(a) Evaluate the partition function of a quantum mechanical N-oscillator system. From Z_(N)(beta)=inte^(-beta E)Omega(E)dE, and using the asymptotic formulae ln Omega(E)~~ln Z(beta^(**))+beta^(**)E, where beta^(**) is determined by (del ln Z(beta^(**)))/(delbeta^(**))=-E, show that if we let E^(')=E-(1)/(2)N?omega (b) E^(')=N(?omegae^(-beta//?omega))/(1-e^(-beta h omega)) (c) S=Nk[((E)/(N?omega)+(1)/(2))ln((E)/(N?omega)+(1)/(2))-((E)/(N?omega)-(1)/(2))ln((E)/(N?omega)-(1)/(2))]

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(a) Evaluate the partition function of a quantum mechanical

N

-oscillator system. From

Z_{N} (?) = ? e^{? ?E} ? (E) d E

, and using the asymptotic formulae

ln ? (E) ? ln Z (?^{?}) + ?^{?} E

, where

?^{?}

is determined by

\frac{? l n Z ( ? ^{?} )}{? ? ^{?}} = ? E

, show that if we let

E^{?} = E ? \frac{1}{2} N ? ?

(b)

E^{?} = N \frac{? ? e ^{? ? ? ?}}{1 ? e ^{? ? ? ?}}

(c)

S = N k [(\frac{E}{N ? ?} + \frac{1}{2}) ln (\frac{E}{N ? ?} + \frac{1}{2}) ? (\frac{E}{N ? ?} ? \frac{1}{2}) ln (\frac{E}{N ? ?} ? \frac{1}{2})]

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