b/A particle in a box.
One of our principles of quantum mechanics is unitarity (p. 969),
which says, in part, that probability is conserved. Normally we
would interpret this to mean that a wavefunction stays normalized if
it was originally normalized. In this example, the wavefunctions are
not normalizable, but we still expect the fluxes of particles balance
out. We have
flux = (probability density)(group velocity)
= Ψ^2 ·
p
m
=~
mkΨ^2 ,so that if we want the total incident flux to equal the total outgoing
flux, we need
k=kR^2 +k′T^2 ,
which is straightforward to verify.Infinite potential well
In sec. 13.3.3, p. 897, we analyzed the one-dimensional particle
in a box. There was nothing wrong with those results, but it is of in-
terest to see how they fit into the framework of the time-dependent
Schr ̈odinger equation. If we want the walls of the box to be com-
pletely impenetrable, then we should describe it using a potential
such asU(x) =
∞, x < 0
0, 0 < x < L,
∞, x > L,
shown in figure b/1. Because the potential is infinite outside the
box, we expect that there is no tunneling, and zero probability of
finding the particle outside.
In general when we do the cut-and-paste technique, we expect
both the wavefunction and its first derivative to be continuous where
the pieces are joined together. But because we have already solved
this problem by more elementary methods, we know that there will
be kinks in the wavefunction at the walls of the box,x= 0 andL.
The kink is a point where the second derivative∂^2 Ψ/∂x^2 is unde-
fined, and it’s undefined because it’s infinite. The second derivative
is essentially the kinetic energy operator, and normally it would not
be possible to have the kinetic energy be±∞. But in this prob-
lem, itisreasonable to have a kinetic energy of−∞, because the
potential energy is +∞.
Within the box, for a fixed energyE=~ω, the possible wave-
functions will be those of a free particle, which we have already
found. There are two possibilities, of the form
Ψ 1 =ei(kx−ωt)
Ψ 2 =ei(−kx−ωt),972 Chapter 14 Additional Topics in Quantum Physics