Understanding quantum tunnelling

The recent claim from Stark Industries Lockeed Martin that a compact fusion reactor could be built soon is, if true, a breakthrough in engineering rather than basic physics. The basic physics of fusion has been known for some time, and a key element of understanding it is quantum tunnelling. Nuclei have a positive electric charge, and since like charges repel, there is an energy barrier to be overcome. Once the barrier is overcome, the strong nuclear force takes over¹. One way of overcoming the barrier is ‘quantum tunnelling’, and, weird though it sounds (and indeed is) the maths and physics of that is quite well understood.

Here’s how it works.

The first thing to understand is that quantum particles – and all particles are quantum particles once you are as small as an atom – behave, in some ways, like waves as they travel around. The big question “Is it a wave or a particle?” was answered by quantum mechanics, and the answer was “No it’s a quantum.” Waves and particles are both concepts that work well in everyday life, and emerge from quantum mechanics as good approximations to how things really behave, but the reality at the atomic scale is neither one nor the other. Or in some ways, it is both.

Waves and exponentials

Next, we need to look at the mathematics behind waves. If you don’t want to follow the maths, I suggest skipping to the summary at the end, it’s hopefully still interesting. But I hope you stay, and at least check out the animated gif because it’s lovely.

The most common wave in this context is a sine wave. This is a somewhat mysterious mathematical function, written sin(x), and sin(x) will oscillate up and down between 1 and -1 forever as x increases. This always used to bug me. “Sin” was probably the first time I came across maths that used words instead of numbers and symbols, and it unnerved me. Anyway what was a sine really?

I became happier with sines (and cosines) when I realised they could be written as a power series, very much like the exponentials that I wrote about here. In fact the series for sines and cosines are closely related to the series for an exponential. If some variable, call it y, is the exponential of x, the expression for it goes like this:

y = eˣ = exp(x) = 1 + x + x²/2! + x³/3! + x⁴/4! … and so on forever.

If x is bigger than zero, then y grows exponentially as x increases. If x is less than zero, you get an exponential decay toward y = 0 as x gets more and more negative. The symbols in there are described in more detail here, and it’s quite easy to convince yourself this works by trying some values of x in a calculator.

The series expansion for y = sin(x) is

y = sin(x) = x - x³/3! + x⁵/5! - … and so on forever.

and cosine is

y = cos(x) = 1 - x²/2! + x⁴/4! - … and so on forever.

It might not be obvious that these series build up oscillating functions like signs and cosines, but you can again (taking the experimentalist’s approach to maths) see it happen on a calculator, or in this animation showing how the sine function builds up as you add more and more terms in the series above:

It starts with y = x, a straight line. Then y = x - x³/3!, which looks a bit more curvy, and so on. The N in the plot indicates the number of terms from the series that have been included.

You might notice that the three series, exponential, sine and cosine, look fairly similar, and they can indeed be related to each other, in quite a special way, as long as you don’t mind introducing an imaginary number, i, which is the the square root of minus one. So long as you accept that i² = -1, then you can work out that

eⁱˣ = exp(ix) = cos(x) + i sin(x )

(Try it out for the first few terms if you don’t believe me. Also, apparently that exponent isn’t showing up correctly for everyone. It is meant to say e-to-the-i-times-x, or exp(ix).)

So there’s an odd thing here. The function eˣ (or exp(x)) is an exponential growth with positive x, an exponential decay with negative x, and an oscillating wavefunction if x is imaginary!

Imagine this

Back to the physics, the equation which describes quantum particles² is the Schrödinger equation³,and its solutions are wavefunctions that, in some approximation you can write as eⁱªˣ, (or exp(iax)) where x is the distance travelled, and a turns out to be a number proportional to the square root of the kinetic energy of the wave (or particle, or quantum).

What happens when a wave like this hits an energy barrier, like the one caused by the electrostatic repulsion between nuclei? Well, the wave has not got enough energy to go over the barrier. If this were a classical particle, say a golf ball running up a hill on a golf course, that would be the end of the story. The ball would run up the hill, slowing down as kinetic energy was converted into potential energy. At some point the kinetic energy runs out, the ball stops and rolls back again.

But if it is a quantum golf ball, it obeys Schrödinger’s equation. It is described by a wave oscillating with the exponential of i times the square root of the energy, E, multiplied by the distance x. At the point where the classic golf ball stops, E becomes zero, and then negative. The square root of a negative number gives you another imaginary number, another i. This multiplies the i we already have, gives a negative-but-real number, and turns the exponential from being an oscillation into being an exponential decay!

This means the probability of finding the golf ball (which is determined by the wavefunction) drops rapidly, but is not zero. So if the hill is not infinitely high or infinitely thick (few are) there is a small probability, the tail of the exponent, that the ball will make through to the other side, as though it has tunnelled through.

In summary

Quantum tunnelling is real. It is an important factor in many physical phenomena, such as the rate of nuclear fusion, many chemical reactions, and a lot of technology (scanning tunnelling microscopy is a favourite of mine, but enough for now.) There is even a possibility that the universe itself might at some point tunnel through to a lower energy state, depending on what the mass of the Higgs boson is exactly. The maths we use to understand quantum tunnelling involves connecting oscillating trigonometric functions to exponentials, via imaginary numbers, and plugging it all into Schrödinger’s equation. And it works.

Whether controlled fusion will ever work, of course, remains to be seen.

¹ The force that holds nuclei together despite the electrostatic repulsion between the protons they contain.

² If they aren’t going too fast.

³ Yes, the cat guy.

Jon Butterworth has written a book about being involved in the discovery of the Higgs boson, Smashing Physics, available here . Some interesting events where you might be able to hear him talk about it etc are listed here. Also, Twitter.