Schrödinger Wavepacket
A Gaussian wavepacket launched at a potential barrier. Real and imaginary parts oscillate; probability density |ψ|² stays positive. Watch tunneling (some amplitude passes through a barrier higher than the particle's energy), spreading (the wavepacket disperses over time), and reflection. The 1D Schrödinger equation, propagated numerically.
tunneling — passing where you shouldn't
A particle classically reflects from a potential barrier higher than its kinetic energy. Quantum mechanics says otherwise: the wavefunction has exponentially-decaying support inside the barrier, and a small amplitude reaches the other side. Even particles "below" the barrier can appear on the other side.
The 1D Schrödinger equation in dimensionless units:
iℏ ∂ψ/∂t = -ℏ²/(2m) ∂²ψ/∂x² + V(x) ψ
Tunneling probability for a rectangular barrier of height V₀ and width L, for a particle with energy E < V₀:
T ≈ exp(-2L · √(2m(V₀-E)/ℏ²))
This exponential dependence on barrier width is why:
• Alpha decay: alpha particles tunnel out of nuclei. A 1 femtometer change in barrier width changes the half-life by orders of magnitude.
• Scanning tunneling microscope: electrons tunnel from a sharp tip to a sample. The tunneling current's exponential dependence on tip-sample distance gives atomic-resolution imaging.
• Josephson junctions: Cooper pairs tunnel across thin insulators. The basis of all superconducting electronics, including the qubits in IBM and Google quantum computers.
• Nuclear fusion: hydrogen nuclei in the sun tunnel through their mutual Coulomb barrier. Without tunneling, no fusion → no sunshine → no life.
Pick "square barrier" with k₀ low enough that the wavepacket's energy is below V₀ height. Run it. Watch a small piece of probability mass appear on the right side, having tunneled.