202 The Poetry of Physics and The Physics of Poetry
hence, there is an increase in the size of the wave packet. This surprising
result is a unique feature of quantum mechanics not to be found in
classical physics.
Barrier Penetration
Perhaps the most baffling quantum mechanical effect of all is barrier
penetration or tunneling. Let us consider a particle under the influence of
a spherically symmetric force with a radius r = r 0 so that the particle
is trapped within the space r < r 0. The potential energy of the particle
under the influence of the force is –Vo which means the particle is
trapped unless it has kinetic energy greater than or equal to Vo. The
nature of the force is to create a barrier that prevents particles with total
kinetic energy less than Vo from leaving the region r less than r 0. From a
classical point of view if a particle’s total energy is less than Vo, the
maximum potential energy of the force, it will remain exclusively in the
region r < ro. It will not be able to pass into the region r > ro. The nuclear
force that traps the protons and neutrons (nucleons) within a nucleus, is
this kind of force. If the classical laws of physics applied, one would
never expect a proton or neutron to escape the nuclear barrier. There are
cases, however, of unstable nuclei where the nucleons escape their
nucleus.
This effect can be explained in terms of quantum mechanics. The
equations governing the behaviour of the wave function, ψ, demand that
the ψ become very small very quickly in the forbidden region. However,
ψ does not become identically zero in this region as it would within the
framework of classical physics. Because ψ is not identically zero in the
forbidden region there is a very small but finite probability that the
particle will enter this region. If the particle passes through this region to
values of r > ro it will then be free of the nucleus because now its total
energy is positive and hence, its wave function in this region no longer
needs to remain small. In this way nucleons are able to tunnel through
the barrier created by the nuclear force and escape the nucleus. An
equivalent event on the macroscopic level would be the exit of a student
from a schoolroom by walking through the walls leaving them intact.
The probability of the tunneling of a particle is very small because the
wave function in the forbidden zone is so small. Given the potential
energy due to a given force one can use the Schrödinger equation to
calculate the probability of the penetration of this potential barrier. The