Quantum Harmonic Oscillator
Quantum States
0
1
2
3
4
5
6
7
Left-click: toggle · Right-click: show only. Fill shows phase evolution.
Harmonic Oscillator
The quantum harmonic oscillator models a particle in a parabolic potential well, like a mass on a spring. It is the most important quantum system in chemistry because the bottom of almost any bound potential is locally parabolic, so it describes small-amplitude vibrations of molecules in their equilibrium geometry.
Energy levels are evenly spaced: . The wavefunctions are Hermite polynomials multiplied by a Gaussian envelope, so they decay smoothly into the classically-forbidden region beyond the turning points.
Time Evolution:
Each stationary state picks up a phase , so the phasor associated with state n rotates at rate . Time here is measured in ground-state periods .
Tip: Click phasor squares to toggle states on and off, and change the potential to compare qualitatively different spectra (evenly spaced, quadratic, nearly degenerate, dissociating).