Nanomechanics is barely mechanical quantummechanics

Math

Let us define "quantumness" as the ration of the energy quantisation (the minimum allowed energy steps) to the average thermal energy in a single degree of freedom:
Quantumness: $$Q = \frac{\Delta E}{E_T}$$
First we'll need the thermal energy:
Equipartitioning: $$E_T = \frac{1}{2}k_BT \quad$$
The size of the energy quanta depends on the system under consideration.

Reciprocative linear motion

The uncertainty relation: $$\Delta x \Delta p \geq h \quad$$ Newton: $$\Delta E = \frac{\Delta p^2}{2m} \quad$$
Quantumness: $$Q = \frac{h^2}{k_B} \frac{1}{m \Delta x^2 T}$$

Reciprocative circular motion

The uncertainty relation: $$\alpha \Delta L \geq h \quad$$ Newton: $$\Delta E = \frac{\Delta L^2}{2I} \quad$$
Quantumness: $$Q = \frac{h^2}{k_B} \frac{1}{I \alpha^2 T}$$

Values

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Discussion

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