We 
will start now quantum statistic. We will
discuss Fermi Dirac and Bose Einstein statistics.
So far we have discussed classical statistics
or Maxwell Boltzmann statistics. Since, all
the known particles are either fermions or
bosons so Fermi Dirac and Bose Einstein these
two statistics are exact statistics and in
special condition they will reduce to Maxwell
Boltzman statistics, that we will discuss.
Coming back to Fermi Dirac and Bose Einstein
statistics, we consider a system of ‘N’
identical particles, described by a wave function
Psi 1, 2, 3…. N where ‘1’ denotes the
coordinates of particle ‘1’, ‘2’ denotes
the coordinates of particle 2 and so on. Now,
if we interchange the positions of any of
the two particles, suppose we are interchanging
the positions of particle ‘1’ and particle
’2’, the wave function must either remain
the same or changes sign.
If we operate an operator P12 on the wave
functions Psi and this operator exchanges
the coordinates of particles 1 and 2. We are
operating an operator P12 on the wave function
Psi and this operator exchanges the coordinates
of particles 1 and 2. If we do it, we get
back the original wave function with a plus
or minus sign.
So it turns out that whether the wave function
remains the same or changes its sign is a
function of the nature of the two identical
particles that are exchanged.
For particles with integral spin (such as
nucleus of helium 4, photon etc), the wave
function remains the same. In this case the
wave function is called a symmetric wave function,
such particles are known as bosons. So bosons
have integral spins.
On the other hand, for particles with half
integrals spin (such as electron, proton,
etc), the wave function is called antisymmetric
wave function and such particles are known
as fermions. Basically what it said is for
half integral spin particles we get minus
of Psi after operating the exchange operator.
As I said since all known particles are either
fermions or bosons, so these two statistics
are only exact statistics.
Since all known particles are either fermions
or bosons, which are indistinguishable particles
these two statistics that is Fermi Dirac (we
call it FD) and Bose-Einstein statistics (we
call BE) are the exact distributions.
We shall see, however, in the case of high-temperature
and/or low density both these distributions
(that is FD and BE) go over into the Boltzmann
or classical distribution.
Now, we will do the derivation of these two
statistics. Like before we consider let EJ
which is function of number of particles N
and volume V, the energy states available
to a system containing ‘N’ particles.
Epsilon k are the molecular quantum states,
n k is function of Ej here with number of
molecules in the k-th molecular state when
the system itself in the j-th state with energy
Ej.
So we define a three quantities, first quantity
is Ej, Ej is the energy states available to
a system containing N particles. Epsilon k
is the molecular quantum states and nk is
the number of molecules in the k-th molecular
state when the system itself in the j-th state
with energy Ej.
So, the energy of the system in the j-th state
we define this as Ej is sum over k nk times
epsilon k and N is total number of particle
in the j-th state is sum over k nk. We know,
partition function Q is nothing but sum over
j e to the minus beta Ej and we can write
this one like this. So Ej we are replacing
by epsilon i times sum over i epsilon i times
ni, so i is a dummy variable here and here
star signifies the restriction sum over k
nk equals to N, sum over k nk is nothing but
total number of particle in that state N.
So what we did here, here basically what we
did, we had sum over j e to the minus beta
Ej, we replace it by sum nK and then star
e to the minus beta epsilon i ni, here (these
are the basic) they basically represent the
same quantity because summing over the states
of the system is equivalent to summing over
the occupation numbers of each molecular level
subjected to the condition sum over k nk is
N.
So they represent this sum of suppose this
is our equation 1, sum of equation 1 and sum
of equation 2, subjected to the condition
that sum over k nk gives number of particles.
Now, we also know, partition function theta
which is function of V, T and mu is sum over
N goes from 0 to infinity e to the beta mu
N then Q N, V, T, we already derived that.
So we can write it like sum over N goes from
0 to infinity lambda to the N where lambda
is e to the beta times mu.
Here we substitute e to the beta mu by lambda
and then Q N, V, T we know that it is sum
over nk e to the minus beta sum over i ni
times epsilon i . We can write further this.
It can take a lambda inside the summation
here, so we can write lambda is sum over i
ni, so again i is a dummy variable here, so
sum over i ni gives N and then the second
term was there, we can write theta (V, T,
mu) is like this.
In the next step what we did is theta is sum
over n goes from 0 to infinity and sum over
nk and then lambda sum over i ni and e to
the minus beta i epsilon i, we can write like
product of lambda e to the minus beta epsilon
k to the nk and here k varies from 1, 2, 3.
Since, we are summing over all values of N,
each nk ranges over all possible values. So,
equation 3 can be written as now we are replacing
sum over N by n1, sum over n 2 etcetera where
the minimum value of n1 is 0 and maximum value
for n1 is n1 max, similarly, for n2 the minimum
value is 0 and 
the maximum value of n2 is n2 max and so on.
We can further simplify the above equation
like sum over n1 goes from 0 to n1 max lambda
e to the minus beta epsilon 1 to the n1 times
n2 goes from 0 to n2 max lambda e to the minus
beta epsilon 2 to the n2 and so on.
So we get a very simple expression theta is
equal to product sum over nk goes from 0 to
nk max lambda e to the minus beta epsilon
k to the nk, suppose this is our equation
number 4. So far we have not imposed any restriction
or any assumption or any approximation.
For fermions there is a restriction. For Fermi-Dirac
statistics since no two particles can be in
the same quantum state because of Pauli’s
exclusion principle, the maximum possible
values of n1 max is 1, similarly, for n2 max
the maximum possible value of n2 max is 1
and so on.
Thus, equation 4 becomes this. The actual
derivation is in equation 4 we had theta is
product over k and then sum over nk, nk goes
from 0 to nk max lambda e to the minus beta
epsilon k to the nk and for FD statistics
we can write theta FD is like this.
I have shown the actual derivation of how
we arrived at equation 5 below here. If you
just impose the condition that maximum value
of n1, n2 etc is 1, then if you expand this
equation 4, you get sum over n1, n1 goes from
0 to 1 lambda e to the minus beta epsilon
1 to the n1 times n2 again n2 goes from 0
to 1 lambda e to the minus beta epsilon 2
to the n2 and so on.
Now we substitute n1 equals to 0 in the next
line we get first term is 1 and the second
term is we substitute n1 equals to 1 we get
lambda e to the minus beta epsilon 1. For
the second term if we substitute n2 equals
to 0 we get one term here and then we have
pass if we substitute n2 equals to 1 we get
lambda e to the minus beta epsilon 2 and so
on.
So we get 1 plus lambda e to the minus beta
epsilon k to the nk. This is our partition
function or our Fermi-Dirac distribution.
Now for BE statistics or for bosons there
is no such restriction in the number of particles
found in a given state, again we will start
with start from equation number 4 and we will
derive at the partition function for Bose-Einstein
statistics. In Bose-Einstein statistics, on
the other hand nk can be 0, 1, 2, etc. Since,
there is no restriction on the occupancy of
each state. Therefore, n1 max equals to infinity,
n2 max equals to infinity and so on, equation
4 becomes like this. Actual derivation I am
showing now. This is our equation 6, equation
6 is the partition function for Bose Einstein
statistics. From Equation 4 we have the actual
derivation I am showing now.
Equation 4 is theta is pi k sum over nk, nk
goes from 0 to nk max and then we have lambda
e to the minus beta k limit epsilon k to the
nk. So we can write theta like this. If we
expand it we get theta is 1 plus lambda e
to the minus beta epsilon 1 plus lambda e
to the minus beta epsilon 1 to the 2 plus
so on and then for the second term we get
1 plus lambda e to the minus beta epsilon
2 plus lambda e to the minus beta epsilon
to the 2 plus and so on.
It is like 1 plus x plus x to the 2 plus x
to the 3 like this. This is nothing but we
can write for FD statistics, we get some product
of k 1 plus e to the minus beta epsilon k
and in case of BE statistics we get theta
BE is sum over product of k 1 minus e to the
minus beta epsilon k, here we considered lambda
e to the minus beta epsilon k is less than
1.
So in general we can write, theta FD, BE is
product over k and then 1 plus minus lambda
e to the minus beta epsilon k plus minus 1.
Where plus sign is for FD statistics and minus
sign is for BE statistics. So we have got
the partition function or distribution function
for FD and BE statistics.
Next we calculate average number of particles,
N average, is nothing but we can call N is
sum over k nk 
and we know how to calculate average number
of particles, this is KB times T del ln theta
by del mu at constant V and T. Now lambda
is e to the beta mu, so d lambda is beta e
to the beta mu d mu and we can write this
is nothing but beta lambda d mu, so d mu is
nothing but 1 by beta lambda d lambda. Average
number of particles is nothing but KBT del
ln theta by del lambda times 1 by 1 beta times
lambda. So we get average number of particles
is nothing but lambda del ln theta by del
lambda at constant volume and temperature.
This is 
the average number of particles and we have
the expression for theta.
For FD statistic, this is our expression.
If we expand it we get ln 1 plus lambda times
e to the minus beta epsilon plus ln l plus
lambda e to the minus beta epsilon 2 and so
on. This is ln FD, so del ln theta FD by del
lambda at constant V and T we get 1 by 1 plus
lambda e to the minus beta epsilon 1 for the
first term and then we get e to the minus
beta epsilon 1 plus 1 by 1 plus lambda e to
the minus beta epsilon 2 times e to the minus
beta epsilon 2 and so on.
We 
get del ln theta FD by del lambda at constant
V and T we get sum 
over k 
e to 
the minus beta epsilon k by 1 plus lambda
e to the minus beta epsilon k, this is the
value of del ln theta FD by del lambda.
Average number of particles for FD statistics
is sum over k lambda e to the minus beta epsilon
k by 1 plus lambda e to the minus beta epsilon
k or we can further simplify this one like
this, so this is N FD. Similarly, for BE statistics,
N BE is sum over k lambda e to the minus beta
epsilon k by 1 minus lambda e to minus beta
epsilon k. In general again like before we
can write N FD 
or 
BE 
is sum over k 
lambda 
e 
to the beta epsilon k by 1 plus 
minus 
lambda e to the minus beta epsilon k. Now
here 
again like 
before the plus sign is for FD statistics
and minus sign is for BE statistics. Thank
you.
