BLOCH THEOREM

Bloch assumed that electrons move in a perfect periodic potential. He considered one dimensional array of lattice. The potential of electron at positive ion site is zero and is maximum in between. So long any line passing through the centers of positive ions, the potential variation must be as shown in below figure.

So Bloch gave a condition which is

𝚿(x+Na)=𝚿(x) .............................................................................................................(1)

It is considered as boundary condition.

Consider Schrodinger wave equation for one dimensional lattice.

(d²𝚿(x)/dx²) + (2m/ħ²)*[E-V(x)]*𝚿(x) = 0  .................................................................(2)

The Schrodinger equation for an electron in the potential at x+a is

[d²𝚿(x+a)/d(x+a)²] + (2m/ħ²)*[E-V(x+a)]*𝚿(x+a) = 0  ..............................................(3)

Because of periodicity,

[d/d(x+a)] = d/dx  ; V(x+a) = V(x)

With  this, eqn (3) reduces to

[d²𝚿(x+a)/dx²] + (2m/ħ²)*[E-V(x+a)]*𝚿(x+a) = 0  ...................................................(4)

This is Schrodinger  equation at x+a.

as 𝚿 at x+a is also obeying Schrodinger wave equation as 𝚿 at x there should exist a relation between 𝚿(x+a) & 𝚿(x).

Let    𝚿(x+a) = A𝚿(x)..................................................................................................(5)

𝚿(x+2a) = A²𝚿(x) [i.e. A𝚿(x+a) = A.A𝚿(x) = A²𝚿(x)]

𝚿(x+na) = Aⁿ𝚿(x)

from eqn(1),  Aⁿ =1 [i.e by using bloch condition]

Aⁿ =exp(2πij) [i.e. exp(2πij) =1 for j=01,2............]

or

A=exp(2πij/n)

Therefore,  𝚿(x+a) = exp(2πij/n)*𝚿(x) --------------------------------------------------(6) [ from eqn 5]

𝚿(x) can be written in terms of other function Uk(x )

𝚿(x) = exp(ikx)*Uk(x) where k=(2πj/n) ..................................................................(7)

From eqns (6) & (7),

exp[ik(x+a)]*Uk(x+a) = exp(2πij/n)*exp(ikx)*Uk(x)

exp[ika]*Uk(x+a) = exp(2πij/n)*Uk(x)

noting that  Ka = 2πj/n,

we can write that  Uk(x+a) = Uk(x) ..........................................................................(8)

Conclusion

Bloch Theorem is a mathematical theorem and it gives us the form of electron wave function in a periodic potential.

 𝚿(x) = exp(ikx)*Uk(x) represents Plane Wave

Thus, electron in a one dimensional lattice behaves a a plane wave.It only gives Wave nature of electron.

GIBBS CANONICAL ENSEMBLE

(SYSTEM IN CONTACT WITH A HEAT RESERVOIR)

Canonical ensemble describes those systems that are not isolated but are in contact with a heat reservoir.

The system under consideration together with a heat reservoir forms a closed system and then system of interest is taken as sub system of this closed system.

Any sub system of an isolated system in statistical equilibrium can be represented by a canonical ensemble.

The probability density of a canonical ensemble depends both on Energy 'E' and temperature 'T'

It is given by

ρ(E,T) = Ae-E/𝜏

Partition function for canonical ensemble

z=(1/h³ⁿ.n!)∫exp(-E(q,p)/KTdΓ

The statistical energy of a system in a canonical ensemble is given by

𝜎 = logZ  +(E/𝜏);  𝜎 = logZ  +T[∂(log Z)/∂T]

Thermodynamical entropy 'S' of a system in a canonical ensemble is given by

S=K𝜎

S=KlogZ + (E/T); U=KT2[∂(log Z)/∂T]

The entropy at absolute zero in a canonical ensemble can be expressed as

 S=K.log gₒ

gₒ is statistical weight of ground state