Understanding the Uncertainty Principle with Quantum Fourier Series | Space Time

Articles, Blog

Understanding the Uncertainty Principle with Quantum Fourier Series | Space Time

Understanding the Uncertainty Principle with Quantum Fourier Series | Space Time

Thanks to The Great Courses Plus for supporting PBS Digital Studios. Sometimes intuitive large-scale phenomena can give us incredible insights into the extremely unintuitive world of quantum mechanics Today, the humble sound wave is going to open the door to really understanding Heisenberg’s uncertainty principle and, ultimately, quantum fields and Hawking radiation. One of the most difficult ideas to swallow in quantum mechanics is Werner Heisenberg’s famous uncertainty principle. It expresses the fundamental limit on the knowability of our universe. We’ve discussed it in earlier videos on quantum mechanics But it’s time we looked a little deeper. See the apparent weirdness of the uncertainty principle hints at an even weirder underlying reality that gives rise to it. The universe we experience seems to be constructed of singular particles with well-defined properties, but this intuitive mechanical reality is emergent from an underlying reality in which the particles that form matter arise from the combination of an infinity of possible properties. And forget matter – the vacuum itself can be thought of as constructed from the sum of infinite possible particles. If we fully unravel this idea, we’ll be on the verge of tackling things like Hawking radiation. But as you’ll see today in that unraveling, we are led unavoidably to Heisenberg’s uncertainty principle. The uncertainty principle is most often expressed in terms of position and momentum We cannot simultaneously know both position and momentum for a quantum system with absolute precision Try to perfectly nail down a particles position, and we have complete uncertainty about its momentum And it’s not just because our measurement of position requires us to interact with the particle therefore changing its momentum No, the uncertainty principle exists alongside this observer effect. It’s instead a statement about how information we are ever able to extract from a quantum system To understand the origin of the uncertainty principle We don’t need to know any quantum mechanics At least not to start with, see, quantum mechanics is a type of wave mechanics – a very weird type. However, it turns out that something like the uncertainty principle Arises in any way of mechanics, so let’s choose a type of wave that’s a little more intuitive – sound waves. You can describe a sound wave just as the intensity of the wave as it passes by, so intensity changing over time. It can take really any shape, that shape determines what the wave sounds like to our ears. The sound wave for a simple pure tone, like a middle C, is a sinusoidal wave with the frequency determining the pitch of the ton. The sound wave from, say, an orchestra is extremely complex, but, amazingly, it can always be broken down into a combination of many simple sine waves of different frequencies. This is Fourier’s theorem, after French mathematician Jean-Baptiste Joseph Fourier. It states that any complex sound wave can be decomposed into a number of sine waves of different frequencies, each with a different strength, stacked on top of each other or superposed. In fact instead of representing a sound wave in terms of intensity changing with time, you can also represent it in terms of its frequency components: each with its own weighting or strength. When you switch between a time and a frequency representation you’re doing a Fourier transform. In fact digital audio equipment stores, manipulates, and transmits sound in its frequency representation. In the physics of sound, time and frequency have a special relationship, because any sound wave can be represented in terms of one or the other: we call them Fourier pairs. Also, sometimes, conjugate variables. Okay, so we can make any shape sound wave with a series of sine waves of different frequencies. For example, you can build a wave packet by adding frequency components with the right phases to destructively interfere everywhere, except within a small region. The tighter you want to make that time window for the wave packet, the more frequency components you need to use. In fact, to get those steep edges of the wave packet you need to add higher and higher frequencies, because the high frequency components are the ones that give you rapid changes in intensity. So, what if you try to compress the wave packet to a single spike, a blip of sound, that exists for only one instant in time ? Is it even possible to make an instantaneous spike at one point in time, out of a bunch of sine waves that themselves extend infinitely through time. In fact, it is. However, to get a spike at one point in time you need to use infinitely many different frequency sine waves, each of which exists at all points in time. So then, if we make a sound that is perfectly located in time it doesn’t have a frequency or it has all frequencies. At the same time, a sound wave with a perfectly known frequency is a simple traveling sine wave that extends infinitely in time, so the time of its existence is undefined. That sounds an awful lot like a frequency time uncertainty principle for sound waves. Now, it’s not really a statement about the fundamental knowability of a sound wave as is Heisenberg’s uncertainty principle. It’s more a statement about the sampling of frequencies needed to produce a given wave packet. But the underlying idea is the same. So, how does this relate to the quantum world? Well, before we get back to quantum fields let’s think about the wave function – the solution to the Schrodinger equation that contains all of the information about a quantum system. Like the sound wave, it oscillates through space at a particular frequency. To keep things simple we’re just going to consider a wave function that doesn’t vary in time, it only changes with position in space. This is more like a standing sound wave inside an organ pipe, rather than the traveling sound wave from earlier. So, position, rather than time, becomes the first of our Fourier pair. The second of this pair is momentum, not frequency. See, momentum is sort of the generalization of frequency for what we call a matter wave. In the early days of quantum mechanics it was realized that photons are electromagnetic wave packets, whose momentum is given by their frequency. Louis de Broglie extended this idea to particles, and his de Broglie relation generalizes the relationship between frequency and momentum of a matter wave. We now call matter waves wave functions, and we can describe them in terms of position or momentum, just as a traveling sound wave can be expressed in terms of time or frequency. So, any particle, any wave function can be represented as a combination of many locations in space with accompanying intensities. Think of it as the particle being smeared of possible positions, or as a combination of many momenta with accompanying intensities, in which case the particle would be smeared in momentum space, and, of course, this means that position and momentum have the same kind of uncertainty relation that time and frequency had in the sound wave. But, what does it even mean for a particle to be comprised of waves of many different positions or momenta ? To answer this we need one more bit of physics – the interpretation of the wave function itself, known as the Born rule. The magnitude of the wave function squared is the probability distribution for the particle. If we’re expressing the wave function in terms of position, then applying the born rule tells us how likely we are to find the particle at any given point when we make a measurement. Or, put another way, the range of positions in which the particle is likely to be located were we to look. If we apply the born rule to the momentum wave function, then we learn the range of momenta the particle is likely to have. So, if we measure a particles position then from our point of view its wave function is highly localized in space: we know where the particle is. The resulting particle wave packet, now constrained in position, can only be described as a superposition of waves with a very large range of different momenta, via a Fourier transform. The result is a very fat momentum wave function that gives a wide range of possible momenta. The more precisely we try to measure the position, the narrower we make its position wave function and, so, the less certain we become about its momentum as that momentum wavefunction gets wider. This is all super abstract, but a concrete example is single slit diffraction. If we increase our certainty of the position of a particle by narrowing the slit we also increase the uncertainty of its momentum, as it passes the slit. This results in an increasing spread in final locations. Check out Veritasium’s excellent video to see this in action. So, that’s exactly the uncertainty principle. It’s a statement, about how much of a quantum systems information is accessible at a fundamental level. It’s an unavoidable outcome of describing particles as the superposition of waves, waves that can be represented in terms of either position or momentum. The fact that both can’t be known simultaneously with perfect precision is a property of the nature of the wave function itself. Precision in one is actually constructed by the uncertainty in the other. OK, so what does this old-school quantum mechanics have to do with quantum field theory and Hawking radiation? Well, the key to understanding these things is to be able to switch between thinking about quantum fields in terms of position versus momentum. See, a single particle, a quantum field vibration perfectly localized at one spot in space can also be described as infinite oscillations in momentum space, spanning all possible momenta. But each of these oscillations in momentum space are equivalent to particles with highly specific momenta. The uncertainty principle therefore tells us, that they must be completely unconstrained in position. So, a perfectly specially localized particle is equally an infinite number of momentum particles that themselves occupy all locations in the universe. It’s only by manipulating quantum fields in this strange momentum space, by adding and removing these spatially infinite particles, that we can describe how the quantum vacuum changes to give us a phenomena, like Unruh and Hawking radiation, which you’ll soon understand as some of the weirdest behaviors of space-time. Thanks to the Great Courses Plus for supporting PBS Digital Studios. The Great Courses Plus is a digital learning service that allows you to learn about a range of topics from Ivy League professors and other experts from around the world. Go to www.thegreatcoursesplus.com/spacetime, and get access to the library of different video lecture about science, math, history, literature, or even how to cook, play chess or become a photographer. New subjects, lectures and professors are added every month. Now, our recent discussions about the quantum world are leading up to some pretty mind-blowing episodes. To help prep yourself even better, you could check out Benjamin Schumacher series – Quantum Mechanics, which includes a great episode on the uncertainty principle. Help support the series and start your free one-month trial by clicking on the link below or going to thegreatcoursesplus.com/spacetime OK, so, I traveled a bit over the past two weeks until we missed one of our comment responses. Today we’re gonna cover our episode on robots that sacrifice themselves for science, as well as our episode on citizen science. First, thanks to everyone who pointed out the editing error at the end of the suicide robots episode, where we accidentally included both of the two takes of the last line. Totally unintentional But, hey, it was a pretty good line right. Also, thanks to everyone who noticed my mispronunciation of Enceladus in that episode. You know those words you’ve never heard said that have been pronouncing wrong in your head forever. Yeah, this isn’t one of them. I think I was just recording through lunchtime and really craving enchiladas. A few of you who’ve been involved in citizen science projects that we didn’t cover. One of my favorites is the exoplanet search program embedded in the EVE Online game. Players killed time during warp journeys by scanning light curves of distant stars for the characteristic dips in brightness, due to transiting alien planets. sakurasleight suggested, that [email protected] uses computing cycles to mine Bitcoin instead of look for alien signals. A couple of you pointed out a better alternative, the cryptocurrency, GridCoin, which you mine by dividing computing cycles to BOINC research programs Seems a bit more useful than Bitcoin’s useless cryptographic calculations. Daniel Soltesz reminded me of the coolest object found in Galaxy Zoo, discovered by a Dutch school teacher Hanny van Arkel. Hanny’s Voorwerp Is a weird blob of light right next to a spiral galaxy. It’s hypothesized to be the light echo from a dead quasar, that was once in that galaxy, so the cloud of gas ionized by the last burp of energy from an active supermassive black hole in the middle of the spiral galaxy just before it ran out of food. It is the first of its kind discovered. Regarding our zero-point challenge answer Jaden Andrews asked how to prevent the geckos tail falling off when you harness them for wall climbing. Well, Jaden, the trick is to assume geckos with infinitely high tensile strength. As a couple of you pointed out, we already assumed the geckos of zero mass, so who’s to say those rare zero mass geckos don’t have infinite tensile strength. Perhaps you just haven’t tried enough geckos.

100 thoughts on Understanding the Uncertainty Principle with Quantum Fourier Series | Space Time

    For a free copy of my books (they are public domain), please visit: https://wildcard72.wordpress.com/books/ and it goes along the lines of this song: https://www.youtube.com/watch?v=RIGgn1s3AvI

  2. A gedunkenexperiment plus Heisenberg's Uncertainty Principle proves why it is impossible for a macroscopic system to travel backwards in time. Consider: suppose we message to great precision the location x of a particle. Then we go back time to do the same for the particle's momentum, measuring the momentum with great precision. Then we combine the results and voilà! We have for the same time measured position and momentum to a precision that exceeds that which Heisenberg's Uncertainty Principle allows!

  3. I’m sure u get this all the time but thank you for putting in the time and energy to explain this stuff so well. I couldn’t handle college just bc of all the jumping through hoops for certain grades and unnecessary classes, but with the knowledge I’ve gotten from channels like yours my craving for new information is always satiated. I wish I had the endurance to put up with college but bc of people like you my love of learning doesn’t require thousands of dollars and 10 college success or financial preparedness classes

  4. So position is tied to space and momentum is tied to time
    Momentum = Mass x [Velocity;Distance/Time] (p=mv)

    Velocity is distance over time but mass compresses time…

  5. Is everything infinite? Infinite big, infinite small, always smaller particle discovered, earth, solar system, milky way, universe, multiverse of universes, multiverse of multiverses and so on?

  6. But does this mean that because the apprehend size of a particle is constrained by the size of the universe and its therefore the possible variation of waves that its wave packet is represented by? Were our particles less defined in the smaller earlier universe? I love this show!

  7. Look, we are made of molecules, which are made of atoms, which is made of subatomic particles then at last the fundamental elementary particles. So, if the fundamental particles are having a defined momentum and a uncertain position then the atoms location is uncertain so are we. Then how are we able to map movements of an aircraft or a vehicle. You can visibly see it, when it has momentum you are not looking at fuzzy ball of uncertainty you are looking at a solid object.

  8. If a particle is known to move in a straight line in a vacuum, does that mean that consecutive near instantaneous measurements of the momentum will yield the same direction? Another question, if you measure the momentum of a particle, then (as near as possible to the next instant) measure its velocity, are you obtaining a close approximation for both?

  9. The simplest way to understand the uncertainty principle is that to measure momentum you need two things, mass and velocity, if you know the position with 100% precision then you can't get the velocity at all, because the particle as far as you're concerned is stationary, and since you need two spacial points to measure velocity, you're missing one.

  10. When a particle behaves like a wave then its impossible to find its position and momentum at same time, but when electron behaves like a particle being observed, can we then calculate its momentum and position?

  11. You really derailed too much into sound waves for too long. I get you were trying to use sound waves as a premise but just got me lost in your trail of thought.

  12. in classical cimistry, the objects have a quantum gap, and inside the quantum gap you have a "infinity" probably and "uncertainty"

  13. And here I was thinking that learning Fourier Transform in Uni was gonna be completely useless

  14. Richard Feynman a Nobel Laureate and contributor to quantum mechanics said, "I can safely say, no one understands quantum mechanics."

  15. We know that units of both momentum and energy contains mass then I don't understand how they are defined for any massless species like photons??

  16. This is an invitation to see a theory on the nature of time! In this theory we have an emergent uncertain future continuously coming into existence relative to the spontaneous absorption and emission of photon energy. The future is unfolding with each photon electron coupling or dipole moment relative to the atoms of the periodic table and the wavelength of the electromagnetic spectrum. This is part of a universal process of energy exchange that forms the ever changing world of our everyday life.

  17. Excelent as always.
    A little remark. Fourier analysis is valid for any periodic function, not just sound waves. IE: any function can be expressed as an infinite sum of weighted sinusoidal functions. In fact Fourier original work was related with heat propagation, not sound waves. I'm sure you know all this, but just for clarification.
    Thank you very much. I have learn a lot from this channel.

  18. A police man stops a physicist
    – Sir, did you know how fast you were going?
    – Yes, it was way too fast. I'm really sorry
    – Did you know that this is the exact stop where an old lady the other day got run over by a speeding driver?
    – Is that here? Then I have no idea how fast I was going.

  19. Pretty good. At least a better understanding than most physicists, but a more accurate explanation is that without the principle what would the particle be made of? It's got to be made of either position or momentum uncertainty in order to be be measured or even described as a pure particle/momentum and not a wave function. This principle maintains the principle of energy conservation. We can't have one without the other (uncertainty that is).

  20. A cop pulls over a physicist and asks, "do you know how fast you were going?"
    The physicist replies, "no, but i can tell you exactly where we are."
    The cop says, "you were going 75 miles per hour."
    The physicist throws his hands in the air, and cries, "oh great, now we're lost."

  21. Back to watching this for the second time. Making even more sense. Audio is a great way to explain the uncertainty principal. Was using it the other day

  22. Hey, how do we know particles are in superpositions without measuring them yet? Like, if we measure they collapse, but how do we know that was the moment they collapsed?

  23. Another excellent video. The electron is described as a “fundamental “ particle with no internal structure and a point mass. However, when you look at it from a Fourier construction, in a certain sense it does have an internal structure, a wave structure. When you construct the wave packet, the electron is constrained to be “in” the wave packet somewhere. Anyway, the question I have is that when you fire the electron out of your electron “gun”, the electron gun recoils (conserving momentum) and the electron travels in the other direction, although it is actually a construction of “momenta “. When the electron travels thru the slit, and changes direction, how is momentum conserved? Does the electron exchange momentum with the slit? If the electron was only comprised of one momentum, it probably would not interact with the slit, just pass thru. But since the electron is comprised of a construction of momenta, it does interact with the slit. However, wouldn’t it still need to “exchange” momentum with the slit?

  24. So, the metaphor I got from that graph is that the area covered by the known values and uncertainty is constant regardless of how well we know either value

  25. Hello, I will copy my question from another video, hoping it might get some responses here.

    Edward Teller, on measurements and observers:
    "A measurement is defined by an irreversible process which does not allow an original state to be reconstructed from the final state".

    Hasn't every particle in existence at one time or another been "measured" in such a sense. When electron's position is measured in its travel from the initial state to its final state, can it ever reverse to be "unmeasured", and if yes under what conditions.

    To be more to the point, if one measures the position of an electron in the atom at one time, and then walk outs of the room, then another person comes in and revisits the same atom. Has the electron's wave function forever been "collapsed" and it no longer exists as a probability cloud in that atom from now on, from when person no. 1 measured it – so that to person no. 2 electron is no longer a probability cloud, a wave function but something else then, a real god-given particle? If to a person no. 2 that electron is still simply a probability cloud, at what point did the electron become "unmeasured"?

    Thanks, whoever, whenever.

  26. A particle has an exact position, velocity, and energy at a given moment. Its just that we observers cannot know those things using current technology. Our current observation methods impart energy which changes the thing being observed. Why does quantum mechanics equate our inability to measure these things with the fundamental nature of reality? Our ignorance of the full nature of a particle in no way influences the nature of that particle.

  27. If electron behaves as wave means will all mass of electron converts into energy and propagate in the form of wave.Can u plz clarify my dought

  28. After watching this many times and thinking about hup as well as the plank equation I'm not convinced this is an accurate interpretation either. It seems hup is a consequence of describing particles and waves simultaneously as statistical phenomenon. Some information is unknown because the nature of the broad description of statistical representation.

  29. "Uncertainty principle" is an oxymoron. Heisenberg may have identified a lack of principle which implies either that there are natural forces unknown or an inherrent chaos.

  30. can someone help me with this math problem? Anyone who does will get a follow, a like, and a bunch of comments. I just can't hold it in anymore.

    (17661.9478 *5.1974632+q/e² ) – 76.4685/t = (Sq/d xr²)/((5.19)^3xn¹n²)-9.547±.000147)+(xyz)*(stfu)ž­‡™

  31. No links to Veritasium or the Benjamin Shoemaker's series (I'm guessing based on his pronunciation of shoemaker, I'm spelling that wrong.)

  32. What if, Alice measured the y direction of particle A at 9am, and Bob measured the x direction of particle B right at the exact same time as Alice, which is also 9am. Of course A and B are entangled. After that, Alice walk to Bob, tells Bob about the result. So, at this moment, Alice and Bob knows the exactly both x and y directions of particle A and B at 9am, so the uncertainty principle broken?

  33. I'm not sure about the sound wave analogy here. Sound waves propagate through a medium of particles, and the localized distortion or turbulence of the particles would be visible if time froze, relative to particles that weren't "carrying" the wave. Right?

  34. What is the best book for reading… Quantum Mechanics and theory of relativity for undergraduate…… Pls?……

  35. Ok. So If any single measurement cannot know both speed and position at
    the same time, then why not have 2 individuals measure at the same time.
    One measures the speed and the other measures the position-at the same
    time. Problem solved.

  36. …nope, I still don't understand how a (hypothetical) particle that is nailed to a specific location in space, and thus is perfectly still, could possibly have any momentum other than zero.

  37. Your video reeeeeeeeeeeally helps me a lot to understanding the QED and even Fourier transform, thaaaaaaanks!

  38. Quantum mechanics has become so difficult, even mathematicians don't get it. Where is the Einstein to lead us out of this partly Einstein created quagmire?

  39. The statement about digital audio being represented in the frequency domain isn't true. Simple time domain representations are heavily used (CD audio, WAV files, and the signal paths through studio equipment are like this), and any frequency domain representations used in compression or (re-)synthesis must be windowed (in some slightly generalised sense) to allow for finite latency! Indeed, when translated back into the audio domain, this was the topic of the episode, when translated back into the audio domain.

  40. The HUP illustrates how the identity of energy and information become conflated in extreme situations. Time stops at the event horizon of a black hole, and photons don't experience the passage of time, because time is never what you think it is, until that time rolls round again. The obvious implication is that 42 is as good as it gets, explaining the failure of modern physics to produce any significant progress in half a century.

  41. He looks like a newly arrived Syrian refugee so his explanation voice and videos
    2nd thing most of their particle and wave physics was not usefull in developing one electronic product or sensors it was development of let's say "spectral analysis of different metals in different organic compound and electromagnetic current or voltage spectrum . And some smart design engineers for example like developing smart processor design after developing more small but more powerfully semiconductor and and Heisenberg uncertainty principle "god is not uncertain about anything in universe each has a preplanned fixed pattern most of the time space and time position where he don't want to get mad shaking particle and wave all the time which no one but only he is observing at that time.

  42. I think the biggest problem with this show is that the thumbnail looks like it will have awesome CGI but it's just some nerd talking for ages.

  43. The wave particle duality can explain many phenomena if all subatomic particles oscillate between matter and energy at the speed of light with a cycle time of zero.

    John Andrew [email protected]

  44. Do we know for a fact that the particle itself moves rather then having the appearance of moving like a pixel in a tv??

  45. How can the human body or any system be explained by quantum physics we don't just disappear and re appear like any other matter?!

  46. This guy gets me more and more confused every time. Make your video and concept different and simpler. I've seen other people explaining everything you said and understood them. You don't really explain anything. Bad teaching.

  47. Anyone else having trouble focusing on the subject when your faced with the problem of how large can a human head really get?

Leave a Reply

Your email address will not be published. Required fields are marked *