3.21 Sspd_chapter 1_part 9_conclusion_removal of degeneracy  (Page 2/4)

Broadly speaking there are two broad categories of particles:

BOSONS and FERMIONS.

Bosons are virtual particles and any number of Bosons can be accommodated at the same energy E at the same time t. One example is Photons. These have a spin angular momentum of nћ where n= 0, 1, 2,…..Photons have spin angular momentum 0ћ.These are identical particles which cannot be distinguished. They have no distinguishing characteristics. Simultaneously two or more such particles can have the same spatial coordinates as well as same momentum (x,y,z, p _x , p _y , p _z ). This implies that in phase space ( a combination of spatial space and momentum space), the elemental phase space (Δx ΔyΔz Δp _x Δp _y Δp _z ) can accommodate two or more particles, in the case photons. This is another way of saying that Heisenberg Principle of Uncertainty is not applicable to Bosons. These follow Bose-Einstein Statistics. A Helium Atom with 2 protons, 2 neutrons and 2 electrons have 0ћ spin angular momentum hence they behave like Bosons and at liquid He temperature that is at 4K they become superfluid as we will see in later section.

The credit for the discovery of this particle goes to Satyendra Nath Bose. The story of this discovery is as follows.

At the turn of 19-20 ^th century Black Body Radiation was an enigma. There was no correct theoretical formulation of the same. Rayleigh- Jean formulation ended in Ultra Violet Catastrophe and if the shorter wavelength could be explained then longer wavelengths could not be explained.

In 1910, Max Planck came up with an empirical formulation:

<E>= ћω/[Exp(ћω/(kT)) – 1] 1.75

This empirical formulation had its roots in the quantum nature of electro-magnetic energy. Once the photon or energy packet nature of electromagnetic waves was accepted, the above equation could be theoretically derived and that is precisely what S.N Bose did.

This resulted in a completely new behavior and a new probability distribution namely:

F(E) = 1/[Exp(E/(kT)) – 1] 1.76

Here F(E) is the Probability of Occupancy at energy level E.

This formulation was derived for indistinguishable photon particles whose number is not conserved and submitted to a British Journal by S.N.Bose in 1924. But it was rejected. Next he sent it to Einstein who translated it and submitted it to a German Journal and it was published. Subsequently he extended this to material particles such as ² He ₄ where total spin angular momentum is zero and hence behaves as Boson. Therefore this new probability distribution is called Bose-Einstein Statistics. Einstein predicted the fifth state of matter Bose-Einstein Condensate which was subsequently achieved in 90’s.

Photons are indistinguishable whereas electrons are distinguishable due to the four quantum numbers.

What does indistinguishability and distinguishability imply.

1.8.2.1. DISTINGUISHABILITY AND INDISTINGUISHABILITY.

Let us consider a system of two particles ,1 and 2. One of them is in state a and the other in state b. If the particles are distinguishable then the pair in states a and b and the pair in state b and a are also distinguishable. The two possibilities are: