Classically, we know that knowledge of 
and 
at a
given time specifies the state of the system. Quantum mechanically,
the state of a system is specified when the probability distribution
of all observables are determined. We still do not know what is the
minimal amount of information needed to specify the quantum state
. The answer to this question depends on the complexity
of the system and the observables. As we mentioned at the end of the
last section, we will focus on systems where all observables may be
derived from the position and momentum of a single particle. For
these systems we may ask whether we need specify both 
and 
to determine the quantum state.
Perhaps, just one or the other will do. Or perhaps, some other
quantity will specify both.
To address these issues consider the thought experiment carried out with photons in Figure 2.
Photons from a distance source S pass through a filter F,
enter a semicircular chamber of radius R and eventually register on
the photographic plate P. The thickness of the filter varies so that the
intensity of photons entering the chamber at y = 0 is proportional to
. The geometry is arranged so that
. The amplitude of the entering wave
then obeys
We have included a proportionality constant and an arbitrary phase
factor because (1) only determines 
up to an arbitrary
complex phase. In this situation we are fortunate and may determine
the phase by assuming continuity of the phase of the wave as it
crosses the filter so that 
, where 
is the x-component of
the incoming momentum. Knowing the phase factor, we then have
Finally, if we wish to interpret 
as the probability
distribution 
for the location of individual phonons as
they cross the filter, we may determine 
from the
normalization condition,
thus,
and
After entering the chamber with this wavefunction along x, the
photons then propagate to the photographic plate where they arrive
at the position of angle 
with an intensity given by
The phase factor 
comes about because to arrive on
the photographic plate at angle 
, photons
leaving the plate at x must travel a distance 
less than those which cross at x=0. (See figure.)
Note that the angle 
gives us a measurement of the
x-component of the momentum of each photon as it left the filter to
be recorded on the photographic plate. To arrive at angle 
the x component of the momentum must have obeyed 
where p was the total momentum given by 
. Because the momentum p and wave number
k are so closely related (just through the proportionality constant
), we will often use them interchangeably. We for instance
speak of the x component of 
.
If we now interpret the intensity (4) as a probability for
individual photons we find
For our particular case,
So,
Normalizing, using the same procedure we used for 
gives the final
result
which appears as in the sketch below.
As a result in the localization in x, there is a spread in the
momenta 
in accordance with the uncertainty principle. The
key lesson, here, however comes from the fact that the
distribution in the momentum in the x-direction is centered about
its incoming value 
, as we would expect on physical
grounds. This is a direct result of the phase factor in (2). (The
student should note that the inclusion of a phase in a Fourier
integral of the form (6) always represents a shift in the final
result). Without this phase factor it is impossible to determine
where the distribution 
should be centered.
Thus, 
, which gives none of this phase information, is
insufficient to specify the state of the system.
On the other hand, the function 
specifies both 
and 
and so does specifies the state of
the system entirely. We shall now see that the fact that 
gives both 
and 
is a consequence
of the general physical principle of superposition.
