Question

1-r' Problem 16.12 (30 pts) This chapter examines the two-state system but consider instead the infinite-state system consisting of N non-interacting particles. Each particle i can be in one of an infinite number of states designated by an integer, n; = 0,1,2, .... The energy of particle i is given by a = en; where e is a constant. Note: you may need the series sum Li-ori = a) If the particles are distinguishable, compute QIT,N) and A(T,N) for this system. Hint: Recall that for distinguishable, non-interacting particles, we can write the system partition function as a product of the individual particle partition function... b) Repeat part (a) for the indistinguishable case. Hint: We would need to careful to avoid overcounting here.... c) In cases (a) and (b), what is the probability that a particle will be in its the ground state? d) What is the Helmholtz free energy for the indistinguishable case? e) What is the magnitude of the fluctuations in energy, oệ = ((E – (E))2) = (EP) – (E)2 Note that (E2) = EnPnen f) How do the fluctuations behave as T → 0? Is this intuitive?

Answer 1

OU 382 2- - niale mins m; -2 Problem 16.12 (a) foro single particle, - partition function is Z = Zge-fe; nio - PYE Lo-ßx28 =ltete P te + » Z=1 te bereare.... [Here, gi degeneracy] and girl >= 1 here Foro No distinguishable particles partition function becomes - TQ (TN) = = = 11 - REjn Helmholtz free energy, A=-kt enQ » A=-KT ln (ZN) = -rkt enz » A=- NKT ln (1-e-Bejt A = Nxt bo (i-e-Be). (2) foro indistinguishable particle, @(0,5) = cicergy in and A=-xT lonQ = -xT ln (+/-) As-ka (7" - en M!) ---KT/menz. Crenntn) A = Nkq en Ni_krenz-NET." STA = NKT (lan-enz - 1) / (1) Prob. that the particle is in its ground stake in a Boxo 1-e-RE . le-ße

. (d) Helmholtz free energy for indistingui- -shable case we have colculated above > A=-xrlnQ (N,T) - = - XT ln (170) enZ -1 A = NKT / en N- (e) Probability, fn = o e-pen so, U=<E> = 2 Pn En suas En e-pen Zeßen I Ene-Pen C-en) I En effen I En e-ken ou p = I line-fen (-En) Ze-ßen 7X E er egen a b = (s en e-penja (the pen 2 zerben <632-<62) COEL KÉZY- <E)2 -- ay -- , к.- Во 2mann - constant. so, r = - 30 C%), HEFREI - To = ko prev] [ 20 = cv] a specific heat at constant rolume.

(A) re = K8 th Cv po, as T-O, i.l., for tro, SE ~o.. fluctuation becomes zero.