Difference between revisions of "Nanomechanics is barely mechanical quantummechanics"
(symbolic for linear) |
(symbolic math roughly finished) |
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| + | = 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: <br> | ||
| + | Quantumness: <math> Q = \frac{\Delta E}{E_T} </math> <br> | ||
| + | First we'll need the thermal energy: <br> | ||
| + | Equipartitioning: <math> E_T = \frac{1}{2}k_BT \quad</math> <br> | ||
| + | The size of the energy quanta depends on the system under consideration. | ||
== Reciprocative linear motion == | == Reciprocative linear motion == | ||
The uncertainty relation: <math> \Delta x \Delta p \geq h \quad</math> | The uncertainty relation: <math> \Delta x \Delta p \geq h \quad</math> | ||
| − | Newton: <math> \Delta E = \frac{\Delta p^2}{2m} \quad | + | Newton: <math> \Delta E = \frac{\Delta p^2}{2m} \quad</math> <br> |
| − | + | Quantumness: <math> Q = \frac{h^2}{k_B} \frac{1}{m \Delta x^2 T} </math> | |
| − | + | ||
| − | Quantumness: <math> Q = \frac{h^2}{k_B} \frac{1}{m \Delta x^2 T} </math | + | |
== Reciprocative circular motion == | == Reciprocative circular motion == | ||
| + | |||
| + | The uncertainty relation: <math> \alpha \Delta L \geq h \quad</math> | ||
| + | Newton: <math> \Delta E = \frac{\Delta L^2}{2I} \quad</math> <br> | ||
| + | Quantumness: <math> Q = \frac{h^2}{k_B} \frac{1}{I \alpha^2 T} </math> | ||
| + | |||
| + | = Values = | ||
| + | ... | ||
| + | = Discussion = | ||
| + | ... | ||
Revision as of 17:59, 6 February 2016
Contents
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: [math] Q = \frac{\Delta E}{E_T} [/math]
First we'll need the thermal energy:
Equipartitioning: [math] 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] \Delta x \Delta p \geq h \quad[/math]
Newton: [math] \Delta E = \frac{\Delta p^2}{2m} \quad[/math]
Quantumness: [math] Q = \frac{h^2}{k_B} \frac{1}{m \Delta x^2 T} [/math]
Reciprocative circular motion
The uncertainty relation: [math] \alpha \Delta L \geq h \quad[/math]
Newton: [math] \Delta E = \frac{\Delta L^2}{2I} \quad[/math]
Quantumness: [math] Q = \frac{h^2}{k_B} \frac{1}{I \alpha^2 T} [/math]
Values
...
Discussion
...
