B The top panel of the figure shows a series of snapshots in the
motion of two pulses on a coil spring, one negative and one positive, as
they move toward one another and superpose. The final image is very
close to the moment at which the two pulses cancel completely. The
following discussion is simpler if we consider infinite sine waves rather
than pulses. How can the cancellation of two such mechanical waves be
reconciled with conservation of energy? What about the case of colliding
electromagnetic waves?
Quantum-mechanically, the issue isn’t conservation of energy, it’s con-
servation of probability, i.e., if there’s initially a 100% probability that a
particle exists somewhere, we don’t want the probability to be more than
or less than 100% at some later time. What happens when the colliding
waves have real-valued wavefunctionsΨ? Now consider the sketches of
complex-valued wave pulses shown in the bottom panel of the figure as
they are getting ready to collide.
C The figure shows a skateboarder tipping over into a swimming
pool with zero initial kinetic energy. There is no friction, the corners are
smooth enough to allow the skater to pass over the smoothly, and the
vertical distances are small enough so that negligible time is required for
the vertical parts of the motion. The pool is divided into a deep end and a
shallow end. Their widths are equal. The deep end is four times deeper.
(1) Classically, compare the skater’s velocity in the left and right regions,
and infer the probability of finding the skater in either of the two halves if
an observer peeks at a random moment. (2) Quantum-mechanically, this
could be a one-dimensional model of an electron shared between two
atoms in a diatomic molecule. Compare the electron’s kinetic energies,
momenta, and wavelengths in the two sides. For simplicity, let’s assume
that there is no tunneling into the classically forbidden regions. What is
the simplest standing-wave pattern that you can draw, and what are the
probabilities of finding the electron in one side or the other? Does this
obey the correspondence principle?
Section 13.3 Matter as a wave 917