Brewsters Law

When light is transferred from glass surface it may be polarized. It was explained by Brewster. According to Brewster the light reflected from a surface is completely polarized if the reflected beam and the beam refracted into material form a right angle. If the incident beam is polarized in plane of incidence there will be no reflection at all. Only if the incident beam is polarized normal to the plane of incidence it will be reflected.

Reason for above property

In the reflecting material, the light is polarized transversely and we know  that it is motion of charges in material which generates emergent beam which we call as reflected beam.The source of this so called reflected light is not simply that the incident beam is reflected; our deeper understanding of this phenomenon tells us that the incident beam drives an oscillation of charges in the material, which in turn generates reflected beam.

From below fig, it is clear that only oscillations normal to paper can radiate in direction of reflection and consequently reflected beam will be polarized normal to plane of incidence. If the incident beam is polarized in plane of incidence there will be no reflected light.

This phenomenon is readily demonstrated by reflecting a linear polarized beam from a flat piece of glass. If glass is turned to different angles of incidence to polarized beam, sharp attenuation of reflected intensity is observed when angle of incidence passes through Brewsters angle. This attenuation is observed only if plane of polarization lies in plane of incidence. If plane of polarization is normal to plane of incidence, the usual reflected intensity is observed at all angles.

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.