mally exceed the bounds set by the uncertainty relations by many
orders of magnitude, so there will typically be a vast amount of
overkill in this randomization. This is a general rule for reasoning
about quantum-mechanical measurements: they always completely
randomize the quantum-mechanical phase of the thing being mea-
sured. This provides a more physical justification for our more ab-
stract mathematical proof in example 8 on p. 978 that phase is not
an observable.
In our example of the double slit, what will be the effect of this
randomization of the electron’s phase? In our usual description of
the double slit, we assume that the circular waves emerging from one
slit are in phase with those that come out through the other one,
so that the double-slit interference pattern has a maximum in the
center. But if, for example, one of the waves has its phase inverted,
then all the maxima of our interference pattern will become minima
and vice versa. If the phase is randomized, then the positions of the
maxima and minima are randomized as well, and thus if we try to
collect data on enough electrons to see an interference pattern, we
will not see maxima and minima at all.
One subtle question about this description is the following. The
randomization of the phase by the measurement appears to have
erased the information about the phase relationship between the
parts of the wave in the two slits. But how can this be, since one of
our principles of quantum mechanics (p. 990) is that time evolution
is always unitary, so no information is ever supposed to be lost? The
resolution of this paradox is that the phase information still exists,
but it has been taken away from the electron and flowed out into
the observer and the environment. This is similar to the classical
paradox of what happens to the (classical) information written on
a piece of paper when we burn the paper: the information still
exists, and could in principle be reconstructed by observing all the
molecules and tracing their trajectories back in time using Newton’s
laws.14.9.2 Decoherence
Starting around 1970, physicists began to realize that ideas in-
volving a loss of coherence, or “decoherence,” could help to explain
some things about quantum mechanics that had previously seemed
mysterious. The classical notions of coherence and coherence length
were described in sec. 12.5.8, p. 823, and quantum-mechanical de-
coherence was briefly introduced on p. 885.
One mystery was the fact that it is difficult to demonstrate wave
interference effects with large objects. This is partly because the
wavelengthλ=h/p=h/mvtends to be small for an object with
a large mass. But even taking this into account, we do not seem
to have much luck observing, for example, double-slit diffraction
of very large molecules, even when we use slits with appropriate
Section 14.9 Randomization of phase 999