# Estimation of nanomechanical quantisation

## Contents

# Basics

The quantized restriction to discrete kinetic energies stems from:

A) sufficient deprivation of freedom. Less freedom bigger quanta.

- straight movement on a bilaterally closed rail
- rotation is always limited to 360°
- oscillation = through soft restoring force limited straight movement

B) sufficiently small mass and moment of inertia. Halving doubles energy quanta. Robotic system = mechanic coupled to a big massive system with only one (controlled) degree of freedom -> energy quanta get to be unmeasurably small. If quantisation is desired: release small part in limited freedom.

C) sufficient cooling. The colder the smaller the thermal energy parcels get in relation to the energy quanta. Divisible heat energy pacrels fall under indivisible energy quanta. The first energy level above the uncertainty zero point energy can than only seldom be reached and filled. The freedoms of movement "freeze". At room temperature the situation is in most cases classical since (divisible heat energy pacrels >> under indivisible energy quanta) thus the discrete quantum values are fine enough to fit the heat energy distribution nicely.

# Estimations

[math] \langle E_{therm} \rangle =3/2 \cdot k_B T[/math]

Following: zero point energies (need not be equal but are similar in size to quanta)

## Quantisation in rotation (angular momentum)

[math] (2 \pi \cdot rad) \Delta L \gt= h \qquad \Delta L \gt= h/(2 \pi) = \hbar \qquad E = L^2/(2I) \qquad \Delta E \gt= \hbar^2/(2I)[/math]

## Quantisation captured in a box (similar to oszillation)

[math] \Delta x \Delta p \gt= h \qquad E = p^2/2m \qquad \Delta E \gt= h^2 / (2m \cdot \Delta x^2)[/math]