Chapter 6 showed that a free Gaussian wave packet,

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · Wave mechanics in one dimension · Wave mechanics in one dimension

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch The harmonic oscillator

Problem

\Psi(x,0)=(2\pi s^2)^{-1/4}e^{-x^2/4s^2}

, has

\Delta x\,\Delta p=\hbar/2

exactly at

t=0

, for every value of the width

s

: the uncertainty relation is saturated whatever

s

is chosen. Yet this chapter finds that the oscillator's ground state has one particular width,

s^2=\hbar/2m\omega

, not an arbitrary one. If the uncertainty relation alone permits any

s

, what additional requirement singles out the oscillator's own value of

s