A car's suspension spring (quarter-car model) acts as a harmonic oscillator with…

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 car's suspension spring (quarter-car model) acts as a harmonic oscillator with effective mass

m = 15\,\mathrm{kg}

and angular frequency

\omega = 9.42\,\mathrm{rad\,s^{-1}}

(a bounce frequency of about

1.5\,\mathrm{Hz}

). Model the body's vertical motion after a bump as a coherent state of this oscillator with mean displacement

x_0 = 2.0\,\mathrm{cm}

from equilibrium, and take the width of the state to be the oscillator's ground-state width

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

. ((a)) Suppose the spring's potential carries a small quartic correction,

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

with

\lambda>0

unknown. Show that for this coherent state the fractional discrepancy between

\langle V'(\hat x)\rangle

and

V'(\langle\hat x\rangle)

is at most

3(\Delta x/x_0)^2

, whatever

\lambda

is, and estimate this bound. ((b)) Estimate the mean phonon number

\bar n

and the fractional number spread

\Delta n/\bar n

of this coherent state. Say which of the chapter's conditions for a trajectory these numbers settle, and what further information the spreading condition would need.

Related theory

Quantum Mechanics

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