A skyscraper of effective mass sways like a harmonic oscillator with angular…

PHYSICS · Multi-part · 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 skyscraper of effective mass

m=1.0\times10^{7}\,\mathrm{kg}

sways like a harmonic oscillator with angular frequency

\omega=1.26\,\mathrm{rad\,s^{-1}}

(a sway period of

5\,\mathrm{s}

). Model a gust-driven sway as a coherent state with mean displacement

x_0=5.0\,\mathrm{cm}

from vertical, with the width of the state equal to the ground-state width

\Delta x=\sqrt{\hbar/(2m\omega)}

. ((a)) If the potential carries a small quartic correction,

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

with

\lambda>0

, the fractional discrepancy between

\langle V'(\hat x)\rangle

and

V'(\langle\hat x\rangle)

for a coherent state is at most

3(\Delta x/x_0)^2

, whatever

\lambda

is. Estimate this bound. ((b)) Estimate

\bar n

and

\Delta n/\bar n

for the sway. Which conditions for a trajectory do these numbers settle, and what would the spreading condition need that is not given here?