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Quantum Tunnelling

A Gaussian wavepacket evolving under the time-dependent Schrodinger equation, solved by Crank-Nicolson so the total probability is conserved to round-off at every step. The terrain is the potential V(x); the luminous curtain is the probability density, coloured by the quantum phase. A classical ball with the same mean energy is launched alongside: it always reflects off a barrier taller than its energy, while the quantum packet partly tunnels through. Raise, widen and sculpt the barrier and watch the transmitted fraction change; the resonant double barrier transmits almost perfectly at special energies.

Figure 1. Crank-Nicolson solution of the TDSE; potential terrain V(x) and the phase-coloured probability curtain ∣ψ∣2|\psi|^2, with a classical-ball contrast.

WHAT TO TRY

  • Watch the Gaussian wavepacket hit the barrier: part reflects and part tunnels through to the far side, even though classically it lacks the energy to cross. Total probability stays conserved.
  • A taller or wider barrier lets less through: the transmitted amplitude falls off exponentially with the barrier, the sensitivity that makes scanning tunnelling microscopy work.
  • Follow the transmitted lobe: it emerges on the right as a smaller packet, the quantum particle that appeared on the forbidden side.