A suspended carbon-nanotube resonator is modelled as a single mode of effective mass…

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · Quantum dynamics and its formulations · Quantum dynamics and its formulations

The Structure of Quantum Mechanics, Ch Coherent states, Ehrenfest's theorem and the classical limit

Problem

A suspended carbon-nanotube resonator is modelled as a single mode of effective mass

m=1.0\times10^{-20}\,\mathrm{kg}

and angular frequency

\omega=2\pi\times100\,\mathrm{MHz}

, undamped and at zero temperature. Its potential is not quite harmonic,

V(x)=\tfrac12m\omega^2x^2+\lambda x^4

, with

\lambda=m\omega^2/2x_D^2

and

x_D=10\,\mathrm{nm}

, the displacement at which the quartic term would equal the quadratic one. Classically, such an oscillator swinging with amplitude

A

has angular frequency

\omega(A)=\omega\big(1+3\lambda A^2/2m\omega^2\big)

. The tube is prepared in a coherent state caught at rest at

\langle\hat x\rangle=A=1.0\,\mathrm{nm}

and then watched for

1.0\,\mathrm{ms}

. (a) Find the width

\Delta x

and the mean quantum number

\bar n

of the state, taking the width to be the harmonic ground-state width. (b) Evaluate the force ratio

|V'''(A)|(\Delta x)^2/|V'(A)|

and the action in units of

\hbar

, and say whether the force and action conditions are met. (c) A coherent state of amplitude

A

has

A^2=2\hbar\bar n/m\omega

. Use this to turn

\omega(A)

into a drift

\delta\omega

of the level spacing per level, then estimate

t_{\rm spread}

and compare it with the watching time. (d) Does the tube's motion have a trajectory over the

1\ \mathrm{ms}

it is watched? Say which condition decides.