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Partition function for a single particle system and for a quantum harmonic oscillator.

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Write down an expression for the partition function of a single-particle system. Show that the mean energy (E) is given by

(E) = d ln Z / dB,

Where Z is the partition function and Beta = 1/(k_B)(T).

A one-dimensional quantum harmonic oscillator is in thermal equilibrium with a heat bath at temperature T. It has energy levels given by

E_n = (n + 1/2)hw

n = 0,1,2....

Show that its partition function is given by

Z = 1/(2sinh(hwB/2)

Hence, derive expressions for (E) and the entropy. How do these behave in the limits (k_B)(T) >> hw? Give a physical interpretation of your results.

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