An elementary treatment of the quantum harmonic oscillator is proposed. No previous knowledge of
linear differential equation theory or Fourier analysis are required, but rather only a few basics
of elementary calculus. The pivotal role in our analysis is played by the sole particle localization
constraint, which implies square integrability of stationary-state wavefunctions. The oscillator
ground-state characterization is then achieved in a way that could be grasped, in principle, even by
first-year undergraduates. A very elementary approach to build up and to characterize all
higher-level energy eigenstates completes our analysis.