A proton of mass is held in a harmonic trap of angular frequency and occupies the…

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Wave mechanics in one dimension · Wave mechanics in one dimension

The Structure of Quantum Mechanics, Ch The oscillator as an algebra

Problem

A proton of mass

m = 1.67\times10^{-27}\,\mathrm{kg}

is held in a harmonic trap of angular frequency

\omega = 2.0\times10^{14}\,\mathrm{rad\,s^{-1}}

and occupies the number state

|n=3\rangle

. ((a)) Using

\hat x = \sqrt{\hbar/2m\omega}\,(\hat a + \hat a^\dagger)

and counting which terms in

\hat x^2

survive between

\langle 3|

and

|3\rangle

, write

\langle 3|\hat x^2|3\rangle

in terms of

\hbar

m

\omega

and

n=3

. ((b)) Evaluate it numerically.