B.3.a. Number Representation For One State
Consider the occupation of an 1-particle
-particle state in which
must be a non-negative integer. With the understanding that we are dealing
only with the occupation of
we can suppress the subscript a for the sake of clarity.
For convenience, we shall assume the set
to be orthonormal, i.e.,
denoting the hermitian conjugate, the number
operators are defined by
Taking the hermitian conjugates of (2,3)
Thus, (3a)´(3) gives
Similarly, (4a)´(4) gives
Without loss of generality, we can set all
to be real and non-negative. Hence,
and, from (6),
Thus, (3,4) become
Now, from (9), we have
for all n
Hence, we have the commutator relation
Note that the results (9-11) are obtained
under the implicit assumption that there is no upper limit to n.
Hence, it is applicable to bosons.
Conversely, we can say that bosonic behavior will be observed if we
demand the commutator relation (11).
For fermions, each state can be occupied at
most once., i.e., n can only take on
real and non-negative, we have
for all allowable n.