Quantum harmonic oscillator

Explore quantised energy, uncertainty and wave packets oscillating inside a parabolic potential.

Coherent state · α = 1.50Running

Potential V(x)Probability |ψ|²Re(ψ)Turning points

Time0.00 cycles

Normalisation1.000000

⟨x⟩2.121 x₀

⟨p⟩0.000 p₀

⟨E⟩2.750 ħω

Δx Δp0.500 ħ

Energy probabilities

Expectation values

Phase space

Energy comes in steps

The oscillator can only have energies Eₙ=(n+½)ħω. Even the ground state retains ½ħω of zero-point energy because exact position and momentum cannot both vanish.

Stationary and moving states

A single energy state has a fixed probability density and exactly n nodes. Superpositions evolve because their components acquire phase at different rates.

The classical limit

A coherent state is a minimum-uncertainty Gaussian whose centre follows the classical sinusoidal motion. Highly excited states also reveal the correspondence between quantum and classical behaviour.

Numerical model: Normalised eigenfunctions are generated with a stable recurrence rather than large Hermite polynomials. Integrals use a dense dimensionless position grid. Probability beyond the turning points is quantum-mechanically allowed.