Consider a simple quantum system consisting of twenty independent simple harmonic oscillators each with frequency . The

Consider a simple quantum system consisting of twenty independent simple harmonic oscillators each with frequency ω. The energy of this system is just the sum of the energies of the 20 oscillators. When the energy of this system is U(n) = (n + 10)ћω, show that its entropy is S (n) = k_B ln((n + 19)!/(19! n!)). What is the entropy of the system in thermal equilibrium for n = 1, 2, and 3? Assume that ћω ≅ 0.196 eV ≅ 3.14 × 10⁻²⁰ J, corresponding to the vibrational excitation mode of O₂. Beginning in the state with U(1) = 11ћω, by how much does the entropy increase if the energy is increased to U(2) = 12ћω? Use this result to estimate the temperature of the system by 1/T ≡ ΔS/ΔU. Repeat for the shift from U(2) to U(3), and estimate the heat capacity by comparing the change in U to the change in estimated T. Compare the heat capacity to k_B to compute the effective number of degrees of freedom. Do your results agree qualitatively with the analysis of Example 8.3?