|ψ|² (hue = phase arg ψ) V(x) ⟨E⟩ Re ψ Im ψ Units: ħ = m = 1 (length ℓ, energy ε = ħ²/mℓ², time τ = mℓ²/ħ)

Quantum Lab

Watch the time-dependent Schrödinger equation evolve exactly (split-step Fourier). Click, drag and measure.

Speed ×1.0

Scenario

Momentum k₀ Packet width σ Start position x₀ Potential height V₀ Width w Spring constant ω

Measuring samples x from |ψ|² (Born rule) and collapses ψ to a narrow bump there.

Re ψ Im ψ Potential Energy levels

In draw mode, drag across the plot to paint V(x); the eigenstates update live. Edges of the box are hard walls.

Reflected R—

Inside barrier—

Transmitted T—

T + R + inside—

Theory T (packet-averaged)—

Plane-wave T at ⟨E⟩—

Superposition: pick two states

Click a state to load it. Pick two with the superposition option to see beating at ΔE.

⟨x⟩—

⟨p⟩—

Δx—

Δp—

Δx·Δp (≥ ħ/2 = 0.5)—

∫|ψ|²dx (norm)—

Worst norm drift—

What am I seeing?

The wave function ψ(x,t) obeys iħ ∂ψ/∂t = −(ħ²/2m) ∂²ψ/∂x² + V(x)ψ. Here it is integrated by the split-step Fourier method: half a potential step, a full kinetic step in momentum space (FFT), half a potential step. Each piece is unitary, so total probability stays 1.
The filled curve is the probability density |ψ|²; its colour is the phase of ψ (hue wheel). A moving packet cycles through hues along its length.
Tunnelling: when ⟨E⟩ < V₀ part of the packet still crosses the barrier. The theory line integrates the exact plane-wave transmission T(k) over the packet's momentum distribution.
Bound states: eigenstates are found by finite-difference diagonalisation (bisection + inverse iteration). A superposition of two oscillates with period 2π/ΔE.
Coherent state: in a harmonic well a Gaussian of width 1/√(2ω) slides back and forth like a classical ball without spreading; change σ for a "breathing" squeezed state.
Uncertainty: Δx·Δp ≥ ħ/2 always; Gaussians saturate it. Measuring position collapses ψ and blows up Δp.
The box has hard walls at x = ±40; grid 1024 points, dt = 0.005.