Question

The mean energy \(\epsilon\) is related to Z by \(\langle\epsilon\rangle = kT^2 \frac{\partial \ln Z}{\partial T}\). i. Show that the mean energy of a harmonic oscillator is given by $\langle\epsilon\rangle = \frac{h\nu}{2} + \frac{h\nu}{\left(e^{\frac{h\nu}{kT}} - 1\right)}.$ ii. What is the mean energy in the low-temperature limit? What does this correspond to physically? iii. What is the mean energy in the high-temperature limit? How does this compare with the classical prediction?

          The mean energy \(\epsilon\) is related to Z by
\(\langle\epsilon\rangle = kT^2 \frac{\partial \ln Z}{\partial T}\).
i. Show that the mean energy of a harmonic oscillator is given by
$\langle\epsilon\rangle = \frac{h\nu}{2} + \frac{h\nu}{\left(e^{\frac{h\nu}{kT}} - 1\right)}.$ 
ii. What is the mean energy in the low-temperature limit? What does this
correspond to physically?
iii. What is the mean energy in the high-temperature limit? How does this
compare with the classical prediction?

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The mean energy ϵ is related to Z by
⟨ϵ⟩ = kT^2 (∂ln Z)/(∂ T).
i. Show that the mean energy of a harmonic oscillator is given by
⟨ϵ⟩ = (hν)/(2) + (hν)/((e^(hν)/(kT) - 1)).
ii. What is the mean energy in the low-temperature limit? What does this
correspond to physically?
iii. What is the mean energy in the high-temperature limit? How does this
compare with the classical prediction?

Added by Garrett M.

University Physics with Modern Physics

Hugh D. Young 14th Edition

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Solved by Expert Hubert Agamasu

Step 1

To show that the mean energy of a harmonic oscillator is given by (e) = h/2, we need to substitute the given relation (e) = kT^2 * ln(Z) / T into the equation and simplify it. (e) = kT^2 * ln(Z) / T We know that Z = e^(-E/kT), where E is the energy of the Show more…

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Chegg policy is up to 4 parts, anything less will be downvoted. The mean energy (e) is related to Z by (e) = kT^2 * ln(Z) / T i. Show that the mean energy of a harmonic oscillator is given by (e) = h/2 li. What is the mean energy in the low-temperature limit? What does this correspond to physically? iii. What is the mean energy in the high-temperature limit? How does this compare with the classical prediction?

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