11.3 How Many System Eigenfunctions?

The fundamental question from which all of quantum statistics springs is a very basic one: How many system energy eigenstates are there with given generic properties? This section will address that question.

Of course, by definition each system energy eigenfunction is unique. Figures 11.1-11.3 give examples of such unique energy eigenfunctions for systems of distinguishable particles, indistinguishable bosons, and indistinguishable fermions. But trying to get accurate data on each individual eigenfunction just does not work. That is much too big a challenge.

Quantum statistics must satisfy itself by figuring out the
probabilities on groups of system eigenfunctions with similar
properties. To do so, the single-particle energy eigenstates are best
grouped together on shelves of similar energy, as illustrated in
figures 11.1-11.3. Doing so allows for more
answerable questions such as: “How many system energy
eigenfunctions

That question is answerable with some clever mathematics; it is a big
thing in various textbooks. However, the suspicion is that this is
more because of the neat

mathematics than because of
the actual physical insight that these derivations provide. In this
book, the derivations are shoved away into {D.56}. But
here are the results. (Drums please.) The system eigenfunction
counts for distinguishable particles, bosons, and fermions are:

where

where

For example, 5! = 1

This section is mainly concerned with explaining qualitatively why these system eigenfunction counts matter physically. And to do so, a very simple model system having only three shelves will suffice.

The first example is illustrated in quantum-mechanical terms in figure
11.4. Like the other examples, it has only three
shelves, and it has only

Now the question is, how many energy eigenfunctions are there for a
given set of shelf occupation numbers

Some example observations about the figure may help to understand it.
For example, there is only one system eigenfunction with all 4
particles on shelf 1, i.e. with

This is represented by the white square at the origin in the left graph of figure 11.5.

As another example, the darkest square in the left graph of figure
11.5 represents system eigenfunctions that have shelf
numbers

is the one depicted in figure 11.4. It has particle 1 in single-particle state

such eigenfunctions, since there are 4 possible choices for the particle that goes on shelf 1, times a remaining 3 possible choices for the particle that goes on shelf 3, times 8 possible choices

Next, consider a system four times as big. That means that there are
four times as many particles, so

If the system size is increased by another factor 4, to 64 particles,
the number of states with occupation numbers

These general trends do not just apply to this simple model system; they are typical:

The number of system energy eigenfunctions for a macroscopic system is astronomical, and so are the differences in numbers.

Another trend illustrated by figure 11.5 has to do with the
effect of system energy. The system energy of an energy eigenfunction
is given in terms of its shelf numbers by

so all eigenfunctions with the same shelf numbers have the same system energy. In particular, the squares just below the oblique cut-off line in figure 11.5 have the highest system energy. It is seen that these shelf numbers also have by far the most energy eigenfunctions:

The number of system energy eigenfunctions with a higher energy typically dwarfs the number of system eigenfunctions with a lower energy.

Next assume that the system has exactly the energy of the oblique
cut-off line in figure 11.5, with zero uncertainty.
The number of energy eigenstates !

for details.) The maximum number of
energy eigenstates occurs at about most probable set of occupation numbers.

If
you pick an eigenfunction at random, you have more chance of getting
one with that set of occupation numbers than one with a different
given set of occupation numbers.

To be sure, if the number of particles is large, the chances of
picking any eigenfunction with an exact set of occupation
numbers is small. But note how the spike

in figure
11.6 becomes narrower with increasing number of particles.
You may not pick an eigenfunction with exactly the most
probable set of shelf numbers, but you are quite sure to pick one with
shelf numbers very close to it. By the time the system size reaches,
say, 1

Since there is only an incredibly small fraction of eigenfunctions that do not have very accurately the most probable occupation numbers, it seems intuitively obvious that in thermal equilibrium, the physical system must have the same distribution of particle energies. Why would nature prefer one of those extremely rare eigenfunctions that do not have these occupation numbers, rather than one of the vast majority that do? In fact, {N.23},

So the most probable set of shelf numbers, as found from the count of eigenfunctions, gives the distribution of particle energies in thermal equilibrium.It is a fundamental assumption of statistical mechanics that in thermal equilibrium, all system energy eigenfunctions with the same energy have the same probability.

This then is the final conclusion: the particle energy distribution of a macroscopic system of weakly interacting particles at a given energy can be obtained by merely counting the system energy eigenstates. It can be done without doing any physics. Whatever physics may want to do, it is just not enough to offset the vast numerical superiority of the eigenfunctions with very accurately the most probable shelf numbers.