Unobservability of phase and
normalization
When we say that phase and
normalization don’t count as
knowledge of a system, we’re
saying something very mathe-
matically specific: that Ψ and
cΨ represent the same state,
wherec 6 = 0 is a complex num-
ber; the magnitude ofcwould
only affect the normalization,
and its argument would only
affect the phase. We do not
mean, for example, that wave-
functions like sinx and cosx
are indistinguishable. The sine
and cosine give different prob-
ability distributions, so they
are distinguishable. For ex-
ample, the sinx wavefunction
gives zero probability of detec-
tion atx = 0. See also prob-
lem 17, p. 1012 and example 8,
p. 978.
Linear algebra application
Wavefunctions can be de-
scribed by vectors in a vector
space (p. 964). A state is a
one-dimensional subspace of
the vector space, i.e., the set of
all wavefunctions of the form
cΨ for some fixed Ψ.
14.4 The underlying structure of quantum me-
chanics, part 1
So far we have been building up the structure of quantum mechanics
by casually laying one brick on top of another, but at this point it
will be advantageous to pause and consider the broader blueprint.14.4.1 The time-dependent Schrodinger equation ̈
For simplicity, our discussion of the Schr ̈odinger equation in sec-
tion 13.3.6, p. 905, was limited to standing waves, allowing us to
avoid explicitly discussing how the wavefunction changed with pass-
ing time. Let’s consider the generalization to the full time-dependent
case.
Classically, suppose I show you a picture of a baseball next to
a tree, and I ask you how long it will take to hit the ground. You
can’t tell, because you also need information about the ball’s initial
velocity. That is, the future time-evolution of the systemx(t) de-
pends not just on the initial positionx(0) but also on its initial time
derivativex′(0).
But if I show you a uranium atom in its lowest energy state, you
don’t need to know any other information to predict everything that
can be predicted about its future decay. Whereas the baseball could
be thrown downward in order to make it reach the ground more
quickly, nobody knows of any way to prepare the uranium nucleus
in such a way that it is any more likely to decay sooner. Knowing
the initial wavefunction Ψ(0) to be that of the ground state lets us
say as much as can be said about the future time-evolution Ψ(t),
and it’s neither necessary nor helpful to know the time derivative
Ψ′(t).
This is an example of a more general idea about the interpre-
tation of quantum mechanics, which is that the wavefunction is a
complete description of any system. There isn’t more information
that can be known about the system. This principle seems to be
widely agreed upon by physicists, but doesn’t seem to have a stan-
dard name. (The phase and normalization of the wavefunction are
not considered to give any information, since the phase is unobserv-
able, and the normalization can be standardized so that the total
probability is 1. See the sidebar for more detail.)Wavefunction fundamentalism
All knowable information about a system is encoded in its wave-
function (ignoring phase and normalization).An example of an idea that would violate this principle is the pilot
wave theory proposed by de Broglie around 1927, and improved
by Bohm in the 1950’s. In this theory, an electron-particle is a
separate object from an electron-wave, with the particle surfing the966 Chapter 14 Additional Topics in Quantum Physics