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Before we take a look at what tunnelling means in quantum physics, it is helpful to take a look at what happens in classical physics when a classical particle meets a barrier.  Consider the situation in the following figure:

This shows a classical object (e.g. a bowling ball) rolling along a level, frictionless surface with a total energy E.  In its path is a slope of height h.  The question is, under what conditions can the object get over the slope and continue on its journey.

In order to get to the top of the slope, the object needs to be able to acquire the potential energy required to do so.  For a slope of height h, the potential energy required is

where m is the mass of the object.  Now, this potential energy needs to come from the initial energy E of the object.  Thus, the condition for our classical object to clear the slope is

If this condition holds, then the object makes it to the top of the slope with energy to spare, and makes it over the top, regaining the potential energy mgh as it falls back to the ground.  If the condition is not met, then the object will only make it to a certain distance up the slope, and will not be able to clear it.

We can look at this situation in a slightly different way, as shown in the following figure:

In this figure, we are noting that the slope represents a “potential barrier” to the motion of our object, since the object needs to be able to have a certain minimum energy (V in the figure above) before it can clear the barrier.  Thus, in the figure above, if the object has an energy less than V, then it cannot clear the barrier.  If the energy is greater than V, then the object can clear the barrier.

The point is that in classical physics, the situation is clear cut.  The object can either clear the potential barrier, or it can’t.  In quantum physics, things are not so clear cut.  This arises because, as we have seen in an earlier article, quantum objects can have some of the properties of waves, as well as properties of particles.

Consider the situation in the following figure:

This shows a similar situation to the previous figure, but this time we are concerned with a quantum object approaching a potential barrier.  The energy of the quantum object is assumed to be lower than the “height” of the potential barrier.

In the classical situation, we know that the object should not be able to penetrate the barrier.  To express it another way, the probability that the classical object can penetrate the barrier is zero.  However, for the quantum object, a different situation applies.  In this case, there is a non-zero probability that the object can penetrate the barrier.  Thus, if we were to fire a large number of quantum objects at the potential barrier, some of them would penetrate the barrier.  This is referred to as “tunnelling”.

Tunnelling is analogous to the situation where light waves (or any electromagnetic radiation) hit a medium with a different refractive index from that of the ambient medium.  In this case, both reflection and transmission of the light wave occurs.  If you stand by a window, then people can see you on the other side of the window – light waves are transmitted through the glass.  On the other hand, you will also be able to see your reflection in the window – light waves are reflected as well as transmitted.

The probability that a quantum object is transmitted through the potential barrier depends of course on the magnitude of the potential energy V.  If V is large compared to the quantum object’s energy E, then the probability is small.  If V and E are comparable, then the probability can be significant.

An example of a phenomenon that can be explained through this tunnelling effect is alpha decay, a mode of radioactive decay in which an atomic nucleus expels an alpha particle ( a helium nucleus, with two protons and two neutrons).  In the case of alpha decay, the potential barrier is similar to that shown in the following figure:

The horizontal axis represents distance from the centre of an atomic nucleus, and the vertical axis represents alpha particle energy.  The main curve shows the potential barrier that an alpha particle must breach in order for alpha decay to occur.  The section of the potential barrier at distances greater than the nuclear radius reflects the repulsive electrostatic force that exists between the positively charged alpha particle and nucleus.  The section within the nucleus reflects the nuclear binding forces within the nucleus.

In classical terms, if an alpha particle within the nucleus has a smaller total energy than the peak of the potential barrier (at the nuclear radius), then the alpha particle cannot escape and alpha decay does not occur.  However, in the quantum world, there is a probability that the alpha particle could tunnel through the potential barrier and thus escape in the form of alpha decay.

The mathematical theory of alpha particle tunnelling is found to give excellent agreement with observed data for radioactive isotopes that decay through alpha decay.  In particular, it helps to explain why it is that the radioactive half lives of alpha emitters can assume a wide range of values – from many billions of years (e.g. Thorium-232) through to fractions of a microsecond (e.g. Polonium-212).  The theory also helps to explain why it is that alpha decay is generally only observed in nuclei with large atomic numbers (greater than about 80).

For this reason, the theory of alpha particle tunnelling is an important confirmation of the existence of the tunnelling in quantum mechanical systems.

THE END