I first learned about quantum tunneling from
science fiction, specifically a short story by Larry Niven. The
idea is often tossed out there as one of those "quantum is weird and
almost magical!" concepts. It is
surely far from our daily experience.

Imagine a car of mass \(m\) rolling along a road toward a small
hill. Let’s make the car and the road
ideal – we’re not going to worry about friction or drag from the air or
anything like that. You know from
everyday experience that the car will roll up the hill and slow down. This ideal car’s total energy is conserved,
and it has (conventionally) two pieces, the kinetic energy \(p^2/2m\) (where \(p\) is
the momentum; here I’m leaving out the rotational contribution of the tires),
and the gravitational potential energy, \(mgz\), where \(g\) is the gravitational acceleration
and \(z\) is the height of the center of mass above some reference level. As the car goes up, so does its potential
energy, meaning its kinetic energy has to fall.
When the kinetic energy hits zero, the car stops momentarily before
starting to roll backward down the hill.
The spot where the car stops is called a classical turning point. Without some additional contribution to the
energy, you won’t ever find the car on the other side of that hill, because the
shaded region is “classically forbidden”.
We’d either have to sacrifice conservation of energy, or the car would
have to have negative kinetic energy to exist in the forbidden region. Since the kinetic piece is proportional to \(p^2\), to have negative kinetic energy would require \(p\) to be imaginary (!).

However, we know that the car is really a quantum object, built
out of a huge number (more than \(10^27\)) other quantum objects. The spatial locations of quantum objects can
be described with “wavefunctions”, and you need to know a couple of things
about these to get a feel for tunneling. For the ideal case of a free particle with a
definite momentum, the wavefunction really looks like a wave with a wavelength \(h/p\), where \(h\) is Planck’s constant. Because
a wave extends throughout all space, the probability of finding the ideal free
particle anywhere is equal, in agreement with the oft-quoted uncertainty
principle.

Here’s the essential piece of physics: In a classically forbidden region, the wavefunction
decays exponentially with distance (mathematically equivalent to the wave
having an imaginary wavelength), but it can’t change abruptly. That means that if you solve the problem of a
quantum particle incident on a finite (in energy and spatial size) barrier from
one side, there is always some probability that the particle will be found on
the far side of the classically forbidden region.

This means that it’s technically possible for the car to “tunnel”
through the hillside and end up on the downslope. I would not recommend this as a
transportation strategy, though, because that’s incredibly unlikely. The more massive the particle, and the more forbidden
the region (that is, the more negative the classical kinetic energy of the
particle would have to be in the barrier), the faster the exponential decay of
the probability of getting through. For
a 1000 kg car trying to tunnel through a 10 cm high speed bump 1 m long, the
probability is around exp(-2.7e20). That
kind of number is why quantum tunneling is not an obvious part of your daily
existence. For something much less
massive, like an electron, the tunneling probability from, say, a metal tip to
a metal surface decays by around a factor of \(e^2\) for every 0.1 nm of tip-surface
distance separation. It’s that
exponential sensitivity to geometry that makes scanning tunneling microscopy
possible.

However, quantum tunneling is very much a part of your
life. Protons can tunnel through the repulsion
of their positive charges to bind to each other – that’s what powers the sun. Electrons routinely tunnel in zillions of
chemical reactions going on in your body right now, as well as in the photosynthesis
process that drives most plant life.

## 1 comment:

Great post, the clearest explanation of tunneling I've ever read. You have a talent for explaining complex subjects to people with no special training in that field.

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