Archive for the 'Good Physics' category

The Elegance of Uncertainty

Nov 15 2013 Published by under Good Math, Good Physics

I was recently reading yet another botched explanation of Heisenberg's uncertainty principle, and it ticked me off. It wasn't a particularly interesting one, so I'm not going disassemble it in detail. What it did was the usual crackpot quantum dance: Heisenberg said that quantum means observers affect the universe, therefore our thoughts can control the universe. Blah blah blah.

It's not worth getting into the cranky details. But it inspired me to actually take some time and try to explain what uncertainty really means. Heisenberg's uncertainty principle is fascinating. It's an extremely simple concept, and yet when you realize what it means, it's the most mind-blowingly strange thing that you've ever heard.

One of the beautiful things about it is that you can take the math of uncertainty and reduce it to one simple equation. It says that given any object or particle, the following equation is always true:

\[sigma_x sigma_p ge hbar\]

Where:

  • \(sigma_x\) is a measurement of the amount of uncertainty
    about the position of the particle;
  • \(sigma_p\) is the uncertainty about the momentum of the particle; and
  • \(hbar\) is a fundamental constant, called the reduced Plank's constant, which is roughly \(1.05457173 times 10^{-34}frac{m^2 kg}{s}\).

That last constant deserves a bit of extra explanation. Plank's constant describes the fundamental granularity of the universe. We perceive the world as being smooth. When we look at the distance between two objects, we can divide it in half, and in half again, and in half again. It seems like we should be able to do that forever. Mathematically we can, but physically we can't! Eventually, we get to a point where where is no way to subdivide distance anymore. We hit the grain-size of the universe. The same goes for time: we can look at what happens in a second, or a millisecond, or a nanosecond. But eventually, it gets down to a point where you can't divide time anymore! Planck's constant essentially defines that smallest unit of time or space.

Back to that beautiful equation: what uncertainty says is that the product of the uncertainty about the position of a particle and the uncertainty about the momentum of a particle must be at least a certain minimum.

Here's where people go wrong. They take that to mean that our ability to measure the position and momentum of a particle is uncertain - that the problem is in the process of measurement. But no: it's talking about a fundamental uncertainty. This is what makes it an incredibly crazy idea. It's not just talking about our inability to measure something: it's talking about the fundamental true uncertainty of the particle in the universe because of the quantum structure of the universe.

Let's talk about an example. Look out the window. See the sunlight? It's produced by fusion in the sun. But fusion should be impossible. Without uncertainty, the sun could not exist. We could not exist.

Why should it be impossible for fusion to happen in the sun? Because it's nowhere near dense or hot enough.

There are two forces that you need to consider in the process of nuclear fusion. There's the electromagnetic force, and there's the strong nuclear force.

The electromagnetic force, we're all familiar with. Like charges repel, different charges attract. The nucleus of an atom has a positive charge - so nuclei repel each other.

The nuclear force we're less familiar with. The protons in a nucleus repel each other - they've still got like charges! But there's another force - the strong nuclear force - that holds the nucleus together. The strong nuclear force is incredibly strong at extremely short distances, but it diminishes much, much faster than electromagnetism. So if you can get a proton close enough to the nucleus of an atom for the strong force to outweigh the electromagnetic, then that proton will stick to the nucleus, and you've got fusion!

The problem with fusion is that it takes a lot of energy to get two hydrogen nuclei close enough to each other for that strong force to kick in. In fact, it turns out that hydrogen nuclei in the sun are nowhere close to energetic enough to overcome the electromagnetic repulsion - not by multiple orders of magnitude!

But this is where uncertainty comes in to play. The core of the sun is a dense soup of other hydrogen atoms. They can't move around very much without the other atoms around them moving. That means that their momentum is very constrained - \(sigma_p\) is very small, because there's just not much possible variation in how fast it's moving. But the product of \(sigma_p\) and \(sigma_x\) have to be greater than \(hbar\), which means that \(sigma_x\) needs to be pretty large to compensate for the certainty about the momentum.

If \(sigma_x\) is large, that means that the particle's position is not very constrained at all. It's not just that we can't tell exactly where it is, but it's position is fundamentally fuzzy. It doesn't have a precise position!

That uncertainty about the position allows a strange thing to happen. The fuzziness of position of a hydrogen nucleus is large enough that it overlaps with the the nucleus of another atom - and bang, they fuse.

This is an insane idea. A hydrogen nucleus doesn't get pushed into a collision with another hydrogen nucleus. It randomly appears in a collided state, because it's position wasn't really fixed. The two nuclei that fused didn't move: they simply didn't have a precise position!

So where does this uncertainty come from? It's part of the hard-to-comprehend world of quantum physics. Particles aren't really particles. They're waves. But they're not really waves. They're particles. They're both, and they're neither. They're something in between, or they're both at the same time. But they're not the precise things that we think of. They're inherently fuzzy probabilistic things. That's the source uncertainty: at macroscopic scales, they behave as if they're particles. But they aren't really. So the properties that associate with particles just don't work. An electron doesn't have an exact position and velocity. It has a haze of probability space where it could be. The uncertainty equation describes that haze - the inherent uncertainty that's caused by the real particle/wave duality of the things we call particles.

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