The above discussion centers on ensemble averages over systems whose dynamics is primarily quantum-mechanical. In this section we consider the classical ensemble average of Lectures 3-7, and how their nature is perturbed by quantum effects. There are two perturbations, one stemming from the symmetry of the wave function, the other nominally stemming from the uncertainty principle.
Principle 12 of Lecture 10 required that the wave function for identical particles obey the symmetry condition
where for real particles 
is 0 or 
. This condition ensures that
and 
refer two different parts of the same wave-function; i.
e. the two 
's refer to the same state of the system. If we follow the
prescription that an ensemble is supposed to be a complete, non-repeating
list of the states of the system, then phases (states, identified by their
particle positions and momenta) which differ from each other only by the
exchange of identical particles are simply different ways of labelling the
same state of the system. For example, in the classical ideal gas, the
phases 
and 
are indistinguishable, and should only be counted once in the
ensemble average.
In a set of N identical particles, the phase space integral
overcounts states, because each distinct state of the system can be
represented as approximately N! different points
in phase space. Heuristically, the N! arises from combinatorial
arguments. Suppose a certain state of the system puts a particle at a,
another particle at b, a third particle at c, etc. Within the
phase integral, any of the N particles could be at a, any of the N-1
particles not at a could be at b, any of the N-2 remaining
particles could be at c, etc. The phase space integral is completely
systematic, so the integral puts each particle at a, each of the
remaining particles at b, and so on, so that a single state 
of the system is generated N! times in N! different ways by
the phase space integral. The phase space integral thereby overcounts states
of the system. The factor N! can be divided away, restoring agreement
between the phase space integral and the corresponding quantum calculation.
[Note that I said 'approximately', because there are complications if two
particles are at the same location. If there is a significant likelihood that
more than one particle is in the same quantum state, a more sophisticated
analysis is required.
It is sometimes asserted that the N! overcounting effect could not have been obtained before the invention of quantum mechanics, or that quantum mechanics can be used to derive the N! correction to the phase space integral. Neither of these arguments is precisely correct. On one hand, the N! correction was known well before quantum mechanics was invented[3]; we've shown in Lecture 5 why this term would be expected for classical particles which cannot be told apart. Gibbs set down the N! term as a factor which must necessarily be present to explain the entropy of mixing of gases, as discussed in a text section yet to be written.
The above arguments do not derive the factor N!. Quantum mechanics shows that a classical system has N! times as many phase space points as a quantum system has states, This relation does not reveal whether the correct ensemble average should divide the phase space integral by N!, or whether the quantum sum of states should be multiplied by N!. The derivation of the N! factor relied on the assertion that each quantum state was to appear once and only once in the sum over allowed states. This assertion is strictly statistical mechanical, not something derived from quantum mechanics. .
The other quantum correction to the phase space integrals can be described heuristically in terms of the Heisenberg uncertainty principle. I stress that a sound derivation of this correction -- the Kirkwood-Wigner theorem, treated below -- does not use the uncertainty principle. Indeed, naive arguments based on the uncertainty principle do not obtain correct numerical factors. Proceeding heuristically:
The phase-space integral identifies states by the positions and momenta of the
individual particles. For the one-particle system, 
and 
refer classically
to two different states of the system.
The uncertainty principle instructs us that we cannot measure a position and
its canonically conjugate momentum simultaneously with complete accuracy. Two
states in which the positions and momenta of the particle differ from each
other by too small an amount cannot be distinguished. For the one-particle
system, in order for the two abovementioned states to be distinguished, one
must have
If 
, then and
refer to the same state of the system.
Eq. 11.21 differs from the more common forms of the Heisenberg
uncertainty principle in the presence of h rather than 
as a bound.
In the usual form ( 
) of the principle, the
s are the half-widths at half-heights of the uncertainty in x and
p of a state which has minimized the joint uncertainty in x and p. It
happens, as is effectively proven by the Kirkwood-Wigner theorem, that ``half
width at half height'' is not the correct criterion for distinguishing between
two statistico-mechanical states, the correct criterion being given by eq.
11.21. In a 6N-dimensional phase space, states would be
indistinguishable if they lie within the same volume of extent 
, so by
dividing by we nominally turn the phase space integral into a count of
phase space states.
With these corrections, a classical ensemble average becomes
while the canonical partition function is written
Observe that state counting issues do affect the partition function but do not change the value of the ensemble average of any variable.
