Dr. Neelabh Srivastava(Assistant Professor)

Department of PhysicsMahatma Gandhi Central University

Motihari-845401, BiharE-mail: neelabh@mgcub.ac.in

Programme: M.Sc. Physics

Semester: 2nd

Classical and Quantum Statistics: MB, BE & FD Statistics

Lecture NotesPart – 2 (Unit-III & IV)

PHYS4006: Thermal and Statistical Physics

Need for Quantum Statistics

2

Classical statistics due to Maxwell-Boltzmann explained the

energy and velocity distribution of the molecules of an ideal gas

but it failed to explain the observed energy distribution of electrons

in the so called ‘electron gas’ and that of photons in the ‘photon

gas’.

Moreover, M-B distribution function suffers from the following

two major objections –

i. First the particles are assumed to be distinguishable though

electrons or other elementary particles are indistinguishable.

ii. Any number of particles was allowed to occupy the same

quantum state while many particles particularly electrons,

obey Pauli’s exclusion principle which does not allow a

quantum state to accept more than one particle.

3

In classical statistics,

So, the probability of a cell containing two or more particles is

negligible and the particles thus can be treated as distinguishable.

But, all natural systems obey uncertainty principle and minimum size

of a cell in phase space in such a case is h3 so that number of cells gi is

limited by the value of h3.

When occupation index , the particles cannot be treated as

distinguishable and thus classical statistics cannot therefore be applied.

These difficulties were resolved by the use of Quantum statistics and

can be divided as:

i. Bose-Einstein (BE) statistics

ii. Fermi-Dirac (FD) statistics

1i

i

g

n

1i

i

g

n

Fermi-Dirac (FD) StatisticsThe basic assumptions of FD statistics are:

• It is applicable for identical and indistinguishable particles.

• Total energy and total number of the particles of the entire

system is constant.

• Spin of the particles should be half integral spin i.e.

• Particles must obey Pauli’s exclusion principle, the number of

particles in each sublevel will be one or more i.e. gi ≥ ni.

• Functions associated with the particles should be anti-

symmetric.

• All particles which obey F.D. statistics are known as Fermions

(electron, proton, neutron, 3He, neutrino, muons, all hyperons

(Λ, Σ, Ξ, Ω) etc).

4

.........,2

3,

2

1

5

Fermi-Dirac distribution law

Consider a system of Fermions having total number of particles N.

We assume that these particles can be divided into different quantum

groups or levels having energies ε1, ε2…….. εk and the number of

particles in these levels are n1, n2, ………. nk, respectively and let gi

represent the degeneracy of the energy level εi.

Since the particles are indistinguishable and only one fermion can

accommodate in one cell.

So, the number of meaningful ways in which ni particles are arranged

in gi quantum states is given by -

i

i

n

gC

!

)1()1)((

i

iiii

n

nggg or

)!(!

!

iii

i

ngn

g

6

Thus, the total thermodynamic probability for an assembly of

fermions (or particular macrostate) is

k

i iii

ik

ngn

gnnnW

1

21)!(!

!),(

Most probable state:

taking natural logarithms

k

i

ii

k

i

ii

k

i

ngngW111

))!ln(()!ln()!ln()ln(

Using Stirling’s approximation in above equation, we get

k

i

ii

k

i

iiii

k

i

i

k

i

iii

k

i

ngngngnnngW11111

)()ln()()ln()!ln()ln(

Differentiating on both sides, we get

7

k

i

iii

k

i

ii ngdnndnWd11

)ln()()ln())(ln(

k

i

i

i

ii dnn

ngWd

1

)(ln))(ln(

In addition, the system must satisfy two auxiliary conditions:

i. Conservation of total number of particles:

ii. Conservation of total energy of the system:

Applying Lagrange’s method of undetermined multipliers

0)()()(ln111

k

i

ii

k

i

i

k

i

i

i

ii dndndnn

ng

For the most probable state, 0))(ln( Wd

k

i

idndn1

0

k

i

iidndU1

0

8

0)(ln1

k

i

ii

i

ii dnn

ng

Proceeding as before

i

i

iii

i

ii

n

ng

n

ng

ln0ln

1

1)(1

i

i

eg

nfe

n

g

i

ii

i

i

Now, as we know thatTk

1

The other multiplier, α is related to the chemical potential μ

kT

E

kT

F

where EF is Fermi energy

FD distribution

function

Bose-Einstein (BE) Statistics

The basic assumptions of BE statistics are:

• It is applicable for identical and indistinguishableparticles.

• Any number of particles can occupy a single cell inthe phase space.

• The size of cell can not be less than h3 where h isPlanck’s constant.

• The number of phase space cells is comparable withthe number of particles i.e., occupation index,ni/gi=1.

9

• It is applicable to particles with integral spin angular

momentum in units of h/2π. All particles which obey

B.E. statistics are known as Bosons (photon, phonon, all

mesons etc).

• The wave function of the system is symmetric under the

positional exchange of any two particles.

• Total energy and total number of particles of the entire

system is constant.

10

11

Let us consider a system of bosons having total number of particles

N. Suppose these identical, indistinguishable and non-interacting

particles are to be distributed among gi different quantum states each

having energy εi.

So, the number of meaningful arrangements for which ni

indistinguishable particles are to be distributed among gi cells

without any restriction on the number of particles is given by -

Number of ways =

!1!

!1

ii

ii

gn

gn

Bose-Einstein distribution law

So, the thermodynamic probability for a macrostate is then:

k

i ii

iik

gn

gnnnnW

1

21!1!

!1),,(

12

Taking natural logarithm:

k

i

i

k

i

iii

k

i

gngnW111

))!1ln(()!ln())!1(ln()ln(

Using Stirling’s approximations, we get

k

i

i

k

i

ii

k

i

i

k

i

ii

k

i

iiii

k

i

ii

gggn

nngngngnW

111

111

)1()1ln()1(

)ln()1()1ln()1()ln(

i

k

i ii

i dngn

nWd

1

ln))(ln(On further solving, we get

Using Lagrange’s method of undetermined multipliers wherein we

multiply with two constraints α and β, respectively, one get

0ln111

k

i

ii

k

i

i

k

i

i

ii

i dndndngn

n

13

Proceeding as before:0sin0ln

ii

ii

i dncegn

n

Since, ni and gi are very large therefore, ni+gi-1 = ni+gi

ii

ii

i

ii

i

n

ng

gn

n

exp

expln

1

11

i

i

eg

ne

n

g

i

i

i

i

Hence, expression for the most probable distribution of particles among

various energy levels of a system obeying B-E statistics is given by -

)(1

lawondistributiEinsteinBosee

gn

i

ii

14

In general, combining all the three statistics, we can write the expression

as

kT

i

i

ieg

nf

/)(

1)(

where, constant takes values 0, -1 and 1 for M-B (classical), B-E and

F-D statistics.

15

Plot of fFD versus ε for

fermions at two temperatures

T1 and T2 (> T1)

Plots of fBE versus ε for

bosons at two different

temperatures T1 and T2 (> T1)

For εi=μ, fFD=1/2 whereas fBE→∞.

At low temperatures, when the f(ε) is nearly a step function, the

distribution function is said to be strongly degenerate. At very high

temperatures, when the step like character is lost, it is said to be

nearly non-degenerate.

1. Distribution of Bosons is skewed towards lower energy states, whereas

fermions are skewed towards higher energy states, i.e. higher probability of

finding bosons in low level energy states and fermions in higher energy

states.

2. At sufficiently high energies, classical and quantum results are identical.

16

at lower energies,

fBE > fMB > fFD

• Thus, it is obvious that the quantum statistics (B-E and F-D) will tend to classical one (M-B) onlywhen

then

• Under high temperature and low particle density,quantum statistics tend to classical statistics.

17

1/)(

kTie

1i

i

g

n

Degeneracy

A system is said to be –

• Strongly degenerate when

• Weakly degenerate when

• Non-degenerate when

18

1i

i

g

n

1i

i

g

n

1i

i

g

n

References: Further Readings

1. Statistical Mechanics by R.K. Pathria

2. Statistical Mechanics by K. Huang

3. Statistical Mechanics by B.K. Agrawal and M. Eisner

4. Thermal Physics (Kinetic theory, Thermodynamics and

Statistical Mechanics) by S.C. Garg, R.M. Bansal and C.K.

Ghosh

5. Heat, Thermodynamics and Statistical Physics by C.L. Arora

19

Assignment1) Calculate the number of different arrangements of

10 indistinguishable particles in 15 cells of equal apriori probability considering that one cell containsonly one particle.

2) A system has 7 particles arranged in twocompartments. The first compartment has 8 cellsand the second has 10 cells (all cells are of equalsize). Calculate the number of microstates in themacrostate (3, 4) when particles obey F-D statistics.

3) The molar mass of lithium is 0.00694 and itsdensity is 0.53×103 kg/m3. Calculate the Fermienergy and Fermi temperature of the electrons.

20

4) In a system of two particles, each particle can be in anyone of three possible quantum states. Find the ratio of theprobability that the two particles occupy the same state tothe probability that the two particles occupy differentstates for MB, BE and FD statistics.

5) Three spin-1/2 fermions are to be distributed in twolevels having energies E1 & E2, respectively. Each levelcan accommodate a maximum of two fermions withopposite spins, +1/2 and -1/2. Determine the number ofmicrostates and macrostates.

6) Show that if f is the FD distribution function

is a maximum at the Fermi level. Also, show that

is symmetric about the Fermi level.21

E

f

E

f

22

For any questions/doubts/suggestions and submission of

assignment

write at E-mail: neelabh@mgcub.ac.in