# 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}{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 [math]\Delta E[/math] depends on the system under consideration.

To see quantum behaviour the system must be bounded thus reciprocative motion is considered.

## Reciprocative linear motion

The uncertainty relation: [math] \Delta x \Delta p \geq h \quad[/math]

Kinetic energy: [math] \Delta E = \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 = \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 N_{2}: [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: Thermodynamic_beta and Boltzmann factor
- Wikipedia: Rotational_spectroscopy