n/The electrical, nuclear,
and total interaction energies for
an alpha particle escaping from a
nucleus.the probability of making it through the barrier is
P=e−^2 rw= exp(
−
2 w
~√
2 m(U−E))
.
self-check H
If we were to apply this equation to find the probability that a person can
walk through a wall, what would the small value of Planck’s constant
imply? .Answer, p. 1063
Tunneling in alpha decay example 17
Naively, we would expect alpha decay to be a very fast process.
The typical speeds of neutrons and protons inside a nucleus are
extremely high (see problem 20). If we imagine an alpha particle
coalescing out of neutrons and protons inside the nucleus, then
at the typical speeds we’re talking about, it takes a ridiculously
small amount of time for them to reach the surface and try to
escape. Clattering back and forth inside the nucleus, we could
imagine them making a vast number of these “escape attempts”
every second.
Consider figure n, however, which shows the interaction energy
for an alpha particle escaping from a nucleus. The electrical en-
ergy isk q 1 q 2 /rwhen the alpha is outside the nucleus, while its
variation inside the nucleus has the shape of a parabola, as a
consequence of the shell theorem. The nuclear energy is con-
stant when the alpha is inside the nucleus, because the forces
from all the neighboring neutrons and protons cancel out; it rises
sharply near the surface, and flattens out to zero over a distance
of∼ 1 fm, which is the maximum distance scale at which the
strong force can operate. There is a classically forbidden region
immediately outside the nucleus, so the alpha particle can only
escape by quantum mechanical tunneling. (It’s true, but some-
what counterintuitive, that arepulsiveelectrical force can make it
more difficult for the alpha to getout.)
In reality, alpha-decay half-lives are often extremely long — some-
times billions of years — because the tunneling probability is so
small. Although the shape of the barrier is not a rectangle, the
equation for the tunneling probability on page 907 can still be
used as a rough guide to our thinking. Essentially the tunneling
probability is so small becauseU−Eis fairly big, typically about
30 MeV at the peak of the barrier.The correspondence principle forE > U example 18
The correspondence principle demands that in the classical limit
h→0, we recover the correct result for a particle encountering
a barrierU, for bothE < UandE > U. TheE < Ucase
was analyzed in self-check H on p. 907. In the remainder of this
example, we analyzeE>U, which turns out to be a little trickier.Section 13.3 Matter as a wave 907