Nanomechanics is barely mechanical quantummechanics

Math

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

Reciprocative linear motion

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

Reciprocative circular motion

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

Values

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Discussion

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