The energy levels of a harmonic oscillator with frequency

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(20 points) For a collection of N, 3D quantum harmonic oscillators of frequency o and total energy U, compute the following: (use the microcanonical ensemble approach, this is a completion of the derivation that we started in class) a. Derive the expression for the entropy as a function of M and N assuming that N and M are large. Recall that M is the number of energy quanta. b. Calculate the temperature as a function of energy. c. Determine the energy as a function of temperature. d. Find the formula for the heat capacity. e. Describe the condition that defines the high temperature limit and determine the high temperature limit of the heat capacity. f. Using a computer, graph the heat capacity as a function of temperature. Use kt/ħu on the x-axis and C/3Nk on the y-axis. Go from 0 to 2 on the x-axis. g. Find heat capacity data for lead, aluminum, and diamond. Compare the data to your graph. Discuss any discrepancies. Cite your data source(s).
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The energy levels of a harmonic oscillator with frequency  are given by

En=(n+1/2)ℏ ω, n=0,1,2,…
The total energy of N uncoupled and distinguishable oscillators has total energy
U=3N/2 ℏ ω + M ℏ ω
The number of micro states can be calculated by the number of ways to distribute M identical object
among N different states.
Imagine that N different oscillato...

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