By Vijay Damodharan - Natural Sciences Student @ Christ College, Cambridge

People sometimes play basketball, football, or table tennis using a wall as their opponent. It may be for practice, or just out of boredom. When the ball strikes the wall, it dutifully returns the ball back to us. But what if it didn’t? What if the ball passed straight through? It seems like an absurd question to ask; however, it occurs more often than one might think in quantum mechanics.

Quantum particles are able to ‘tunnel’ through potential barriers and this effect is essential for many physical processes, (radioactive decay and nuclear fusion), biological processes, (photosynthesis and respiration), and it is also critical for the operation of many electronic components.

First, what do we mean by a wall? If we have a brick wall and we hit it with a tennis ball, the ball will definitely come back to us. If we instead throw a bowling ball onto a thin plastic sheet, chances are the ball will break through the plastic wall. Even a brick wall can be broken through if the object is sufficiently massive and travelling fast enough (such as a truck).

We can generalise this by thinking in terms of energy. The ball has a certain amount of kinetic energy, KE, which depends on its mass and the velocity with which it is travelling. The wall is a solid material which requires a certain amount of energy to fracture. We can think of the wall as something that absorbs energy, and if we give it more energy than it can absorb, then it breaks. The amount of energy needed to break the wall can be thought of as a potential energy, V, associated with the wall.

In other words, a wall is something that provides a (potential) energy barrier to objects. If in a region KE ≥ V, the particle can exist there, otherwise if KE ≤ V, it cannot. The latter is called the *classically forbidden region*.

We can generalise this to cases such as gravity. We can think of the earth as providing a gravitational potential energy barrier, GPE. For a rocket to escape from the earth’s gravity, KErocket ≥ GPE must be satisfied.

Quantum particles are facing potential barriers all the time. Inside the nucleus of an atom, protons, and neutrons (generally called nucleons) are held together by a strong nuclear force, which provides a potential energy barrier. From our discussion so far, we know that the nucleons cannot leave unless they have enough energy to overcome this barrier.

Alpha decay is a common process by which an atom ejects two protons and neutrons from its nucleus. So how is it that at one moment the nucleons are trapped, unable to move, and the next they suddenly leave? Is this not like a ball suddenly deciding to go through the wall, instead of returning back to us?

Quantum tunnelling is one of the quantum effects which is best understood mathematically. However even without doing any detailed calculations, we can get an idea of where it comes from by exploring some fundamental postulates of quantum mechanics.

De Broglie famously hypothesised that all particles have wave-like properties, which was later confirmed experimentally. As a result, we can describe particles by a wave function Ψ(x,y,z). Like any mathematical function, it will just take in some numbers (x, y, z coordinates) as an input, and spit out another number as an output. This output by itself doesn’t mean anything.

However, it is found that performing certain operations to the wavefunction does give an output which is physically meaningful. One such example is probability. If Ψ describes an electron, then the probability density, P, of finding the electron at a point is given by taking the absolute value squared of the wavefunction: *P(x)dx *ǀ Ψ* Ψ ǀ^2d*x *.

We know that total probability must always equal one and never higher or lower. This means thatǀ Ψ* Ψ ǀ^2, and therefore Ψ itself must be a smooth and continuous function. As such, the value of the function cannot ‘jump’ - e.g. it cannot be equal to 2 at one point and 20 right next to it, as that would make Ψ undefined at that point, which makes ǀ Ψ* Ψ ǀ^2, and therefore the probability, undefined at that point.

Going back to our problem, we know that in the regions around the nucleon where KE ≥ V is satisfied, the particle can exist, so we expect a non-zero probability of finding the particle there. Hence Ψ must have some finite value, like 0.3.

However, we don’t want the particle to exist in the regions where KE ≤ V, we want the probability of finding it there to be 0, meaning Ψ should be equal to be 0. But we just said that Ψ cannot jump in values! It cannot go from 0.3 (or any finite number) to 0 immediately! Ψ can only decrease gradually little by little as it goes further and further into the barrier as shown in Figure 1. It will only reach zero after going infinitely far into the barrier. All of this is only true however if the potential energy barrier is not infinite, but that’s a problem for another article!

This means that Ψ is non-zero within the barrier. If the barrier has a finite width, such as a wall, a

nd Ψ is non-zero inside the wall, it can have a non-zero value on the other side of the wall by the same argument, as shown in Figure 2. In other words, the particle can appear on the other side of the wall! This is quantum tunnelling.

It may feel slightly unsatisfactory to imagine that quantum tunnelling simply arises as a result of some mathematical manipulations. However, mathematics is an essential tool in quantum mechanics. Often, just by setting up some mathematical postulates, abstract spaces, and operations in the spaces, simply through algebra, we can arrive at physically meaningful results that are proven to hold experimentally. Quantum tunnelling is one such example, however without it many processes wouldn’t occur, and our world would be completely different.

**Further reading:**

1. Krämer, K. (2020). Explainer: What is quantum tunnelling? [online] Chemistry World. Available at: https://www.chemistryworld.com/news/explainer-what-is-quantum-tunnelling/4012210.article

2. August 2020, D.L.-L.S.C. 07 (2020). Physicists watch quantum particles tunnel through solid barriers. Here’s what they found. [online] livescience.com. Available at: https://www.livescience.com/quantum-tunneling-observed-and-measured.html

3. Up and Atom (2018). What is Quantum Tunneling, Exactly? YouTube. Available at: https://www.youtube.com/watch?v=WPZLRtyvEqo.