3.37 Sspd_chapter 1_part 14_bosons , fermions and physics of laser  (Page 2/3)

(1/3)/(1/4)= (4/3) = 1.333

In the above example distinguishable coins are Fermions and indistinguishable are Bosons.

1.14.2. THE SCATTERING OF DISTINGHUISHABLE AND INDISTINGHUISH ABLE PARTICLES.

In Figure(1.78) the scattering system of two indistinguishable particles is shown. The indistinguishable particles are alpha particles.

Figure 1.78. Scattering of two indistinguishable particles.

We clearly see that an alpha particle is detected at a scattering angle θ ₁ irrespective of the fact that the particle may be scattered at θ ₁ or at θ ₂ . This is because the alpha particles are indistinguishable. Hence

If probability amplitude of detection at θ ₁ is ψ(θ ₁ )

And probability amplitude of detection at θ ₂ is ψ(θ ₂ ),

Then total probability density of detection of alpha particles at θ ₁ is

| ψ(θ ₁ ) + ψ(θ ₂ )| ² …………………………………………………………. 1.138

In the same scattering set up if α- α pair is replaced by α-O ₂ pair the probability density of detection of α particles at θ ₁ is

| ψ(θ ₁ ) - ψ(θ ₂ )| ² ……………………………………………………… 1.139

This is because the scattering at θ ₂ gives the detection of O ₂ which is completely distinguishable from the alpha particle.

Eq.(1.138) and Eq.(1.139) bring out the fundamental difference between bosons and fermions.

1.14.2. SCATTERING OF N PHOTONS LOCALIZED IN THE SAME SPATIAL CELL.

In one elemental phase cell (meaning by at the same energy level and in the same spatial space) only two fermions of opposite spin are accommodated. If a third fermion pushes itself in the cell it will be repulsed. Therefore we say fermions are segregative.

But in an elemental phase cell any number of photons can aggregate. As the number increases the tendency to aggregate increases. A quantitative measure of aggregation tendency comes out while theoretically analyzing Black body Radiation.

Suppose in a cavity the emission and absorption of photons are in equilibrium then the average energy at a given frequency is shown to be:

<E>= ћω/[Exp(ћω/(kT)) -1] ………………………………………….. 1.140

Planck assumed that every black body radiator is composed of harmonic oscillators which oscillate at fundamental and harmonic frequencies only

that is at ω, 2 ω, 3 ω, 4 ω….. and their discrete energies are ћω , 2 ћω, 3 ћω, 4 ћω..

If total number of oscillators is N ⁰ ₀ and if we assume that they are in energy equilibrium then from Maxwell – Boltzmann statistics:

If N ₀ is the number of oscillators at 0 energy then

N ₀ / N ⁰ ₀ = Exp(-0/kT) = 1

N ₁ is the number of oscillators at ћω then

N ₁ / N ⁰ ₀ = Exp[-ћω/kT]

N _N is the number of oscillators at nћω then

N _N / N ⁰ ₀ = Exp[-N ћω/kT] ………………………………………………………… 1.141

Equation(1.141) have been set up according to Maxwell-Boltzmann Statistics which states that at temperature T Kelvin, probability of existence at energy level E is :

P(E) = Exp(-E/kT)…………………………………………………………… 1.142

So the average energy (per oscillator) = total energy/ total number of oscillators

<E>= E _total /N _tot

= [(0× N ₀ ) + (ћω× N ₁ )+ (2ћω× N ₁ )+ (3ћω× N ₃ )+…..]/[ N ₀ +N ₁ + N ₂ + N ₃ +..]……………………………………………………………………………………………….. 1.143

Dividing Numerator and Denominator by N _tot = N ⁰ ₀ ,