Nanomechanics is barely mechanical quantummechanics

The three parameters (temperature, inertia, confinement) that can be used to get something to behave quantum mechanically. Note that an apparent absence of an angular constraint is still a constraint to a finite 360° (or 720° taking the properties of electrons into account).
(wiki-TODO: Make the graphic less silly - no triangle frame - Boltzmann factor - ...)

This is just a rule of thumb estimation with simple algebra
in order to go get a very crude initial estimate.

This page judges "quantummechanicalness" in the sense of
emerging quantizedness (stemming from the uncertainty relationship)
starting to show notable effects.

There's also the topic of sufficient isolation of systems towards the environment (and possibly towards each other)
such that these systems can entangle relative to the environment (and possibly relative to each other).
That is: To allow for multiple classically inconsistent realities to "quantum exist" at the same time.
The right quantitative measure for "quantummechanicalness" in this regard is likely
the maximally possible decoherence time in relation to the typical timescale of the system.
Related: quantum decoherence, mixed states, density matrix, ...

(wiki-TODO: Find a similarly simple rule of thumb estimation for say decorerence time of levitated crystolecules)

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"):

$$Q = \frac{\Delta E_{Quantum}}{E_{Thermal}}$$

Thermal energy per degree of freedom (thermal "quantum")

First we'll need the thermal energy:
Equipartitioning:

$$E_{Thermal} = \frac{1}{2}k_BT = \frac{1}{2 \beta} \quad$$

Energy per quantum (quantum mechanical quantum)

The size of the energy quanta

$$\Delta E_{Quantum}$$

• depends on the system under consideration.
  
• falls out from the spacial restraints (linear or circular) that
  enforce a minimum impulse and thus a minimum energy

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:

$$\Delta x \Delta p \geq h \quad$$
Kinetic energy: $$\Delta E_{Quantum} = \frac{\Delta p^2}{2m} \quad$$
Quantumness: $$\color{red}{Q_{trans} = \frac{h^2}{k_B} \frac{1}{m \Delta x^2 T}}$$

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:

$$\Delta \alpha \Delta L \geq h \quad$$
Kinetic energy: $$\Delta E_{Quantum} = \frac{\Delta L^2}{2I} \quad$$
Quantumness: $$\color{red}{Q_{rot} = \frac{h^2}{k_B} \frac{1}{I \Delta \alpha^2 T}}$$

Values

With the Boltzmann constant:

$$k_B = 1.38 \cdot 10^{-23} J/K$$ we get the
Average thermal energy per degree of freedom: $$E_{T=300K} = 414 \cdot 10^{-23} J$$

rotative (full 360°)

$$L_0 = \hbar = 1.054 \cdot 10^{-34} {kg m^2} / s$$

$$L_0 = I \omega_0 = 2 m r^2 \omega$$

Nitrogen molecule N₂:

$$\quad \color{blue}{2r = 0.11 nm \quad m_N = 2.3 \cdot 10^{-26} kg}$$

$$\omega_0 = 2 \pi f = 7.5 \cdot 10^{11} s^{-1}$$

$$f_0 = 119GHz$$

$$E_0 = I \omega_0^2 /2 = L_0 \omega_0 /2$$

Size of energy quanta:

$$E_0 = 3.95 \cdot 10^{-23} J$$

Quantumness:

$$\color{red}{Q_{rot} \lt 1/100}$$
is rather small thus we have pretty classical behaviour (at room-temperature).

Note that dinitrogen is a single free floating lightweight 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 many millions of atoms.
This is making energy quantization imperceptibly low even at liquid helium temperatures.
That is why "Nanomechanics is barely mechanical quantummechanics".

Getting quantum mechanically behaving nanomechanics would take deliberate efforts:

• See: Quantum dispersed crystolecules and Trapped free particles
• Nanocantilever: Also free mechanical oscillations of stiff nanostructures that are hard to excite thermally (lowest mode one degree of freedom) can behave quite quantum mechanically ...

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

Deviations from equipartitioning theorem & Dulong Petite law

Note that on average it is not kT/2 per degree of freedom.
The equipartitioning theorem naively applied corresponds to the Dulong Petite law in solids.
Three kinetic energy DOFs and three potential energy DOFs makes 6 times kT/2.
Multiplying the Avogardo constant for one mol of matter: 3R = 3 N_A k_B T

Diamond is especially strongly deviating to less than 3kT per atom.
Not only at very low temperatures. Even at room temperature the effect is significant (~factor 3).

Successivlely more accurate models are Einstein model, Debye model, and
using DFT calculations to get actual density of states for phonons.

Related

• Estimation of nanomechanical quantisation
• Trapped free particles
• Pages with math
• Quantum mechanics
• Common critique towards diamondoid atomically precise manufacturing and technology

External links

Qunatummechanicalness in terms of quantizedness:

Wikipedia:

• Thermodynamic_beta and Boltzmann factor
• Rotational_spectroscopy
• Equipartition theorem
• Dulong–Petit law

Quantummechanicalness in terms of decoherence time:

Wikipedia:

• Quantum_decoherence#Timescales
• Quantum_state#Mixed_states
• Density matrix