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How Does Quantum Uncertainty Work?
🕑 Added 2022-12-10 13:00:07 +0000 UTC[This is a transcript of the video.]
If you know one thing about quantum mechanics, then that’s more than most people. And it’s probably that it’s got something to do with the uncertainty principle. But just exactly what does that mean and how does it work? That’s what we’ll talk about today.
Werner Heisenberg was the first to understand that quantum mechanics sets limits to what we can know. Yes, even for string theorists. Some properties of quantum particles can’t be determined precisely at the same time. And it’s not because we’re not good at measuring, it’s just how nature is.
The best known example is the position and momentum of an object. You can think of the momentum as the velocity multiplied with the mass. This isn’t correct for massless particles, but for our purposes it’ll be good enough. Let’s just pretend that massless particles don’t exist, it shouldn’t be too hard.
Quantum mechanics says that you can’t measure both the position and momentum precisely. This means, if you know precisely *where a particle is, you don’t know which direction it’s going. And the other way round, if you know which direction it’s going, you don’t know where it is.
The position of a particle is usually denoted with x and the momentum with p. The uncertainty in each is marked with a large Delta in front of the variable. So Delta x is the position uncertainty and Delta p is the momentum uncertainty. Heisenberg’s uncertainty principle then says that the product of those two is always larger than a constant, h-bar, divided by two.
This means, if one of those uncertainties gets very small, then the other one must be very large, otherwise the inequality would be violated. This is where the joke about Heisenberg comes from where he gets stopped by police who tell him how fast he was driving, and Heisenberg says “Great, now I’m lost!”
But what does it mean that you can’t measure the position and momentum with certainty at the same time. Does it mean if you come with two detectors, a lightning strike will explode one of them? Well, no, the uncertainty principle doesn’t say that you can’t *measure both of those quantities. It’s just that you can’t measure both precisely. If you’ve pinned down one, then the other will have a large spread. This means if you repeat the experiment, the result will be all over the place. It’s the statistical uncertainty of this series of measurements that becomes large.
A good example of this is letting quantum particles go through a narrow opening. The quantum particles act like waves, so this will create an interference pattern on the screen. This pattern is built up of single particles; the particles don’t all collect in one place. And that’s because you know the particles went through the slit, so you know something about their position. Then you can’t tell which direction they are going. But of course, once a single particle hits the screen, you know which direction it went. What the uncertainty principle means is therefore that if you repeat this for many particles, then the outcome has a broad distribution.
Another example is the relation between the uncertainty in energy and time. Consider you have a fluorescent sticker, the kind of stuff that glows in the dark. You put it into light and then, for a few hours afterwards, it will emit light of a particular color, usually it’s some shade of green.
The way this works is that the light which falls on the sticker kicks an electron from its normal energy level into a higher level. In most materials, the higher electron levels are extremely unstable. The electron falls back down immediately, and in that process the photon is emitted again. In fluorescent materials, however, the electrons can stay there for a few hours. It’s called a “meta-stable” state. And only when the electron falls down will the light be emitted again. As a consequence, the sticker glows for some hours.
But just exactly how long do the electrons stay in that higher energy level? Ah, well, you can’t say exactly. You can only talk about the *probability of it dropping down in a certain amount of time. And why is that? It’s because you know something about the energy of the electron. It’s sitting on this meta-stable level. Consequently you can’t know exactly how long it stays there. Such is life, you can’t have it all. This is basically also how nuclear decay works by the way.
Formally, the uncertainty principle has its origin in the properties of waves. It comes about because in quantum mechanics particles can act like waves. What’s a wave? A wave is something that changes periodically in space and time. That could be water waves, electromagnetic waves, sound waves or even a Mexican wave when a sports game gets boring.
Waves have a wavelength, and a frequency that’s inversely proportional to that wavelength. The frequency tells you how much change there is in a certain period of time. If you have a perfectly regular soundwave, then it would sound like a beep at one particular frequency.
But most sound signals aren’t that simple. For example, here is me singing an a. This isn’t just one frequency. You can take this sound apart and you will find that, for one thing I’m not a computer chip, so I don’t exactly hit the a. But also, the way the human voice box works, it has a lot of higher harmonics. This breakdown of sound into all the frequencies it contains is called a spectrum.
This doesn’t only work for sound but for every distribution of anything, really. You can calculate a spectrum for the distribution of people in a room if you want to and I’m sure somewhere there are physicists doing just that. But a more relevant example for us is light. If you have light of one particular wavelength, then that would be one particular color, and the spectrum of this light would have one single peak at one particular frequency.
Technically, for the spectrum to be exactly one frequency, the light-wave would have to keep on waving forever. We never deal with that in reality. In reality, the light we deal with is bunched. It passes by within a certain window of time. First it wasn’t there, then was there, now it isn’t there again. Like money in my bank account.
Physicists call this a “wave-packet”. How do you pack waves? Especially without getting wet. The curious thing about waves is that they can interfere both constructively and destructively. If they interfere constructively, they add up. If they interfere destructively, they can cancel each other out. If you overlay many of them just the right way, you can do this so that outside of some region of space and time, the interference is almost completely destructive, and you are left with a constructive interference just in a small region. A wave-packet is an overlay of different wave-lengths and therefore an overlay of different frequencies, with the correct offset.
Such an overlay of waves by the way is called a superposition. Sounds familiar? Yes, we do use superpositions in quantum mechanics. And this is where the word comes from. It comes from adding different wavelengths together. You super pose them.
Okay, now here’s the thing. As we just saw, we have two ways to describe the same wave-packet. You can either say, it’s this distribution in time. Wasn’t there, was there, isn’t there again. Or you can say, it’s this distribution of frequencies with these offsets. They both describe the same thing, the same wave-packet.
There is a mathematical procedure to calculate one from the other. It’s called a Fourier transformation. Put in a distribution in time, the Fourier transformation will give you the spectrum. Put in the spectrum, the Fourier transformation will give you the distribution in time.
Here’s the kicker. If the one distribution is sharply peaked, say you have a flash of light that’s just a brief moment in time, then that means that it must contain a lot of different frequencies; the spectrum is very spread out. And the opposite is also true, if you have something with a very narrow frequency, then it must be spread out in space.
Now you can ask what’s the relation between the spread in those two distributions, the one in time and the one in frequency. And would you know it, they’re related to each other by an inequality that looks pretty much like the uncertainty principle.
Whoo! But wait, this had nothing to do with quantum mechanics, we were just talking about doing a spectral analysis, what is going on?
What’s going on is that while this looks like the uncertainty principle, it’s not. This is a property of waves. It relates a spread in time with a spread in frequency. To get Heisenberg’s uncertainty principle you further have to identify a frequency with an energy. Or, if you do it in space, you have to identify the wave-vector with the momentum. And this is where the hbar gets in. We learned this from no other than Albert Einstein. Yes, that guy again. He pops up more than toast in a student house.
It's actually what he won the Nobel Prize for. No, seriously. Albert Einstein didn’t win the Nobel Prize for Special or General Relativity. He won it for putting hbar in the right place. And the way he figured it out was from an observation called the photoelectric effect.
If light shines on a metal plate, that can kick electrons out of the plate. One can measure this because it creates a current, so it’s electric. This is why it’s called the photo-electric effect. But for this to work, the frequency of the light must be above a certain threshold. If the frequency stays below the threshold but you increase the brightness of the light, nothing happens.
This was very confusing to scientists in the 19th century. They could understand that it takes energy above a certain threshold to kick out an electron. But if you think that light is a wave, then it’s the brightness that measures this energy. So why would turning up the brightness not kick out electrons?
Along came Einstein, and he said that’s because light is made of chunks of energy. We now call those chunks photons. To kick an electron out of an electron band, the energy of a single photon has to be above a certain threshold. The total energy of the light is the sum of the energy of all those photons. But the energy in each photon is proportional to the frequency. Therefore, if you want to know the total energy of the light you take the number of photons and multiply it with their frequency and now you need a constant to convert the frequency into an energy. That constant is Planck’s constant, h.
And the h with the bar that’s called the reduced Planck’s constant and is h divided by 2 times pi. It’s more commonly used in quantum mechanics because, like my uncle, it makes a lot of pi’s disappear, so it’s more convenient. Why is it called Planck’s constant and not Einstein’s constant? Ah, good question, but I’ll leave this for another video.
Let’s instead step back here and ask what this all means now. We have seen that what’s so seemingly weird about the uncertainty principle, that there are two quantities that you can’t measure precisely at the same time, has really nothing to do with quantum mechanics, it’s just a property of waves. What quantum mechanics does for you is to make this a property of particles. And for that to work, you have to relate time with energy, and space with momentum. If you’re uncertain about any of this then you’re doing it right.
We have also seen that the uncertainty principle doesn’t mean you can’t measure those quantities precisely in one round of an experiment, but that it means if you repeat the experiment, you get a large statistical distribution for at least one of those quantities.
But then couldn’t it be that the quantum particle actually *had this property before it was measured, you just didn’t know what it was? Rather than doing this weird thing where it’s in several states at the same time until you look at it and, poof, it decides on one? Yes, indeed, that’s what is called a hidden variables model, and it’s how Einstein thought quantum mechanics really works. He didn’t like spooky action at a distance and he was certain about his principles.
Comments
Thanks for that link to that series of talks. I'm starting at 1.
Hi Jeffery, I talked about the bra-ket notation long ago here: https://www.youtube.com/watch?v=ctXDXABJRtg The big Delta is is used both for a difference and for variance. If you are referring to a change it's more common to use a small delta I would say, but these conventions differ by community.
Rad Antonov
If I may, I’ll throw one more paper in the mix. It too argues against the Casimir effect as evidence for vacuum energy, but has physics in it, as opposed to the purely mathematical approach from Nikolic. The author is interested in the cosmological constant and although he shows an alternative derivation of the Casimir effect is possible, he concedes that it may be impossible to formulate quantum mechanics without zero point energy. It helps to see both sides of the argument: https://arxiv.org/pdf/hep-th/0503158.pdf#page12
Rad Antonov
🤣 or golf, good one and my pleasure, I learned some things looking into it.
Thanks. I will read Birula's "Photon Wave Function" paper.
Thank you for the link. 260 pages ... It may take me a while ... I will also read the proof you critiqued, it is much shorter 14 pages. Both are bit over my head ... I can read math, but I cannot do math. It is kind of like being a fan of soccer.
Rad Antonov
The cited “proof” claims that the force “does not follow directly from general principles”, whatever those might be and furthermore, his critique “has the origin in general principles of classical mechanics” (pg. 3, penultimate paragraph before “The main proof” section). He then attempts to turn into a QED proof by waiving his hands and writing down some commutation relations that ignore any of the physics, basically concluding light and matter don’t interact. It’s rubbish and as such, has barely been cited. Now, your question about van der Waals forces. In fact, Casimir set out to compute van der Waals forces on polarizable molecules in the presence of a conducting plate. You could say the Casimir effect is a long range, retarded van der Waals force. That’s semantics. The point is, absent vacuum energy, there would be no induced dipole moment to give rise to a force. The review linked below has a good description of the underlying physics on pg. 10-11. https://arxiv.org/abs/quant-ph/0106045
Good explanation. Your mention of delta as "uncertainty" brings up what I think could be a topic of a video, how to read the equations in quantum mechanics. For example, I read delta as "change of" or "change in", not "uncertainty". Also, I watched The Great Courses lectures on quantum mechanics and learned that that vertical line, psi and greater than/less than symbol is a "ket" and "bra" respectively, as in "bra-ket notation". So, now that I know what to look for I can find it, such as in "Quantum Mechanics 1: Foundations" by Green this is explained on page 14 in "Dirac's bra-ket notation"! I bet that many others could use a nice overview because while we've taken math, physics and engineering courses, we haven't done the smart thing and started with quantum mechanics 101 and so missed some of the meaning in the notations used in quantum mechanics. Consider it, at least.
Sabine, You have explained very clearly and understandable that Heisenberg’s uncertainty is in fact the same as the relation of delta frequency to delta time in a wave packet. - However, then you say that the difference of both is the presence of ‘h’. The question now: what is the role of ‘h’? At the first glance, it is a proportionality factor which connects the frequency to energy. E = h*ny. Now, what is the physics of this relation? According to the original idea of de Broglie and the findings of Schrödinger and Dirac, there is a permanent oscillation in an elementary particle – going on at the speed c (which btw is the origin of relativistic dilation). This oscillation has to be an orbital motion as the particle has a spin and a magnetic moment (the latter if charged). The constituents of this particle (which have to be two by fundamental physical causes) have to be bound to each other. If we state the binding field by the expression h*c, then we have fixed what h is and why there is E = h*ny. Classical and quite simple, I think. No quantum mystery. And another goody: if we determine the mass of an elementary particle on the basis of this model, then we get the correct value with high precision (for the electron with 1:300’000). Please compare this to the Higgs mechanism: nothing!
Rad Antonov
No, he didn’t. I’ll explain.
Greggery Peccary
Yes indeed the Principle did come earlier.
How can we distinguish Casimir from Van der Waals ??? "More recently, Nikolic proved from first principles of quantum electrodynamics that Casimir force does not originate from vacuum energy of electromagnetic field, and explained in simple terms why the fundamental microscopic origin of Casimir force lies in van der Waals forces." https://en.wikipedia.org/wiki/Casimir_effect
Rad Antonov
I saw Copenhagen at the Royale Theatre and to be honest, didn’t really appreciate it. I just remember it not being very coherent. Sounds like the movie was better, so I’ll look to stream it after the WCUP. Regarding the eponymous principle, that came much earlier in life than the events depicted in the play. It’s possible he drew on some insights about the human experience, but practically, the principle falls out of his matrix formulation of quantum mechanics by way of non-commuting operators. That approach works for position/momentum and spin. The time/energy uncertainty is a bit of a different animal that was very well explained in the episode.
🥳
Huh, they haven't told me about that. (Though they said they would.) Thanks for pointing out!
Tracey, Thanks for the suggestion, I will keep that in mind!
Happy you like it!
Hey everyone, Sabine's TEDx talk just popped up on my YouTube feed: https://www.youtube.com/watch?v=Yx1k8q6PnWU
It's great when different people cover the same or similar topics (and inevitable with several STEM-Tubers) to help understand the nuances of those topics. I always enjoy the SWTG videos still, because Sabine.
I was thinking that the Pauli exclusion principle would be a good one to cover in this same manner for the same reason. Especially since it accounts for atomic structure, white dwarf stars, and ultimately the distance scale of the universe. Then, Arvin Ash put out a nice video today on the exclusion principle. Still, Sabine could tell this story as well, focusing on different topics than Arvin chose, and inserting the trademark sarcastic humor.
Vacuum polarization is also a measurable effect.
Rad Antonov
The vacuum is full of energy. The Casimir effect is one striking example of that fact. It has been observed in the lab and was found to be in agreement with the theoretical prediction: https://en.wikipedia.org/wiki/Casimir_effect.
Greggery Peccary
There's a play, made into a movie twenty years ago, with James Bond playing Heisenberg (srsly jk), which depicts the [historical] uncertainty of what happened when Heisenberg met Bohr during WWII (before Bohr escaped). The play works with the idea that humans experience the edge of decision, (or the superposition of roles, or the deferral of measurement) constantly, and that insight into this experience is what Heisenberg drew on to reach his understanding of the formulation of the eponymous principle. The filmed version of the play is very worth seeing, if only for Stephan Rea's and Daniel Craig's performances, as well as this idea presented that humans actually experience the notion of uncertainty, even in the form of conjugate measurables, fairly regularly in life.
Another video where Sabine broke down and explained concepts that seem to be taken for granted. I really enjoy these. 😸
The correct way to relate energy with time and momentum with space for a single particle is to have E=i hbar d/dt, and p=-i hbar 3gradient, and then take expectation values for whatever wavefunction, not just for a particle that is at one frequency or wavelength, more general than Einstein's E=h v and de Broglie's p=h lambda, valid for superpositions. It works for photons too! The interesting thing for photons is that the wavefunction (there are several ways to do this, but this is the simplest) is a 6 component vector of E and B fields (modulo some factors involving the average energy or the photon, and taking the real parts of the possibly complex ccomponents). So a quantum mechanical object (the wavefunction) can be written in terms of classical EM fields (this doesn't work for fermions, or at least we haven't figured out how to do it yet, there might be a way to interpret fermion wavefunctions in terms of purely classical geometric quantities having to do with singularities of spacetime) . See for example Open Journal of Microphysics, 2011, 1, 41-52, or any of Birula's papers on photons wave function.
Counting the fairies on a pin-head: "delta_t * delta_E > h_bar" is used in the claim that the vacuum is full of energy because the math says that if you push delta_t below some threshold, delta_E has to rise. But that assumes there are virtual-particles present (whatever that means) and/or 'quantum-foam'. It also assumes that the vacuum is a continuum at the level of Plank's constant. - In math, you can assume anything you like. But in physics, you can only design feasible experiments. When I read about physicists 'going to zero' or 'going to infinity', and/or talking about singularities, I tend to think they MUST BE JOKING.
Rad Antonov
On a serious note, I enjoyed how wave superposition took us full circle to EPR.
Rad Antonov
We can knock string theorists all we want, but they have finally achieved an enviable spot in the zeitgeist. Can anyone name another type of physics nerd that rappers glorify, like Ab-Soul does 26 seconds in on this recent track? https://youtu.be/4WX6YCm8En4
