implies that the waves obey the principle of superposition. In most
cases in nature, we find that the principle of superposition for waves
is at best an approximation. For example, if the amplitude of a
tsunami is so huge that the trough of the wave reaches all the way
down to the ocean floor, exposing the rocks and sand as it passes
overhead, then clearly there is no way to double the amplitude of the
wave and still get something that obeys the laws of physics. Even
at less extreme amplitudes, superposition is only an approximation
for water waves, and so for example it is only approximately true
that when two sets of ripples intersect on the surface of a pond, they
pass through without “seeing” each other.
It is therefore natural to ask whether the apparent linearity of the
Schr ̈odinger equation is only an approximation to some more pre-
cise, nonlinear theory. This is not currently believed to be the case.
If we are to make sense of Schr ̈odinger’s cat (sec. 13.2.4, p. 885),
then the experimenter who sees a live cat and the one who sees a
dead cat must remain oblivious to their other selves, like the ripples
on the pond that intersect without “seeing” each other. Attempts
to create slightly nonlinear versions of standard quantum mechan-
ics have been shown to have implausible physical properties, such as
allowing the propagation of signals faster thanc. (This is known as
Gisin’s theorem. The original paper, “Weinberg’s non-linear quan-
tum mechanics and supraluminal communications,” is surprisingly
readable and nonmathematical.)
If you have had a course in linear algebra, then it is worth noting
that the linearity of the Schr ̈odinger equation allows us to talk about
its solutions as vectors in a vector space. For example, if Ψ 1 repre-
sents an unstable nucleus that has not yet gamma decayed, and Ψ 2
is its state after the decay, then any superpositionαΨ 1 +βΨ 2 , with
real or complex coefficientsαandβ, is a possible wavefunction, and
we can notate this as a vector,〈α,β〉, in a two-dimensional vector
space.Discussion Questions
A The zero level of interaction energyUis arbitrary, e.g., it’s equally
valid to pick the zero of gravitational energy to be on the floor of your lab
or at the ceiling. Suppose we’re doing the double-slit experiment, t/3, with
electrons. We define the zero-level ofUso that the total energyE=U+K
of each electron is positive. and we observe a certain interference pattern
like the one in figure i on p. 878. What happens if we then redefine the
zero-level ofUso that the electrons haveE<0?916 Chapter 13 Quantum Physics