Question

A collection of noninteracting, classical, distinguishable harmonic oscillators are subject to a gravitational field along the direction of oscillation. This motion can be described by the classical Hamiltonian, egin{equation} mathcal{H} = frac{p^2}{2m} + frac{1}{2}momega^2 q^2 + mgq. end{equation} (a) Calculate the partition function, $Z_1$. (b) Calculate the average energy for $N$ such oscillators. Hint: It is easier to use a derivative involving $Z$ rather than a phase space integration. (c) Calculate the specific heat. (d) Go back to the Hamiltonian and rewrite is as egin{equation} mathcal{H} = frac{p^2}{2m} + frac{1}{2}momega^2 (q + x_0)^2 + E_0, end{equation}calculating $x_0$ and $E_0$. Without going into too much detail, calculate the specific heat in this case. (e) Make an argument based on the equipartition theorem that the relation between the values of the specific heats calculated in parts (a) and (d) makes sense.

          A collection of noninteracting, classical, distinguishable harmonic oscillators are subject to a gravitational field along the direction of oscillation. This motion can be described by the classical Hamiltonian,
egin{equation*}
mathcal{H} = frac{p^2}{2m} + frac{1}{2}momega^2 q^2 + mgq.
end{equation*}
(a) Calculate the partition function, $Z_1$.
(b) Calculate the average energy for $N$ such oscillators.
Hint: It is easier to use a derivative involving $Z$ rather than a phase space integration.
(c) Calculate the specific heat.
(d) Go back to the Hamiltonian and rewrite is as
egin{equation*}
mathcal{H} = frac{p^2}{2m} + frac{1}{2}momega^2 (q + x_0)^2 + E_0,
end{equation*}calculating $x_0$ and $E_0$. Without going into too much detail, calculate the specific heat in this case.
(e) Make an argument based on the equipartition theorem that the relation between the values of the specific heats calculated in parts (a) and (d) makes sense.

$A collection of noninteracting, classical, distinguishable harmonic oscillators are subject to a gravitational field along the direction of oscillation. This motion can be described by the classical Hamiltonian, eginequation* mathcalH = fracp^22m + frac12momega^2 q^2 + mgq. endequation* (a) Calculate the partition function, Z1. (b) Calculate the average energy for N such oscillators. Hint: It is easier to use a derivative involving Z rather than a phase space integration. (c) Calculate the specific heat. (d) Go back to the Hamiltonian and rewrite is as eginequation* mathcalH = fracp^22m + frac12momega^2 (q + x0)^2 + E0, endequation*calculating x0 and E0. Without going into too much detail, calculate the specific heat in this case. (e) Make an argument based on the equipartition theorem that the relation between the values of the specific heats calculated in parts (a) and (d) makes sense.$

Added by Deborah D.

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