As a followup question to this question: The relation between energy quanta of an Einstein solid and the equipartition value of heat capacity and this answer https://physics.stackexchange.com/a/133053/55779:
Apparently energy quanta and the equipartition values are related somehow.
I'd like to approach the problem without the formula, since the problem the question is asked in is before the problem where that formula is derived.
How can I know that value of $x\approx3$ without the formula ? I can't even tell that I should be looking for the half maximum. Or can I?
Answer
I'm not sure how one can know that the half maximum corresponds to $kT/\epsilon \approx 1/3$ without resorting to the formula for the heat capacity.
Still, notice that there are only two energy scales, i.e., $\epsilon$ and $kT$, in the problem. Then, whatever (dimensionless) number that determines whether the equipartition holds or fails has to be the ratio $x\equiv kT/\epsilon$. The limits $x \rightarrow 0$ and $x\rightarrow \infty$ should correspond to the two cases, and $x$ would be of order one in the crossover region. This consideration at least lets you make an order-of-magnitude estimation of $\epsilon$.
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