The Harmonic Oscillator

The quantum harmonic oscillator is the canonical bound-state problem: a particle in a quadratic potential

$V(x) = \tfrac{1}{2} m \omega^2 x^2$

The energy levels are quantised:

$E_n = \hbar\omega\left(n + \tfrac{1}{2}\right), \quad n = 0, 1, 2, \dots$

The ground state has energy $E_0 = \tfrac{1}{2}\hbar\omega$ — the zero-point energy, a purely quantum effect.

Time evolution

A Gaussian wavepacket built from these eigenstates breathes coherently in the well: