Calculate the grand partition function for an ideal quantum gas and evaluate it

Question

Calculate the grand partition function for an ideal quantum gas and evaluate it specifically for fermions and for bosom (let s be the spin of each particle). (b) Using the grand partition function, evaluate the average occupation number for fermions and for bosons. (c) (i) Using the expression of for fermions, obtain its behavior for the case of zero temperature and plot it. (ii) What is the ground state of an ideal Fermi gas (construct using implications of the exchange statistics)? (iii) Derive the expression for the Fermi energy as a function of the average density of fermions (iv) How is the Fermi energy related to the chemical potential at zero temperature? (d) (i) Using the expression of for bosons, obtain the bound on the value of the chemical potential mu. (iii) What is the ground state of an ideal Bose gas (argue using exchange statistics)? (iii) How does behave at zero temperature for the zero momentum case? (iv) What is Bose-Einstein condensation? In case you need it for any problem, the area of a d dimensional sphere of radius R is given by A_d = S_d R^d-1 where S_d = 2 pi^d/2/(d/2 -1)!

Explanation / Answer

In classical mechanics identical particles remain distinguishable because it is possible, at least in principle, to label them according to their trajectories. Once the initial position and momentum is determined for each particle with the infinite precision available to classical mechanics, the swarm of classical phase points moves along trajectories which also can in principle be determined with absolute certainty.

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