A particle of mass moves on the half-line only

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

A particle of mass

m

moves on the half-line

x > 0

only, with an infinite wall at

x=0

, and for

x>0

feels the usual harmonic potential

\tfrac12m\omega^2x^2

. Using the full oscillator's known solutions, without solving the equation again, find the allowed energies of this half-oscillator. Then, for

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

, find its ground-state energy and the spacing between its first two levels, both in eV.