10.2 Two Types of Quantum Statistics
517
temperatures. We have also seen that at sufficiently high temperatures we can treat
the energy of our system as essentially classical, so that MB statistics and the
classical limit are essentially equivalent.
10.2 Two Types of Quantum Statistics
There are two cases to consider in the evaluation of the partition function for sets of
indistinguishable particles. The resultant distribution for fermions is known as the
‘Fermi–Dirac distribution’, while that for bosons is known as the ‘Bose–Einstein
distribution’. We should note, moreover, that these distributions are the only exact
distributions.
N -particle wavefunctions are complicated to construct, as the requirement
of antisymmetrization (fermions) or symmetrization (bosons) may place strong
restrictions on the accessibility of states at the specified energy to the particles.
However, because we are dealing with inherently indistinguishable particles, the
only meaningful statistical data are the numbers of particles that occupy the
allowed quantum states. Thus, rather than attempting to construct sets of N-particle
wavefunctions, it will prove more useful to describe the system as a whole in terms
of ‘occupation numbers’, n k , of the available/accessible energy states. In quantum
chemistry, this is referred to as the Fock basis. Each single-particle state corresponds
to a distinct energy k , and the energy states are ordered by increasing energy. We
may introduce the notation |n 1 , n 2 , . . . , n k , . . . to designate the N -particle state for
which n 1 particles occupy the state of energy 1 , n 2 that of energy 2 , and so forth.
Of course, there must be a constraint on the set of n k -values, since the total number
of particles must be N, i.e., the n k are required to satisfy the condition
k
n k = N ,
(10.2.1)
while at the same time the total energy E N of the N -particle state must be given by
k
n k k = E N .
(10.2.2)
If we wished to work in terms of the canonical ensemble, we would have to write
Z(N, V , T ) as
Z(N, V , T ) =
j
e
−βE j =
{n k }
∗
exp
−β
i
n i i
,
(10.2.3)
in which the asterisk on the outer summation is to remind us that only those n k such
that
k n k = N are to be included in it. This restriction is extremely difficult to
517
temperatures. We have also seen that at sufficiently high temperatures we can treat
the energy of our system as essentially classical, so that MB statistics and the
classical limit are essentially equivalent.
10.2 Two Types of Quantum Statistics
There are two cases to consider in the evaluation of the partition function for sets of
indistinguishable particles. The resultant distribution for fermions is known as the
‘Fermi–Dirac distribution’, while that for bosons is known as the ‘Bose–Einstein
distribution’. We should note, moreover, that these distributions are the only exact
distributions.
N -particle wavefunctions are complicated to construct, as the requirement
of antisymmetrization (fermions) or symmetrization (bosons) may place strong
restrictions on the accessibility of states at the specified energy to the particles.
However, because we are dealing with inherently indistinguishable particles, the
only meaningful statistical data are the numbers of particles that occupy the
allowed quantum states. Thus, rather than attempting to construct sets of N-particle
wavefunctions, it will prove more useful to describe the system as a whole in terms
of ‘occupation numbers’, n k , of the available/accessible energy states. In quantum
chemistry, this is referred to as the Fock basis. Each single-particle state corresponds
to a distinct energy k , and the energy states are ordered by increasing energy. We
may introduce the notation |n 1 , n 2 , . . . , n k , . . . to designate the N -particle state for
which n 1 particles occupy the state of energy 1 , n 2 that of energy 2 , and so forth.
Of course, there must be a constraint on the set of n k -values, since the total number
of particles must be N, i.e., the n k are required to satisfy the condition
k
n k = N ,
(10.2.1)
while at the same time the total energy E N of the N -particle state must be given by
k
n k k = E N .
(10.2.2)
If we wished to work in terms of the canonical ensemble, we would have to write
Z(N, V , T ) as
Z(N, V , T ) =
j
e
−βE j =
{n k }
∗
exp
−β
i
n i i
,
(10.2.3)
in which the asterisk on the outer summation is to remind us that only those n k such
that
k n k = N are to be included in it. This restriction is extremely difficult to