Nanomechanics is barely mechanical quantummechanics
Contents
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
The logarithm of the Boltzmann factor ("Quantumness"):
[math] Q = \frac{\Delta E_{Quantum}}{E_{Thermal}} [/math]
First we'll need the thermal energy:
Equipartitioning:
[math] E_{Thermal} = \frac{1}{2}k_BT = \frac{1}{2 \beta} \quad[/math]
The size of the energy quanta [math]\Delta E_{Quantum}[/math] depends on the system under consideration.
Reciprocative linear motion
To see quantum behaviour (in position space) the system must be spatially bounded. Thus reciprocative motion (here in a 1D box) considered.
The uncertainty relation: [math] \Delta x \Delta p \geq h \quad[/math]
Kinetic energy: [math] \Delta E_{Quantum} = \frac{\Delta p^2}{2m} \quad[/math]
Quantumness: [math] \color{red}{Q_{trans} = \frac{h^2}{k_B} \frac{1}{m \Delta x^2 T}} [/math]
Reciprocative circular motion
Here alpha is the fraction of a full circle that is passed through in a rotative oszillation.
For a normal unidirectional rotation alpha must be set to 2pi.
The uncertainty relation: [math] \Delta \alpha \Delta L \geq h \quad[/math]
Kinetic energy: [math] \Delta E_{Quantum} = \frac{\Delta L^2}{2I} \quad[/math]
Quantumness: [math] \color{red}{Q_{rot} = \frac{h^2}{k_B} \frac{1}{I \Delta \alpha^2 T}} [/math]
Values
With the Boltzmann constant: [math] k_B = 1.38 \cdot 10^{-23} J/K [/math] we get the
Average thermal energy per degree of freedom: [math] E_{T=300K} = 414 \cdot 10^{-23} J [/math]
rotative (full 360°)
[math] L_0 = \hbar = 1.054 \cdot 10^{-34} {kg m^2} / s [/math]
[math] L_0 = I \omega_0 = 2 m r^2 \omega [/math]
Nitrogen molecule N2: [math] \quad \color{blue}{2r = 0.11 nm \quad m_N = 2.3 \cdot 10^{-26} kg} [/math]
[math] \omega_0 = 2 \pi f = 7.5 \cdot 10^{11} s^{-1} [/math]
[math] f_0 = 119GHz [/math]
[math] E_0 = I \omega_0^2 /2 = L_0 \omega_0 /2 [/math]
Size of energy quanta: [math] E_0 = 3.95 \cdot 10^{-23} J [/math]
Quantumness: [math] \color{red}{Q_{rot} \lt 1/100} [/math]
is rather small thus we have pretty classical behaviour (at room-temperature).
Note that this is a single free floating molecule. In advanced nano-machinery there are axles made of thousands and thousands of atoms which are in turn stiffly integrated in an axle system made out of millions of atoms. This is making energy quantisation imperceptible even at liquid helium temperatures.
linear
...
general
Vibrations of individual molecules can behave quite quantummechanically even at room-temperature. This is the reason why the thermal capacity of gasses (needed energy per degree heated) can make crazy jumps even at relatively high temperatures. (Jumps with a factor significantly greater than one.)
Discussion
There are three parameters that can be changed to get something to behave more quantum mechanically.
The three options are:
- (1) lowering temperature
- (2) lowering inertia
- (3) decreasing the degree of freedom
Related
External links
Wikipedia:
