# Tunnelling

This post arose as a result of a email discussion with David Deutsch about quantum tunnelling. I will explain roughly what tunnelling is, the relevant parts of quantum mechanics and something about why tunnelling happens.

Potential energy and tunnelling

If you throw a stone in the air, then it starts with some particular speed, it then slows down in the vertical direction until it stops and then falls to the ground. When the stone is moving slowly it doesn’t have much kinetic energy. So where does all the kinetic energy go? Doesn’t this violate conservation of energy? No. An object that is higher off the ground has a different kind of energy: the energy it would gain by falling to the ground. More precisely the system consisting of the Earth and the stone has this energy. When a set of systems has some energy by virtue of its configuration, i.e. – the stone being above the ground rather than resting on it, it is said to have potential energy.

To gain potential energy, an object has to lose some other kind of energy, e.g. – kinetic energy. So if you roll a stone up a hill it can gain as much potential energy as you impart to it. If you roll a stone up a hill with some specific amount of energy it can get to a certain height on the hill. If you don’t give it enough energy it can’t get over the hill. But suppose there is a place on the other side of the hill with the same potential energy. It doesn’t break conservation of energy for the stone to be on the other side, but that doesn’t happen because the stone would have to go through states in which it gained more energy than it had originally.

However, for a system that can go undergo quantum interference in a particular potential, it can be the case that it does something a bit like appearing on the other side of the hill. This is called quantum tunnelling. You have a potential in a particular region. You have a particle with energy less than the energy required to “go over” the potential. But the particle goes through the region in which the potential is “too large” to a place on the other side where it has the same energy it had originally. This is called tunnelling because it superficially looks a bit like the particle digs a tunnel through the hill instead of climbing over it. Tunnelling only happens in quantum mechanics, not in classical physics.

Locality in quantum mechanics

Before I can discuss tunnelling in more detail I will say something about the context of the explanation. Quantum mechanics is used by many physicists to make accurate predictions about the results of experiments. However, most physicists are unwilling to accept its implications and this has led to a bizarre controversy about the “interpretation” of quantum mechanics. Interpretation in this context means “say absolutely anything you want about quantum mechanics without paying attention to whether what you say makes sense, explains experimental results, or is even compatible with other physical theories.” That’s not what people say it means, but in practise they act exactly as if this was the case.

For example, lots of physicists say quantum mechanics is non-local. In reality, the equations of motion for quantum systems, such as the Schrodinger equation and the Heisenberg equation are entirely local. That is, the equations say the value of the wave function or observables in some region is a function of the stuff in and around that region, not of stuff arbitrarily far away. The main experiment that is usually deemed to demonstrate non-locality, the EPR experiment, has an entirely local explanation. And since the explanation of tunnelling is in part a result of quantum mechanics being local, I shall explain why quantum mechanics is local, despite the standard line that it is non-local.

Quantum mechanics says that all of the stuff you see around exists in multiple versions. There is a version of me that is sitting one millimetre to my right of where I am sitting now. I can’t interact with that version of me and so can’t exchange information with him. These versions are sorted into layers of versions of systems, each of which approximately obeys the laws of classical physics. In one layer I am sitting in my current position and the version of the chair that I am sitting on has a specific pattern of pressure on it. There is another version of me sitting one millimetre to the right on another version of the chair with a slightly different pattern of pressure on it. Each version of me goes with some specific version of the chair, and with some specific version of the other systems I’m interacting with, like my computer. A specific layer is called a universe and the collection of all the layers is called the multiverse. (This should not be confused with other uses of the word multiverse, like the string theory multiverse. The quantum multiverse is necessary to explain the results of experiments, whether the string theory multiverse can be tested at all is still unclear.)

In carefully arranged experiments it is possible to tell that different versions of a system existed since you can’t explain the results without them. One example is the EPR experiment that is commonly said to be non-local. I will explain a simplified version of the experiment and a sketch of the local explanation. For more detail see this and this. It is possible to set up two systems with the following properties. (1) Each system has two observables A,B, both of which have two possible values and the probability of getting either value is 50% for both observables. (2) If you measure the same observable on both particles, e.g. – you measure A on particle 1 and A on particle 2, you get the same result. (3) If you measure A on one particle and B on the other then when you compare the results you find that they match 50% of the time. So whether the results match depends on whether you do the same measurement on both systems. And this remains true if you move them far apart, do the measurements and then compare the results. If they were correlated before the measurement then whether the results match wouldn’t depend on whether you measure the same observable on both. What happens is that when you do the measurement there are two versions of each system and the correlations are only established when you compare them. But if there was only one version of each system how would you explain this result? The correlations can’t be established before the measurement, or after it. So they must be established during the measurement, no matter how far apart the measurements are or even if there is no time for information to get from one system to the other during the measurement. So the correlations must be non-local if there is only a single version of each system as a result of the measurement, hence the misconception that the correlations are non-local. In reality, there are multiple versions of each system after the measurement and the correlations are established locally afterward.

Versions and instances

The above account of what quantum mechanics of the implications of quantum mechanics is usually called the many worlds interpretation, and physicists act as if it is an optional add on. It is not an optional add on. It is the only explanation of what quantum mechanics is saying about the world and it is the only explanation of many experiments, not just EPR type experiments. I’m going to give a slightly fuller account of quantum mechanics before I continue. The existence of multiple distinct versions of each system is only one feature of what quantum mechanics says about how the world works.

Suppose that you measure the position of a particle along a particular line with some accuracy x. There is a mathematical operator that describes all of the different versions of the particle measured with that accuracy: it’s called an observable. Let’s call the observable in question D for distance. So D might have values x, 2x… You would find that each of those versions has a probability predicted by quantum mechanics. A particular version of the system after the measurement will have a sharp value of Dx.  If you were to measure an observable M that gives the momentum of the version of the particle to some accuracy p, you would find that if you make the measurement accurate enough there would be more than one state with a non-negligible probability. When a system only has one state with non-negligible probability it is said to be sharp, otherwise it is unsharp. But before you do the measurement of M there is no single fact of the matter about what value M has. What is more there are experiments you could do to make the different M values interact with one another. As a result it doesn’t make sense to call the things with different values of M distinct versions. I will use the term instance to refer to some set of aspects of a system that could become distinct versions whether or not they actually become distinct versions. All versions of a system are instances, but not vice versa. You could also set up the particle so that neither D nor M is sharp but rather, both of them have non-negligible probabilities for multiple different possible values. Experiments in which multiple instances of a single particle interact are often called single particle interference experiments. And tunnelling is an example of interference.

Tunnelling

David Deutsch wrote a tweet:

‘Quantum tunnelling’ is very badly named. It’s mountaineering: In some universes the system has enough energy to cross the barrier. Period.

In a quantum tunnelling experiment, we send instances of a particle toward a potential (the barrier) where the particle has a negligible probability of having an energy above the barrier. But the particle has a non-negligible probability of getting through the barrier. And if you measure the energy of the particle on the other side of the barrier you find that it is lower than the barrier energy with high probability. This is not a purely theoretical issue. It is used in technologies like tunnel diodes and scanning tunnelling microscopy. Since tunnelling is well known precisely because particles with energy less than the barrier get through the barrier it seemed wrong to say that the particle ends up on the other side by having energy above the barrier energy. So David Deutsch’s tweet was wrong, and is contradicted by both theory and experiment.

What’s actually happening in tunnelling is not that instances of the particle with energy above the barrier get through. Rather, instances of the particle with energy below the barrier go into the barrier because there is no way for the particle to get information about the barrier without interacting with it. Some instances of the particle have to go part way into the barrier for the probability of the particle being reflected to be affected by the barrier. And if instances of the particle go into the barrier then a thin enough barrier will let some of them through. In addition some instances of the particle will be propagating forward through the barrier and others backward and this will have an effect on the probability of reflection, so tunnelling is also an interference effect.

The maths explaining why tunnelling can work goes like this. I reproduce the equations for a sum of quantum states of the particle with a particular energy. I construct a state in which the particle starts off in a state in which it has a high probability of being on one side of a potential barrier with a high probability of being far from the barrier and the particle is propagating toward the barrier. I explain that the wave packet acts as we would expect a wave packet propagating from some distance far from the barrier to act. I argue that the distribution of energies does not change significantly as the instances of the particle propagate toward the barrier. I then argue that the amplitude for instances of the particle above the barrier is not significant. See tunnelling4. So, in conclusion, particles don’t tunnel through a potential by having energy higher than the potential.

UPDATE I made a mistake in this post that changes its conclusions.