Single Particle Partition Function - Solid State Physics - Solved Paper, Exams of Solid State Physics

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University of California at Berkeley Physics 112, Fall 2000 Final Examination Name: Solutions Signature: SID: Problem_1 The subject of this problem is an ideal gas of N particles confined to a one-dimensional line segment of length L. The energy of orbital n is En = Cn? /L2, where C is a constant and n is an integer that is greater than er-eqaaise 0. (a) Evaluate the single-particle partition function by approximating the sum by an integral. A useful integral is the following: the integral from 0 to infinity of exp(-x2) with respect to x is (x1/2)/2 . (b) Using the approximation that the N-particle partition function is given by Zn = (Z1)N/ NI, find the free energy F and from this, the entropy. 3 exp (~ Cn*/i? x) rer] = {* exp ( Cr’/L t) dn = (is = LSE /2 (b) F = -t bog Zy = cf og (v1) ~ N dog Z,] = x | Nigh ~N~N Aog (Mp, U)| » ‘Ka (S& = oN tog ( we) 7 1] ; ~(a6/9e), = — Nf-bg(2)- {I - xn] - 2 ta] = N [tog Gite) * 3| (a) Zz, ut gv i