24 Statistics

Chapter 24: Quantum Statistics

Mahdi Pourfath

[email protected]

Quantum Physics

Pourfath (ECE UT) Quantum Statistics Quantum Physics 1 / 14

Why Statistical Mechanics ?

When we know completely the initial condition of a physical system,one may predict exactly its future.

If the system is only partially known, however, we can still makepredictions on its future behavior.

The lack of maximum knowledge of the initial condition of the systemis often due to the extremely large number of degrees of freedom.

Statistical physics is the branch of physics that deals with systemsonly partially known.

Statistical mechanics provides a framework for relating themicroscopic properties of individual atoms and molecules to themacroscopic bulk properties of materials that can be observed ineveryday life.


A Classical System

Consider a system composed of 4 distinguishable particles.

Each particle can have an integer number of energy quanta.

Two units of energy is given to this system.

All possible ways of distributing this energy on this system is given inthe following table.


Classical Statistics

Micro-State

Macro-State 1 2 3 4

2 0 0 00 2 0 0

A 0 0 2 00 0 0 21 1 0 01 0 1 0

B 1 0 0 10 1 1 00 1 0 10 0 1 1

The probabilities of finding a particle with 2, 1, and 0 units of energy:

p(0) = 410 × 3

4 +610 × 1

2 = 610 , p(1) =

610 × 1

2 = 310 , p(2) =

410 × 1

4 = 110


Classical Statistics: Distribution Function

The essential problem in statistical mechanics is to calculate thedistribution of a given amount of energy E over N identical systems.

In classical systems, where particles are distinguishable, thedistribution function is given by the Maxwell-Boltzmann statistics

f (E ) = A exp

(

E

kBT

)


Classical Statistics: Distribution Function

The microstates into which the particles are arranged are classified interms of their energy Ei and the degeneracy they have gi .

Therefore, each level has gi degenerate states into which Ni particlescan be arranged.

The system is assumed to have n independent levels.

The total multiplicity function for this system is given by

Q(N1,N2, . . . ,Nn) =N!

∏ni=1Ni !

n∏

i=1

gNi

i


Classical Statistics: Boltzmann Statistics

There are two physical constraints on the system: the total number ofparticles and the total energy of the system must be conserved:

n∑

i=1

Ni = N

n∑

i=1

EiNi = U

The equilibrium corresponds to the most probable configuration ofthe system. Mathematically, this entails maximizing the multiplicityfunction Q subject to the above constraints.


Multivariate functions

Consider a function of n variables defined as f (x1, x2, . . . , xn), whereφ(x1, x2, . . . , xn) is held constant. The condition of φ is the systemconstraint.

An extremum of f is found when df = 0. If φ is constant thendφ = 0. Therefore, one can write

df + αdφ = 0

α is called a Lagrange multiplier and is simply a multiplicative factor.

The complete differentials can be expanded out in terms of partialderivatives of the coordinate system

(

∂f

∂x1+ α

∂φ

∂x1

)

dx1 + . . .+

(

∂f

∂xn+ α

∂φ

∂xn

)

dxn = 0



It is easier to use lnQ rather than Q

lnQ(N1,N2, . . . ,NN ) = lnN! +n∑

i=1

Ni ln gi −n∑

i=1

lnNi !

Using the Stirling’s approximation in the form ln x! ∼ x ln x − x forlarge values of x :

lnQ = lnN! +n∑

i=1

Ni ln gi −n∑

i=1

Ni lnNi +n∑

i=1

Ni

One can maximize lnQ subject to the conditions that∑

Ni = N and∑

EiNi = U as

∂

∂Nj

lnQ + α∂φ

∂Nj

− β∂ψ

∂Nj

= 0



∂

∂Nj

(

∑

i

Ni ln gi −∑

i

Ni lnNi +∑

i

Ni

)

+

α∂

∂Nj

(

∑

i

Ni

)

− β∂

∂Nj

(

∑

i

EiNi

)

= 0

ln gi − lnNj − 1 + 1 + α− βEj = 0

lnNj = ln gi + α− βEj

lnNj

gi= α− βEj

Nj = gieαe−βEj

The Lagrange multiplier β is equal to 1/kBT .

kB = 8.62 × 10−5eV/K is Boltzmann’s constant.


Quantum Statistics: Indistinguishablity

If there is an appreciable overlapping of the wavefunctions of twoidentical particles in a system, non-classical effects arise fromindistinguishablity of the identical particles:

Measurable results can not depend on the assignment of the labels toidentical particles.In the presence of one in a particular quantum state influences thechance that another identical particle will be in that state.

Indistinguishablity requires that

|ψ(r1, · · · , ri , rj , · · · , rn)|2 = |ψ(r1, · · · , rj , ri , · · · , rn)|2

Particles with symmetric wavefunctions are called Bosons

ψ(r1, · · · , ri , rj , · · · , rn) = +ψ(r1, · · · , rj , ri , · · · , rn)

Particles with anti-symmetric wavefunctions are called Fermions

ψ(r1, · · · , ri , rj , · · · , rn) = −ψ(r1, · · · , rj , ri , · · · , rn)


Quantum Statistics: Indistinguishablity

In a system without interaction, particles move independently. For asystem with two particles one obtains:

ψT (r1, r2) = ψα(r1)ψβ(r2)

ψT does not satisfy the symmetry or antisymmetry conditions. Tosatisfy these relations one can write the wavefunction of the system as

ψS =1√2[ψα(r1)ψβ(r2) + ψβ(r1)ψα(r2)]

ψA =1√2[ψα(r1)ψβ(r2)− ψβ(r1)ψα(r2)]

Two identical particles with antisymmetric wavefuntions (Fermions)can not occupy the same state.


Quantum Statistics: Multiplicity Functions

In classical mechanics particles are distinguishable.

Q(N1,N2, . . . ,Nn) =N!

∏ni=1Ni !

n∏

i=1

gNi

i

In quantum mechanics particles are indistinguishable.

Particles with antisymmetric wavefunctions (Fermions) obey Pauliexclusion principle.

Q(N1,N2, . . . ,Nn) =n∏

i=1

gi !

Ni ! (gi − Ni )

Particles with symmetric wavefunctions (Bosons) do not obey Pauliexclusion principle.

Q(N1,N2, . . . ,Nn) =n∏

i=1

(gi + Ni − 1)!

Ni ! (gi − 1)!


Quantum Statistics: Distribution Functions

f (E ) give the proability that a state of energy E is occupied:

Classic: Boltzmann distinguishable f (E ) = Ae−E/kBT

Quantum: Fermi-Dirac indistinguishable f (E ) =1

e(E−µ)/kBT + 1

Quantum: Bose-Einstein indistinguishable f (E ) =1

eE/kBT − 1


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24 Statistics