Other properties can also be determined from for the particle-in-a-box.
We point them out because they are properties that can be determined for
all of the systems that will be considered. The energy of a particle having a
particular wavefunction has already been discussed. Example 10.12 shows
that the observable position can be determined, although only as an average
value. The average value of the (one-dimensional) momentum can also be
determined using the momentum operator. Figure 10.6, which shows plots
of the first few wavefunctions of the particle in a box, illustrates other fea-
tures of the wavefunctions. For example, there are positions in the box where
the wavefunction should be identically zero: at the limits of the box,x 0
and xa, in all cases. For 1 , those are the only positions where 0.
For larger values of the quantum number n, there more positions where the
wavefunction goes to zero. For 2 , there is one more position in the center
of the box. For 3 , there are two additional positions along with the bound-
aries; for 4 , there are three. A nodeis a point at which the wavefunction is
exactly zero. Not including the boundaries, for nthere are n1 nodes in
the wavefunction.
More information is available from a plot of*, which is related to the
probability density that a particle exists at any particular point in the box (al-
though probability densities are evaluated only for regionsof space, not indi-
vidual points in space).†Such plots for some particle-in-a-box wavefunctions
are shown in Figure 10.7. These plots imply that a particle has a varying prob-
ability of existing in different regions of the box. At the boundaries and at
every node, the probability of the particle existing at that point is exactly zero.
At the boundaries this causes no problem, but at the nodes? How can a parti-
cle be on one side of a node and also the other without having any probabil-
ity of existing at the node itself? That’s like being inside a room and then out-
side a room and never being in the doorway. This is the first of many oddities
in the interpretation of quantum mechanics.
The other thing to notice about the plot of probability densities is that as
one goes to higher and higher quantum numbers, the plot of^2 can be ap-
proximated as some constant probability. This is an example of the correspon-
dence principle:at sufficiently high energies, quantum mechanics agrees with
classical mechanics. The correspondence principle was first stated by Niels
Bohr and puts classical mechanics in its proper place: a very good first approx-
imation when applied to atomic systems in high-energy or high-quantum-
number states (and, for all practical purposes, absolutely correct when applied
to macroscopic systems).
Before we leave this section, we point out that this “ideal” system does have
an application in the real world. There are many examples of large organic
molecules that have alternating single and double bonds, a so-called conju-
gated double bond system. In such cases, the electrons in the double bonds are
considered to move somewhat freely from one side of the alternating system
to the other, acting as a sort of particle-in-a-box. The wavelengths of light ab-
sorbed by the molecules can be very well approximated by applying expres-
sions derived for the particle-in-a-box system. Although it is not a perfect fit
between theory and experiment, it is close enough that we acknowledge the
usefulness of the particle-in-a-box model.10.9 Average Values and Other Properties 295†Sometimes the expression *is written as (^2) , indicating that it is a real (that is,
nonimaginary) value.
^2 n 2
^2 n^ ^5
^2 n 18
^2 n high
Figure 10.7 The plots of^2 illustrate the
correspondence principle: for large quantum
numbers, quantum mechanics begins to approx-
imate classical mechanics. At large n, the particle-
in-a-box looks as if the particle were present in all
regions of the box with equal probability.