In you computed the partition function for a quantum

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(a) From problem 6.20 , the partition function of a quantum harmonic oscillator is given by:

\(Z_{h}=\left(1-e^{-\beta \epsilon}\right)^{-1}\)

where \(\epsilon=h f\). Helmholtz energy in terms of partition function is given by:

\(F=-k T \ln (Z)\)

substitute with the partition function to get:

\(F=-k T \ln \left(\left(1-e^{-\beta \epsilon}\right)^{-1}\right)\)

use \(\ln \left(x^{n}\right)=n \ln (x)\), we get:

\(F=k T \ln \left(1-e^{-\beta c}\right)\)

the Helmholtz free energy is an extensive quantity, so for \(N oscillators we have:

\(F=N k T \ln \left(1-e^{-\beta c}\right)\)