Quantum tunnelling simulation

A ball that lacks the energy to climb a hill bounces back. A quantum particle does not have to: its wavefunction decays inside the barrier rather than stopping dead, so if the barrier is thin enough some of it emerges on the far side.

Interactive, free and runs in your browser. Part of the Quantum Mechanics simulations.

What you will see

A wave meets a rectangular barrier taller than its energy. Inside the barrier the amplitude falls off exponentially (the evanescent wave); beyond it a smaller wave carries on. The transmission and reflection probabilities are shown and always add to 1.

Try this

  1. Make the barrier a little wider and watch the transmitted wave shrink sharply.
  2. Raise the barrier height and see the decay inside it get steeper.
  3. Lower the barrier towards the particle’s energy and watch transmission climb.

Common misconception

The particle does not borrow energy to hop over the barrier. Its energy stays the same throughout; tunnelling happens because the wavefunction is not zero inside the classically forbidden region.

Exam-style question

For a thick barrier, T ≈ e^{−2κa}. If T = 10⁻³ for width a, roughly what is T when the width is doubled?

Doubling a squares the exponential: T ≈ (10⁻³)² = 10⁻⁶. Transmission is extraordinarily sensitive to barrier width, which is what makes the scanning tunnelling microscope work.

Key equations

  • κ = √(2m(V₀ − E)) / ℏ
  • T ≈ e^{−2κa}
  • T + R = 1

Curriculum

AQA A-level physics, IB Physics HL C.3 and E.1.

More quantum mechanics simulations

Other topics