Quantum Mechanics for Engineers 

© Leon van Dommelen 

D.25 Number of boson states
For identical bosons, the number is choose . To see
that think of the bosons as being inside a series of single
particlestate boxes.
The idea is as illustrated in
figure D.2; the circles are the bosons and the thin lines
separate the boxes. In the picture as shown, each term in the group
of states has one boson in the first singleparticle function, three
bosons in the second, three bosons in the third, etcetera.
Figure D.2:
Bosons in singleparticlestate boxes.

Each picture of this type corresponds to exactly one system state. To
figure out how many different pictures there are, imagine there are
numbers written from 1 to on the bosons and from to
on the separators between the boxes. There are then ways
to arrange that total of objects. (There are choices
for which object to put first, times choices for which object
to put second, etcetera.) However, the different ways to order
the subset of boson numbers do not produce different pictures if you
erase the numbers again, so divide by . The same way, the
different ways to order the subset of box separator numbers do not
make a difference, so divide by .
For example, if 2 and 4, you get 5!2!3!
or 10 system states.