By way of rigorous proof, suppose to the contrary that we did
have such an observableOph. By our definition of an observ-
able, it would have to have some set of eigenvalues that were
real numbers. Consider such an eigenvalueφ, which might per-
haps be the argument of the wavefunction in the complex plane,
although we will not need to assume that. LetΨbe the state of
definite phase having the phaseφ, so that
[ 1 ] OphΨ=φΨ.
We can change the phase ofΨto create a new wavefunction.
Let’s retard its phase by 90 degrees, creatingiΨ. SinceΨwas a
state of definite phase, clearlyiΨis as well, and and it must have
some different eigenvalueφ′. Perhapsφ′=φ+π/2, but in any
case we must haveφ′ 6 =φ. Then
[ 2 ] Oph(iΨ) =φ′(iΨ).
But by linearity equation[ 2 ]is equivalent toiOphΨ = iφ′Ψ, or
OphΨ=φ′Ψ, and therefore by comparison with equation[ 1 ],φ=
φ′, which is a contradiction, so we conclude that there cannot be
an observable representing phase.
The result of example 8 was a bit of a foregone conclusion, since
we specifically designed our notion of an observable to be insensitive
to phase. Therefore this argument is subject to the objection that
perhaps there is some way to measure a quantum-mechanical phase,
but our definition of an observable is just too restrictive to describe
it. However, we will see on p. 998 that there are more concrete
reasons why phase cannot be measured.
Time is not an observable example 9
We do not expect to have a time operator in quantum mechan-
ics. This follows simply because an operator is supposed to be
a function that takes a wavefunction as an input, but we typically
can’t tell what time it is by looking at the wavefunction. For exam-
ple, if the electron in a hydrogen atom is in its ground state, then
we could say its energy is zero, so its frequency is zero, the pe-
riod is infinite, and the wavefunction doesn’t vary at all with time.
(We can choose our reference level for the electrical energyUelec
to be anything we like. Even if we choose it such that the energy
of the ground state is nonzero, the only change in the electron’s
wavefunction over time will be a phase rotation, which by example
8 is not observable.)
Of course this doesn’t mean that quantum mechanics forbids us
from building clocks. It just tells us that many quantum mechan-
ical systems are too simple to function as clocks. In particular,
we would be misled if we pictured a hydrogen atom classically in
terms of an electron traveling in a circular orbit around a proton,
in which case it really could act like the hand on a tiny clock. For
further discussion of this idea, see p. 997
Section 14.6 The underlying structure of quantum mechanics, part 2 979