Answer Happy

1. The grand potential of a quantum gas may be written +c = #k97 e) in (1 + $u-e)%) de for fermions and bosons respectively. Use integration by parts to write GE) Фа 10 ele-M) 11 de where G(C) = g(e)de Hence for non-relativistic particles in 3D, for which g) a ¢1/2, show that PV - E for both fermions and bosons Find the corresponding expressions for ultra-relativistic particles in 3D, and for both non-relativistic and ultra-relativistic particles in 2D. (Use the results for gle) for the various cases from sheet 7. question 1.) 2. For non-relativistic particles in 3D, consider the case of nne so that us <0 and : = <1. In this question we will consider the first corrections corrections to the classical ideal gas results. Show that * - -*,79 Volt 2-9/999 +...) for fermions and bosons respectively, where no ** (mk, 7/2w%)/2 Hence find the particle number as a function of , and eliminato z to give PV = N*,T1+ TO ... 4/29. Comment on the physical reason for the sign of the first quantum correction to the ideal law in the two cases, 3. Show that the mean spoed, (w), in a gas of N non-relativistic spin particles at T = 0 is 3 12 gas where w, is the speed of a particle with momentum equal to equal to the Fermi momentum. 4.) Show that the energy of a gas of N ultra-relativistic fermions at T = 0 in a volume V is NEM