Quantum Tunnelling in the Sun

This is a script for Quantum On The Clock, the video is here at https://youtu.be/ksyrIH1LIBw. I am quite proud of it.

Why does the sun actually shine? A weird question since everyone knows it’s nuclear fusion. But, the sun is actually kind of cold. It’s 15 million kelvin but fusion reactors on earth are over 100 million kelvin, so how does it do it? It’s really nothing more than a probability trick.

The answer lies in quantum tunnelling and that’s where we’ll begin.

Look at this ball near a hill. If it doesn’t have enough energy it won’t be able to get over, if it does then it will.

But in contrast if it was a “quantum” ball, the ball could tunnel right through, this is a phenomenon which really goes against intuition, but it will be more rigorously explored now.

But first we need to define some constants, say we had a potential barrier. Its width would be a, the distance the particle must tunnel. It’s height would be V (the potential energy required) and the energy of the particle would be E.

You may have heard of the schrodinger equation and this is where it comes in, this governs the wavefunction of a quantum mechanical system. The wavefunction squared gives probability density, and determined by Schrödinger, the wavefunction diminishes exponentially while in a potential barrier, thus so does the probability as width increases. We can derive this probability from the Schrödinger equation, and we call it transmission probability.

This equation is useful to show as you can see relationships between width, height and mass and original energy and their resulting probability changes.

Although, I will note that that is only for rectangular, constant potential wells and is more generally this, for example for curved potential wells like the one we shall discuss.

For quantum tunnelling to happen in an identifiable way, we need a finite height and also a sufficient number of particles.

So, the sun uses nuclear fusion and for nuclear fusion to happen we need to get nucleons close enough together for something called the sasatrong force to take over from the electrostatic force which acts between like charged protons. The strong force is a fundamental force that holds nucleons together. It’s incredibly strong, orders of magnitude stronger than the electrostatic force but acts at very tiny distances, like velcro.

The energy required to overcome the electrostatic repulsion between like particles is called the Coulomb barrier, and it looks like this.

The potential energy required gets higher and higher as you approach the proton (due to repulsion), and then suddenly goes negative when the strong force takes over. This is where all the quantum tunnelling comes in. See protons are quantum particles, they are waves of probability and therefore have a nonzero probability on the other side of the potential barrier, so they can tunnel.

Now, we can find a rough probability of this happening. The answer is a tiny 1/10²⁸, but remember the point about having a lot of particles?

There are over 10⁵⁶ particles in the core of the sun, so with a tunnelling probability of 1/10²⁸, the number of particles tunnelling is sufficient enough for fusion to happen effectively.

We can’t do fusion like this on earth because we simply don’t have that many particles!

The same principle actually underlies alpha decay, but in the opposite direction.