Consider N identical and indistinguishable quantum harmonic oscillators of frequency ω. They are not interacting each ot

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Consider N identical and indistinguishable quantum harmonicoscillators of frequency ω. They are not interacting each other.The Hamiltonian of a single oscillator is given by H = Ћω (a†a + ½)where a† and a are creation and annihilation operators,respectively. (a) Show that the partition function of a singleoscillator is given by Z1 = e-βЋω/2 /(1- e-βЋω/2 ) (b) Find themean energy per oscillator. (c) Show that when kT >> Ћω, theclassical equipartition rule holds. That is U/N→kT.