Well, well...quantum tunneling at last

Graphic - No title - quantumworld

Consider a roller coaster. A maniac sledger (probably a physicist) starts off down the slope from rest at point A. During the ride that follows there is an interchange of kinetic (K) and potential (U) energies. But the total energy remains constant:

K+U=E (constant)
Since the height (above some reference point, say B) is a direct measure of the potential energy we can regard the track as a plot of the PE as a function of x. It is useful to draw a horizontal line across the figure, going through point A. The ‘height’ of this line (above the reference point B) then gives the PE at A; since the KE is zero at A, this height must also measure the total energy E. At the other points (C,D) through which this line goes the KE must also be zero (because U=E at these points.)

According to energy conservation then, the maniac sledger will reach point C and no further; C is a ‘turning point’ where the motion is reversed. The subsequent motion will comprise repeated oscillations about the lowest point B. If the amplitude of oscillation is small enough (if A is not too far above B) these oscillations will be simple-harmonic.

Were that the end of the story it would not really be worth telling. But there is much more.

First we should realize that underlying the whimsical imagery (maniac sledger and roller coaster) is a powerful and general way of thinking about motion. We can, for example, think of the behaviour of a charged object in an electric field in a very similar way: the height of the surface would then represent the electrostatic potential energy of the object, rather than the gravitational PE that drives our sledger. We then think of motion across the PE surface comprising hills and valleys. In the jargon of the trade a ‘valley’ in this surface is called a potential energy well.

According to classical physics (to which this course is largely restricted) such motion is always subject to the rules of energy conservation. Thus, a sledger (or any other macroscopic object –which obeys the laws of classical dynamics) will be trapped forever in the potential energy well in which he is released. But the rules of quantum physics are more relaxed about energy conservation. So an electron (or any other suitably microscopic object) has some chance of escaping from a potential well in which (from a classical standpoint) it is trapped; it may thus magically appear on the other side of the hilltop, at point D. This is the phenomenon of quantum tunnelling: one might think of the particle as digging its way ‘through’ the potential energy barrier. It happens...and you have probably heard it happen. Can you think when?