o/A particle encounters a
step of height U < E in the
interaction energy. Both sides are
classically allowed. A reflected
wave exists, but is not shown in
the figure.p/The marble has zero proba-
bility of being reflected from the
edge of the table. (This example
hasU < 0, not U > 0 as in
figures o and q).q/Making the step more grad-
ual reduces the probability of
reflection.The particle has enough energy to get over the barrier, and the
classical result is that it continues forward at a different speed (a
reduced speed ifU>0, or an increased one ifU<0), then re-
gains its original speed as it emerges from the other side. What
happens quantum-mechanically in this case? We would like to
get a “tunneling” probability of 1 in the classical limit. The expres-
sion derived on p. 907, however, doesn’t apply here, because it
was derived under the assumption that the wavefunction inside
the barrier was an exponential; in the classically allowed case,
the barrier isn’t classically forbidden, and the wavefunction inside
it is a sine wave.
We can simplify things a little by letting the widthwof the barrier
go to infinity. Classically, after all, there is no possibility that the
particle will turn around, no matter how wide the barrier. We then
have the situation shown in figure o.^6
The analysis is similar to that for any other wave being partially
reflected at the boundary between two regions where its velocity
differs, and the result is the same as the one found on p. 381.
(There are some technical differences, which don’t turn out to
matter. This is discussed in more detail on p. 970.) The ratio of
the amplitude of the reflected wave to that of the incident wave
isR = (v 2 −v 1 )/(v 2 +v 1 ). The probability of reflection isR^2.
(Counterintuitively,R^2 is nonzero even ifU<0, i.e.,v 2 >v 1 .)
This seems to violate the correspondence principle. There is no
morhanywhere in the result, so we seem to have the result that,
even classically, the marble in figure p can be reflected!
The solution to this paradox is that the step in figure o was taken
to be completely abrupt — an idealized mathematical discontinu-
ity. Suppose we make the transition a little more gradual, as in
figure q. As shown in problem 17 on p. 395, this reduces the am-
plitude with which a wave is reflected. By smoothing out the step
more and more, we continue to reduce the probability of reflec-
tion, until finally we arrive at a barrier shaped like a smooth ramp.
More detailed calculations show that this results in zero reflection
in the limit where the width of the ramp is large compared to the
wavelength.
Beta decay: a push or pull on the way out the door example 19
The nucleus^64 Cu undergoesβ+andβ−decay with similar prob-
abilities and energies. Each of these decays releases a fixed
amount of energyQdue to the difference in mass between the
parent nucleus and the decay products. This energy is shared
randomly between the beta and the neutrino. In experiments, the
beta’s energy is easily measured, while the neutrino flies off with-
out interacting. Figure r shows the energy spectrum of theβ+and(^6) As in several previous examples, we cheat by representing a traveling wave
as a real-valued function. See pp. 894 and 970.
908 Chapter 13 Quantum Physics