole weight of the graph is kind of off-center, so to speak.

The idea behind the Fourier transform is that if you follow the center of mass of the wound up graph with frequency f, the position of that center of mass encodes the strength of that frequency in the original signal.

The distance between that center of mass and the origin captures the strength of that frequency, and this is something I didn't really talk about in the main video, but the angle of horizontal corresponds to the phase of the given frequency.

One way to think of this whole winding mechanism is that it's a way to measure how well your signal correlates with a given pure frequency.

So remember, when we say the Fourier transform, we're referring to this new function whose input is that winding frequency, and whose output is the center of mass, thought of as a complex number.

Or more technically, it's a certain multiple of that center of mass, but whatever, the overall shape remains the same.

And the graph I'm drawing is just going to be the x-coordinate of that center of mass, the real part of its output.

If you wanted, you could also plot the distance between the center of mass and the origin, and maybe that better conveys how strongly each possible frequency correlates with the  The downside is that you lose some of the nice linearity properties that I talked about last video.

Anyway, point is, this spike you're looking at here, above the winding frequency of 5, is the Fourier transform's way of telling us that the dominant frequency of the signal is 5 beats per second, and equally importantly, the fact that it's a little spread out around that 5 is an indication that pure sine waves near 5 beats per second also correlate pretty well with the signal.

And that last idea is key for the uncertainty principle.

What I want you to do is think about how this spread changes as the signal persists longer or shorter over time.

You've already seen this at an intuitive level, all we're doing right now is just illustrating it in the language of Fourier transforms.

If the signal persists over a long period of time, then when the winding frequency is even slightly different from 5, the signal goes on long enough to wrap itself around the circle and balance out.

So looking at the Fourier plot over here, that corresponds to a super sharp drop-off in the magnitude of the transform as your frequency shifts away from that 5 beats per second.

On the other hand, if your signal was really localized to a short period of time, then as you adjust the frequency away from 5 beats per second, the signal doesn't really have as much time to balance itself out around the circle.

You have to change the winding frequency to be meaningfully different from 5 before that signal starts to balance out again.

Over on the frequency plot, that corresponds to a much broader peak around the 5 beats per second.

And that's the uncertainty principle, just phrased a little bit more mathematically.

A signal concentrated in time must have a spread out Fourier transform, meaning it correlates with a wide range of frequencies, and a signal with a concentrated Fourier transform has to be spread out in time.

And one other place where this comes up in a really tangible way is Doppler radar.

So with radar, the idea is you send out some radio wave pulse, and the pulse might reflect off of objects, and the time it takes for this echo signal to return to you lets you deduce how far away those objects are.

And you can actually take this one step further, and make deductions about the velocities of those objects using the Doppler effect.

Think about sending out a pulse with some frequency.

If this gets reflected off an object moving towards you, then the beats of that wave get kind of smushed together, so the echo you hear back is a slightly higher frequency.

Fourier transforms give a neat way to view this.

The Fourier transform of your original signal tells you the frequencies that go into it, and for simplicity, let's think of that as being dominated by a single pure frequency.

Though as you know, if it's a short pulse, that means that our Fourier transform has to be spread out a little bit.

And now think about the Doppler shifted echo.

By coming back at a higher frequency, it means that the Fourier transform will just look like a similar plot shifted up a bit.

Moreover, if you look at the size of that shift, you can deduce how quickly the object was moving.

By the way, there is an important technical point that I'm choosing to gloss over here, and I've outlined it a little more in the video description.

What follows is meant to be a distilled, if somewhat oversimplified description of the Fourier tradeoff in this setup.

The salient fact is that time and frequency of that echo signal correspond respectively to the position and velocity of the object being measured, which is what makes this example much more closely analogous to the quantum-mechanical Heisenberg Uncertainty Principle.

You see, there is a very real way in which a radar operator faces a dilemma, where the more certain you can be about the positions of things, the less certain you are about their velocities.

Here, imagine sending out a pulse that persists over a long period of time.

Then that means the echo from some object is also spread out over time.

On its own, that might not seem like an issue, but in practice, there's all sorts of different objects in the field, so these echoes are all going to start to get overlapped with each other.

Combine that with other noise and imperfections, and this can make the locations of multiple objects extremely ambiguous.

Instead, a more precise understanding of how far away all these things are would require having a very quick little pulse confined to a small amount of time.

But think about the frequency representations of such a short echo.

As you saw with the sound example, the Fourier transform of a quick pulse is necessarily more spread out.

So for many objects with various velocities, that would mean that the Doppler-shifted echoes, despite having been nicely separated in time, are more likely to overlap in frequency space.

So since what you're looking at is the sum of all of these, it can be really ambiguous how you break it down.

If you wanted a nice clean sharp view of the velocities, you would need to have an echo that only occupies a very small amount of frequency space.

But for a signal to be concentrated in frequency space, it necessarily has to be spread out in time.

You cannot have crisp delineation for both.

Do you know who else spent some time immersed in the pragmatic world of radio wave transmissions?

A young, otherwise philosophically inclined history major in WWI France, Louis de Broglie.

And this was a strangely fitting post, given his predispositions to philosophizing about the nature of waves, because after the war, as de Broglie switched from the humanities to physics, in his 1924 PhD thesis, he proposed that all matter has wave-like properties.

And more than that, he concluded that the momentum of any moving particle is going to be proportional to the spatial frequency of that wave, how many times that wave cycles per unit distance.

Okay, now that's the kind of phrase that can easily fly into one ear and out the other.

Because as soon as you say matter is a wave, it's easy to just throw up your hands and say physics is just weird.

Even if you're willing to grant that particles behave like waves, in some way, whatever that means, why on earth should the momentum of those particles, the thing we classically think of as mass times velocity, have anything to do with the spatial frequency of that wave?

Now being more of a math than a physics guy, I asked a number of people with deeper backgrounds in physics about helpful intuitions here, and one thing that became clear is that there is a surprising variety of viewpoints.

Now personally, one thing I found to be interesting was just going back to the source, and seeing how de Broglie framed things in his seminal paper on the topic.

You see, there is a sense in which it's not all that different from the Doppler effect, where relative movement corresponds to shifts in frequency.

It has a slightly different flavor, since we're not talking about frequency over time, instead we're talking about frequency over space, and special relativity is going to come into play.

But I still think it's an interesting analogy.

In his thesis, de Broglie lays out what is, in his own words, a crude comparison for the kind of wave phenomenon he has in mind.

Imagine many weights hanging from springs, with all of these weights oscillating up and down in sync, and with most of the mass concentrated towards a single point.

The energy of these oscillating weights is meant to be a metaphor for the energy of a particle, specifically the E equals mc2 style energy residing in its mass, and de Broglie emphasized how the conception he had in mind involves the particle being dispersed across all of space.

The whole premise he was exploring here is that the energy of a particle might have to do with something that oscillates over time, since this was known to be the case for photons, and these oscillating weights are just meant to be a metaphor for whatever that something might be.

With Einstein's relatively new theory of relativity in mind, he pointed out that if you view this whole setup while moving relative to it, all of the weights are going to appear to fall out of phase.

That's not obvious, and I'm certainly exaggerating the effect in this animation.

It has to do with a core fact from special relativity that what you consider to be simultaneous events in one reference frame may not be simultaneous in a different reference frame.

So even though from one point of view, you might see two of these weights as reaching their peaks and their valleys at the same instant, from a different, moving point of view, those events might actually be happening at different times.

Understanding this more fully requires some knowledge of special relativity, so we'll all just have to wait for Henry Reich's series on that topic to come out.

Right here, our only goal is to get an inkling for why momentum, that thing we usually think of as mass times velocity, should have anything to do with spatial frequency.