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 SETI@home 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 point 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.