How Does a Particle Get Through a Wall? Passing a Barrier It Cannot Climb

Classically, a ball rolled at a hill either has enough energy to reach the top and cross or it does not, and there is no third option. In quantum mechanics a particle is described by a wave whose square gives the probability of finding it at each place, and when that wave meets a barrier it does not stop abruptly but decays inside it, so if the barrier is thin enough the wave still has some amplitude on the far side. The particle therefore has a calculable probability of being found beyond a barrier it could not classically cross. Nothing travels through with reduced energy, since a particle detected on the far side has the same energy it started with, which is one reason the picture of digging through is misleading.

The chance falls off sharply with the properties of the barrier:

• •Barrier width, with probability falling exponentially as it thickens
• •Barrier height relative to the particle's energy, with the same exponential sensitivity
• •Particle mass, with lighter particles tunnelling far more readily
• •That mass dependence is why electrons tunnel routinely and atoms almost never do
• •The exponential dependence means small changes in width change the rate enormously
• •That extreme sensitivity is what makes the effect useful for measurement

Fusion in stellar cores depends on the effect entirely. Two hydrogen nuclei are both positively charged and repel each other strongly, and fusing requires bringing them close enough for the attractive nuclear force to take over, which needs enormous energy. The temperature at the centre of the sun, around fifteen million degrees, gives nuclei nothing like enough energy to overcome that repulsion by classical means, and the calculation showing this was a serious problem in astrophysics before quantum mechanics. Tunnelling resolves it, since nuclei can pass through the repulsive barrier with small but nonzero probability, and the enormous number of nuclei involved makes a small probability per encounter sufficient. The rate is exquisitely sensitive to temperature for exactly the exponential reason above, which is why stellar cores are stable.

The effect operates inside living things, which was long doubted and is now reasonably established in at least one case. Enzymes transferring a hydrogen atom or a proton from one molecule to another show reaction rates and temperature dependencies that classical barrier crossing does not predict, and the discrepancy is accounted for by tunnelling of the light particle through the barrier, with evidence coming from comparing rates when the hydrogen is replaced by its heavier isotope. Because the probability depends so sharply on mass, the substitution changes the rate far more than a classical account allows, which is the diagnostic. Proposals extending the idea to photosynthesis, to olfaction and to bird navigation exist and are considerably more speculative, and the general field is young enough that claims should be treated carefully.

The effect has been put to work in several technologies. The scanning tunnelling microscope brings an extremely sharp tip within a nanometre of a conducting surface and measures the current tunnelling across the gap, which varies exponentially with distance, so moving the tip while holding the current constant traces the surface with resolution sufficient to image individual atoms, and the instrument earned a Nobel Prize in 1986. Flash memory stores data by driving electrons through an insulating layer onto an isolated gate, where they remain until deliberately removed. Tunnel diodes exploit the effect for very fast switching. Radioactive decay by emission of a helium nucleus is tunnelling out of the nucleus, and explaining its rates was the first successful application of the idea in 1928. And it sets a limit on how small transistors can be made, since electrons leak across barriers that are too thin.

A particle's wave decays inside a barrier rather than stopping, so if the barrier is thin enough there is amplitude on the far side and a calculable chance of finding the particle there with its energy unchanged. Probability falls exponentially with width, height and particle mass. Fusion in the sun depends on it, since the core is far too cool to overcome nuclear repulsion otherwise.