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transmini
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Homework Statement
In the real world, most oscillators are not perfectly harmonic. For a quantum oscillator, this means that the spacing between energy levels is not exactly uniform. The vibration levels of an ##H_2## molecule, for example, are more accurately described by the approximate formula
$$ E_n \approx \epsilon(1.03n-0.03n^2), \space\space n=0, 1, 2, ...,$$
where ##\epsilon## is the spacing between the two lowest levels. Thus, the levels get closer together with increasing energy. (This formula is reasonably accurate only up to about ##n=15##; for slightly higher ##n## it would say that ##E_n## decreases with increasing ##n##. In fact, the molecule dissociates and there are no more discrete levels beyond ##n\approx15##.) Use a computer to calculate the partition function, average, energy, and heat capacity of a system with this set of energy levels. Include all levels through ##n=15##, but check to see how the results change when you include fewer levels. Plot the heat capacity as a function of ##\frac{kT}{\epsilon}##. Compare to the case of a perfectly harmonic oscillator with evenly spaced levels, and also to the vibrational portion of the graph in Figure 1.13.
Homework Equations
Partition Function: ##Z = \sum_n \space e^\frac{-E_n}{kT}##
Average Energy: ##\bar{E} = \sum_n \space E_ne^\frac{-E_n}{kT}##
Heat Capacity: ##\frac{\partial\bar{E}}{\partial T}##
The Attempt at a Solution
First I tried to determine what ##\epsilon## was but just found ##\epsilon=E_1-E_0=1-0=1## but I don't feel like this could be right, because if ##\epsilon## was just ##1##, why would it be there at all. But rolling with this, I wrote some code in Matlab to find ##Z \approx 1## and ##\bar{E} \approx 1.61x10^{-17} eV##, but the average energy doesn't make sense either, because it's 17 orders of magnitude lower than the first nonzero energy level. Then I didn't type anything for the heat capacity yet, but tried to derive another formula:
$$C = \frac{\partial\bar{E}}{\partial T} = \frac{\partial}{\partial T}(\frac{1}{Z}\sum_n \space E_ne^{-\beta E_n})$$
$$C = \frac{-1}{Z^2}\frac{\partial Z}{\partial T} \sum_n \space E_ne^{-\beta E_n} + \frac{1}{Z}\sum_n \space (-E_n^2e^{-\beta E_n} \frac{\partial\beta}{\partial T})$$
$$\frac{\partial Z}{\partial T} = \frac{\partial}{\partial T} (\sum_n \space e^{-\beta E_n}) = \sum_n \space (-E_ne^{-\beta E_n} \frac{\partial\beta}{\partial T})$$
$$\frac{\partial\beta}{\partial T} = -kT^2$$
$$C = \frac{-1}{Z^2}kT^2(\sum_n \space E_n e^{-\beta E_n})^2 + \frac{1}{Z}kT^2 \sum_n \space E_n^2 e^{-\beta E_n}$$
I haven't tried coding this out yet, since I'm not even sure if the formula is correct.
So my questions here are:
How do I find ##\epsilon## if I do not have it correct already
If my values for ##Z## and ##\bar{E}## are incorrect, is it something with my formulas, ##\epsilon##, or should I double check my code?
Is the formula I have for ##C## a correct derivation from the starting equation?
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