What is the Hamiltonian for a harmonic oscillator?
Give expressions for the average energy and the heat capacity of a (classical) harmonic oscillator?
Give an expression for the energy levels of a quantum harmonic oscillator
At what temperatures does the average energy of a quantum harmonic oscillator deviate from the predictions of classical equipartition?
The Hamiltonian for the classical harmonic oscillator is:
$$ H = \frac{p^2}{2m} + \frac{1}{2} kx^2 $$
where $p$ and $x$ are the position and momentum, $m$ is the mass and $k$ is a parameter. It is straightforward to show that the canonical partition function for this (classical) Hamiltonian is:
$$ Z = \frac{2\pi}{\beta \omega} \qquad \textrm{where} \qquad \omega = \sqrt{\frac{k}{m}} $$
From this result it is similarly simple to show that the average energy and heat capacity are:
$$ \langle E \rangle = k_B T \qquad \qquad C_v = k_B $$
These results are in complete accordance with the prediction classical equipartition would make about a system with two quadratic terms (degrees of freedom) in its Hamiltonian. The results above are derived in the following video:
https://www.youtube.com/watch?v=mkZcKNvfS7Q
The energy levels of a quantum harmonic oscillator are given by:
$$ E_n = \left( n + \frac{1}{2}\right)\hbar\omega \quad n=0,1,2,\dots $$
It is straightforward to show that the canonical partition function for this Hamiltonian is given by:
$$ Z = \frac{e^{-\frac{\beta \hbar \omega}{2}}}{1 - e^{\beta \hbar\omega}} $$
The average energy of this system is thus:
$$ \langle E \rangle = \hbar\omega\left[ \frac{1}{2} + \frac{e^{-\beta \hbar\omega}}{1-e^{-\beta \hbar\omega}}\right] $$
This result accords with the predictions of classical equipartition at high temperatures. At low temperatures, however, there are deviations from what would be predicted from classical equipartition because of quantum effects. These results are discussed and derived in the following video:
https://www.youtube.com/watch?v=JlGS74_VCX8
If you have $N$ coupled or uncoupled harmonic oscillators you can use separation of variables to show that the partition function can be written as a product of $N$ partition functions as shown below:
$$ Z(N) = \prod_{i=1}^N Z_i $$
The $Z_i$ values in the expressions above can then be computed using the expressions for the 1D partition functions in the sections above.
<aside> 📌 SUMMARY: It is straightforward to derive analytic expressions for the partition function of a harmonic oscillator. The average energy of such systems at high temperatures can be predicted using equipartition. At low temperatures, there are deviations from equipartition due to quantum effects.
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