Question

4.92 Mean energy of a harmonic oscillator A harmonic oscillator has a mass and spring constant which are such that its classical angular frequency of oscllation is equal to w. In a quantum- mechanical description, such an oscillator is characterized by a set of discrete states having energies En given by The quantum number n which labels these states can here assume all the integral values A particular instance of a harmonic oscillator might, for example, be an atom vibrating about its equilibrium position in a solid. Suppose that such a harmonic oscillator is in thermal equilibrium with some heat reservoir at the absolute temperature T. To find the mean energy E of this oscillator, proceed as follows: (a) First calculate the partition function Z for this oscillator, using the defini- tion (i) of Prob. 4.18. (To evaluate the sum, note that it is merely a geometric series.) (b) Apply the geueral relation () of Prob. 1.18 to calculate the mean energy of the oscillator. (c) Make a qualitative sketch showing how the mean energy E depends on the absolute temperature T (d) Suppose that the temperature T is very small in the sense that kT Without any calculation whatever, using only the energy levels of i), what can you say about the value of E in this case? Does the result you obtained in (b) properly approach this limiting case? (e) Suppose that the temperature T is very high so that kT>hw. what then is the limiting value of the mean energy E obtained in (b)? How does it depend on T? How does it depend ori

question no 4.22, statistical physics by Reif Volume 5

Answer 1

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