Quantum Statistics
211
Our aim is to determine the number of different, distinguishable
arrangements for each of these distributions. The problem is similar to that of
determining the number of distinguishable ways in which Q identical balls can
be placed in different boxes, so that there are q 1 balls in the first box with
f 1 shelves, q 2 balls in the second box with f 2 shelves, etc. (each shelf corresponds
to an energy level). In the case of classical particles, since they are
distinguishable, the balls can be thought of as having different colours. In the
case of bosons and fermions, the balls are identical in every way including their
colour. However, for fermions, there is the additional restriction that at most
one ball can be placed in each shelf.
Identical classical particles: We first determine the number of ways in which
Q particles can be grouped into distinguishable sets of q 1 , q 2 , .., q i ,..., particles
and then the number of ways in which q i particles can be distributed among
f i states of the i-th cell.
The first particle can be chosen in Q number of ways, the second in (Q –1)
ways, etc. However, since the different orders of choosing the same set of
q 1 particles in the set lead to the same result, there are
P 1 (q 1 ) =
1
1
( –1) (
1)
!
− +
Q Q
Q q
q
=
1
1
!
(
)! !
−
Q
Q q q
(7.1)
number of ways choosing q 1 distinguishable particles from Q classical particles.
Similarly, form the remaining Q – q 1 particles, q 2 distinguishable particles can
be chosen in ways. Proceeding in this way, the total number of ways of choosing
P 2 (q 2 ) =
1
1
2
2
(
)!
(
)! !
−
− −
Q q
Q q q q
(7.2)
distinguishable sets of q 1 , q 2 ,..., q i , ... particles from Q distinguishable particles
is found to be
P 1 (q 1 ) P 2 (q 2 ) ... P i (q i ) ... =
1
2
!
! !
!
i
Q
q q
q
(7.3)
Now each of the q i particles can occupy any one of the f i states so that there
are f i
qi
number of ways of distributing q i distinguishable particles in f i states.
Thus the total number of distinguishable arrangements for the distribution of
q 1 , q 2 , ..., q i , ... sets of distinguishable practices in f 1 , f 2 , .., f i , ... states is
P (q i ) = Q!
1
!
q
i
i
i
i
f
q
∞
=
Π
(7.4)
211
Our aim is to determine the number of different, distinguishable
arrangements for each of these distributions. The problem is similar to that of
determining the number of distinguishable ways in which Q identical balls can
be placed in different boxes, so that there are q 1 balls in the first box with
f 1 shelves, q 2 balls in the second box with f 2 shelves, etc. (each shelf corresponds
to an energy level). In the case of classical particles, since they are
distinguishable, the balls can be thought of as having different colours. In the
case of bosons and fermions, the balls are identical in every way including their
colour. However, for fermions, there is the additional restriction that at most
one ball can be placed in each shelf.
Identical classical particles: We first determine the number of ways in which
Q particles can be grouped into distinguishable sets of q 1 , q 2 , .., q i ,..., particles
and then the number of ways in which q i particles can be distributed among
f i states of the i-th cell.
The first particle can be chosen in Q number of ways, the second in (Q –1)
ways, etc. However, since the different orders of choosing the same set of
q 1 particles in the set lead to the same result, there are
P 1 (q 1 ) =
1
1
( –1) (
1)
!
− +
Q Q
Q q
q
=
1
1
!
(
)! !
−
Q
Q q q
(7.1)
number of ways choosing q 1 distinguishable particles from Q classical particles.
Similarly, form the remaining Q – q 1 particles, q 2 distinguishable particles can
be chosen in ways. Proceeding in this way, the total number of ways of choosing
P 2 (q 2 ) =
1
1
2
2
(
)!
(
)! !
−
− −
Q q
Q q q q
(7.2)
distinguishable sets of q 1 , q 2 ,..., q i , ... particles from Q distinguishable particles
is found to be
P 1 (q 1 ) P 2 (q 2 ) ... P i (q i ) ... =
1
2
!
! !
!
i
Q
q q
q
(7.3)
Now each of the q i particles can occupy any one of the f i states so that there
are f i
qi
number of ways of distributing q i distinguishable particles in f i states.
Thus the total number of distinguishable arrangements for the distribution of
q 1 , q 2 , ..., q i , ... sets of distinguishable practices in f 1 , f 2 , .., f i , ... states is
P (q i ) = Q!
1
!
q
i
i
i
i
f
q
∞
=
Π
(7.4)