How does Stern Gerlach Experiment proved spatial quantization and spin of electron

THE STERN-GERLACH EXPERIMENT

 Procedure implemented:

The silver atoms beam is produced by heating silver in a small electric oven. The beam is passed through an inhomogenous Magnetic Field.

Arrangement to produce inhomogenous Magnetic Field

We have one of pole pieces of the magnet flat with a cylindrical groove and the other in the form of a knife edge, parallel to groove.

The intensity of magnetic field increases as we go towards upper knife edge pole from center and it decreases as we go below towards lower pole.

A photographic plate is arranged to record the configuration of beam after its passage through the field.

The whole arrangement is placed in a vacuum. In absence of magnetic field, a trace of form of a narrow strip is obtained as shown in fig (a).

In presence of inhomogenous magnetic field the strip splits up into two components as shown in fig(b). 

The splitting of silver beam into two components in inhomogenous field verifies existence of electro spin and postulate of space quantization as shown below: 

Silver has an atomic number 47. According to Pauli's exclusion principle, all inner shells and sub shells are completely filled except outer most electron in 5S state.  Thus, it is a monovalent element.
The 5S electron is responsible for magnetic moment of atom.

When all silver atoms possessing a magnetic moment 'μᴊ' pass through inhomogenous magnetic field, they experience different amount of force in vertical direction depending on their orientation and alignment with magnetic field.

If magnetic moment  'μᴊ' can have all possible orientations then beam of Silver atoms consisting of Millions of atoms having all possible orientations will spread out into a broad continuous band on emerging from magnetic field. So a broad continuous patch should be observed on photographic plate.

Experimentally only two narrow strips are obtained on photographic plate. Therefore, predictions of classical physics are not correct in this case.

The two narrow strips show that 'μᴊ' cannot have all possible orientations, but only two possible orientations as shown in below figure

We know that 'μᴊ' is proportional to angular momentum 'J' and hence direction of 'J' relative to a well defined direction is given by

J = [√j(j+1)   ]* h/2Π

There are (2j+1) possible orientations of J. The Stern-Gerlach experiment shows that (2j+1)=2 or j=1/2.

Thus, J=(√3 /2)* (h/2Π)

It is known that angular momentum 'J' of Silver atoms entirely due to spin of its valence electrons. Thus, we conclude that the electron has a spin angular momentum

S = [√s(s+1)]*(h/2Π); where s=1/2.

Thus, Stern and Gerlach found that intial beam split into two distinct parts, corresponding to two opposite spin orientations in magnetic field that are permitted by space quantization.

THE RAINBOW - EXPLANATION

Of all the optical phenomenon in every day life, the rainbow is loveliest.

Reflection of sun light by the rain drops is certainly an essential  element of an explanation but refraction-plays a role, too.

The following figure shows crucial path of light

The circle represents the cross section of a spherical rain drop. For the light ray, the sequence is Refraction, Reflection, and Refraction.

The angle (less than 90o) between the incident direction and the emergent direction is called as "Return Angle".

 A Ray from the sun strikes the spherical rain drop and some light is refracted into the water. Here we may ignore the portion that is reflected by the drops' surface. Next, the Ray proceeds to the far side of drop and is reflected there. Now we may ignore the portion that is refracted. Finally, the ray strikes the underside of drop and is refracted out into the air.

In precisely which direction does the emergent ray travel?

Two rules - i) Snell's Law for Refraction & ii) Equality of the angles of incidence and reflection

 suffice for answering that question. Once the initial point of contact between the rain drop and the ray from the sun has been specified. Also because the index of refraction depends on color, we must specify the color of light.

Lets start considering Red. Working out the complete path for many rays-i.e. for many different initial points of contact-reveals a surprising geometric property: The return angle for red light never exceeds 42.5 deg C, and most rays have a return angle 42 deg.

So return angle will decide the color of light reaching our vision of sight. Hence, different colors emerge from different sets of rain drops and produce a colored Rainbow.