Quantum Harmonic Oscillator
Compare signed oscillator wavefunctions, their probability densities and quantized energies, including nodes and forbidden-region tails.
Free theory articleCompare signed oscillator wavefunctions, their probability densities and quantized energies, including nodes and forbidden-region tails.
Concept & interpretation
The quantum harmonic oscillator models a particle in a quadratic potential well, representing small oscillations around a stable equilibrium. Unlike a classical oscillator confined between its turning points, a quantum stationary state exhibits nonzero probability density tails extending into the classically forbidden region.
Experiment Instructions
Interact with the Quantum Harmonic Oscillator simulation to inspect nodes, turning points, and classically forbidden tails:
- Select State n = 0 and Position q = 0 to observe the ground state with zero-point energy 0.5 and peak density π−1/2 ≈ 0.564189583548.
- Keep State n = 0 and set Position q = 2. Because |q| = 2 exceeds the turning point magnitude 1, the probe region becomes "Classically forbidden" while the probability density remains positive (0.010333492677).
- Change State n to 2 and Position q to 1/sqrt2. Both the wavefunction and probability density vanish at this exact root of H2(q) = 4q2 − 2.