Quantum tunnelling

Watch a wave packet split into reflected and transmitted parts at a barrier that a classical particle cannot cross.

Rectangular potential barrierApproaching barrier

Probability density |ψ|²Real componentPotential barrier

Total probability1.0000

Reflected R0.0%

Transmitted T0.0%

Inside barrier0.0%

Expectation ⟨x⟩−8.0

R, T and barrier probability

Approximate transmission vs barrier width

A wave, not a solid bead

The wave packet spreads and overlaps the classically forbidden region. Interaction with the barrier creates reflected and transmitted components.

No borrowed energy

Tunnelling does not require the particle to borrow energy. Energy is conserved; the wave function has a non-zero amplitude across a finite barrier.

Height, width and mass

Transmission falls approximately exponentially as the barrier becomes taller or wider, and heavier particles generally tunnel less readily.

Numerical note: The dimensionless Schrödinger equation is evolved with an implicit Crank–Nicolson finite-difference method and absorbing edge layers. Boundary and discretisation errors are monitored separately.