The more general uncertainty principle, beyond quantum

You've probably heard of the Heisenberg uncertainty principle from quantum mechanics that the more you know about a particles position the less certain you can be sure of its momentum and vice-versa 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 that 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 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. Similarly think of a musical note 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 shock wave 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 little more accurate here to say that the short signal correlates highly with a wider range of frequency And at 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 bit 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 gonna 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 five beats per second over the course of two 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 were 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 five 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 that center of mass the horizontal corresponds to the phase of the given frequency Now 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 were 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 that 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 signal 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 five is the Fourier transforms way of telling us that the dominant frequency of the signal is five beats per second and equally importantly the fact that it's a little bit spread out around that five is an indication that pure sine waves near five 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 five 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 five 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 five 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 five before that signal starts to balance out again over on the frequency plot that corresponds to a much broader peak around the five 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 fornia 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 that 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 going to be 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 trade-off in this setta? The ceiling in fact is that time and frequency of that echo signal? Correspond respectively to the position and the velocity of the object being measure 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 and 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 gonna 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 this is the Fourier trade-off you cannot have crisp delineation for both and This brings us to the quantum case 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 World War one 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 year and out the other Because as soon as you say matter is a wave It's easier to just throw up your hands and say physics is just weird But really think about this 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 debris frame 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 dubrow 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 mc-squared style energy residing in its mass and debris 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 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 Einsteins 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 Rice's series on that topic to come out Right here our only goal is to get an inkling for y momentum that thing we usually think of as mass times velocity should have anything to do with spatial frequency and The basic line of reasoning here is if mass is the same as energy via e equals mc-squared And if that energy was carried as some kind of oscillating phenomenon similar to how it is for photons then this sort of Relativistic Doppler effect means changes to how that mass moves corresponds to changes in the spatial frequency So what does our general 4ei trade-off tell us in this case? Well if a particle is described as a little wave packet over space then the Fourier transform Where we're thinking of this as a function over space naught over time tells us how much various pure frequencies correspond with this top wave So if the momentum is the spatial frequency up to a constant multiple the momentum is also a kind of wave Namely some multiple of the Fourier transform of the original wave So if that original wave was very concentrated around a single point as we have seen multiple times now that means that its Fourier transform must necessarily be more spread out and Hence the wave describing its momentum must be more spread out and vice versa Notice unlike the Doppler radar case where the ambiguity arose because waves were being used to measure an object with a definite distance and speed What we're saying here is that the particle is the wave? So the spread out over space and over momentum is not some artifact of imperfect measurement techniques It's a spread fundamental to what the particle is Analogous to how a musical note being spread out over time is fundamental to what it even means to be a musical note One pet peeve I have in mainstream references to quantum is that they often treat Heisenberg's uncertainty principle as some? Fundamental example of things being unknowable in the quantum realm as if it is a core nugget of the universe's indeterminacy But really it's just a trade-off between how concentrated a wave and it's frequency representation can be Applied to the premise that matter is some kind of wave and hence spread out All of the stuff about randomness and unknowability is still there But it comes one level deeper in the way that these waves have come to be interpreted You see when you measure these particles say trying to detect if it's in a given region Whether or not you find it there appears to be probabilistic where the probability of finding It is proportional to the strength of the wave in that region, so When one of these waves is concentrated near a point what that actually means is that we have a higher? probability of finding it near that point that we are more certain of its location and Just to beat this drum one more time since that concentration implies a more spread-out Fourier transform Then the wave describing its momentum would also be more spread out so you wouldn't be able to find a narrow range of momenta that the particle has a high probability of occupying I Quite like how if you look at the German word for this principle it might be more directly translated as the unsharp ness relation Which I think more faithfully captures the Fourier trade-off at play here without imposing on questions of no ability When I think of the Heisenberg uncertainty principle what makes it fascinating is not so much that it's a statement about randomness I mean yes That randomness is very thought-provoking and contentious and just plain weird but to me equally fascinating is underpinning Heisenberg's conclusion is that position and momentum have the same relationship as sound and frequency as If a particles momentum is somehow the sheet music describing how it moves through space The channel is made possible primarily by the direct contributions from the community and to finish things off here I've got a little recruiting pitch and a math puzzler from this video's main sponsor the art of problem solving and Before getting to that puzzle I just want to say that this organization was actually very influential on me growing up a Teacher of mine early on give me one of the first books that Richard Rusick founder of art of problem-solving had written And it really opened my eyes to a new side of math at the time I recently had the pleasure of getting to meet with Richard and we were talking about finding talented people to work with since part of problem solving now creates all kinds of math resources and I mentioned how sometimes I'll do these recruitment oriented sponsorships since a good number of three blue one brown viewers work in tech And I'm sure a lot of you would be interested in an organization dedicated to putting out rock-solid math explanations And well here we are if you are interested in working with them Or just learning more go to AoPS.com/3b1b No joke if I wasn't doing this channel this would be one of the top places that I'd look We also wanted to share this little puzzle Which I've left up on the screen because hey we're all math loving people here who enjoy solving problems And if you're curious about the solution to that you can find it at the link, I described

Channel: 3Blue1Brown

Views: 1,213,359

Rating: 4.9565406 out of 5

Keywords: three blue one brown, 3brown1blue, fourier, three, Mathematics, blue, uncertainty principle, quantum mechanics, physics, 3b1b, one, brown, 3 brown 1 blue, three brown one blue, de Broglie, 3 blue 1 brown

Id: MBnnXbOM5S4

Length: 18min 7sec (1087 seconds)

Published: Sat Feb 24 2018

Reddit Comments

[u/the_quassitworsh]

i’m a biochem person, not a physics/math person, and this video was really awesome. i’m sure it wasn’t totally in depth but i feel like i have a much better idea of how this works now. thanks for sharing

[u/ozaveggie]

Good video overall. I wonder if the balls on springs model that he used to motivate position and momentum being fourier transforms of each other was the best way to present the material. Clearly the 'standard' physics way has to do with conjugate variables, generators of translations & what not is kind of technical but I personally didn't think the balls on springs thing was very understandable/enlightening. But then again I am not the target audience so I don't know how others received it.

[u/QQwas]

Equally fascinating, is that underpinning Heisenberg's conclusion, is that position and momentum have the same relationship as sound and frequency. As if a particle's momentum is somehow the sheet music describing how it moves through space.

Man, that is some poetic stuff right there.

[u/Wiscansan]

For anyone looking to learn Calculus, Linear algebra, and a few abstract mathematical concepts. This guy is impossible to beat.

[u/Bondle]

Saving this for later. A must see for me.

[u/Plant2Pipe]

i cant believe he went into the physical side of the fourier trasnform without talking about a 4f correlator. using waves to do the math that we use on waves