3.21 Sspd_chapter 1_part 9_conclusion_removal of degeneracy  (Page 3/4)

ψ I = ψ a (1). ψ b (2) and ψ II = ψ a (2). ψ b (1) 1.77

For distinguishable particles, we cannot say which set is valid. Both may be equally valid. Therefore we have two kinds of linear combination. If the wave function remains unaffected by exchange between the two states then we have symmetric function and is given as follows:

ψ B = (1/√2)[ ψ a (1). ψ b (2) + ψ a (2). ψ b (1)] 1.78

This is the wave function describing Bosons which do not obey Pauli Exclusion Principle and any number of Bosons can stay in the same quantum state.

Bosons have spin angular momentum = nћ where n=0,1,2,3….

If the wave function changes sign on interchange of state then we have anti symmetric wave function and is given as follows:

ψ F = (1/√2)[ ψ a (1). ψ b (2) - ψ a (2). ψ b (1)] 1.79

This is the wave function describing Fermions which obey Pauli Exclusion Principle and no two Fermions can stay in the same quantum state.

Fermions have spin angular momentum = (1/2 + n) ћ where n=0,1,2,3…

(1/√2) has been used for normalization so that the total probability of finding the particle over the volume concerned is Unity.

Now what is the possibility that both particles are in the same quantum state (a).

For distinguishable particles, this new state of overlap of wave function is:

ψ M = ψ a (2). ψ a (1) 1.80

For bosons the wave function is:

ψ _{B =} (1/√2)[ ψ _a (1). ψ _a (2) + ψ _a (2). ψ _a (1)]

= (1/√2)[2 ψ _a (1). ψ _a (2)] = √2 ψ _a (1). ψ _a (2)

Probability density of Bosons in same state:

ψ _B. ψ _B * = √2 ψ _a (1). ψ _a (2). √2 ψ _a *(1).ψ _a *(2)

ψ _B. ψ _B *= 2 ψ _M. ψ _M *

Thus the probability that two bosons are in the same state is twice the probability that two distinguishable particles will be in the same state.

For fermions the wave function is:

ψ _{F =} (1/√2)[ ψ _a (1). ψ _a (2) - ψ _a (2). ψ _a (1)] = 0

Therefore fermions can never be in the same quantum state. Hence fermions always follow Pauli’s Exclusion Principle.

1.8.2.2. BOSE-EINSTEIN CONDENSATE- FIFTH STATE OF MATTER.

In 1924, Einstein had visualized that in a cold enough gas, which does not become liquid or solid, loss of momentum will lead to the spread of the wave function on the container’s spatial dimension scale from Heisenberg Uncertainty Principle. In case of bosons as in case of ² He ₄ the wave functions will overlap leading to all the atoms being confined to the same quantum state leading to super atom. This is called Bose-Einstein Condensate.

In 1995, Eric Cornell, Carl Wieman and their coworkers [Appendix XXXV] at Colorado achieved such a condensate by cooling rubidium gas to 10 ^-7 K. About 2000 rubidium atoms formed a single entity namely super atom 10 microns long and it lasted 10 seconds. In 1997 the Nobel Prize was awarded for this work to Cornell and Wieman.

Still larger condensate was achieved where 10 ⁸ hydrogen atoms were condensed to the same quantum state and matter wave like LASER beam was achieved.

1.8.2.3. A COMPARATIVE STUDY OF MAXWELL-BOLTZMAN[Appendix XXXVI] DISTRIBUTION, FERMI-DIRAC DISTRIBUTION[Appendix XXXVII]AND BOSE-EINSTEIN DISTRIBUTION.

Table.1.8. The three statistical distribution function.