How Does Quantum Uncertainty Work?

[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.