This simulation visualises a one-dimensional Gaussian wave packet launched towards a rectangular potential barrier, drawn as a glowing probability-density profile by a GLSL fragment shader. When the packet meets the barrier it splits into a reflected part and a transmitted part. The transmitted fraction is set by the analytic rectangular-barrier formula T = 1 / (1 + V₀² sinh²(κL) / (4E(V₀−E))), where the decay constant is κ = √(V₀−E) in units with ħ = 1 and 2m = 1.
The E slider sets the incoming particle's energy, V₀ sets the barrier height, and Width sets the barrier thickness L. The readout reports the transmission T, reflection R = 1 − T, the ratio E/V₀, and whether the case is classically forbidden. Launch re-fires the packet and Pause freezes the animation. Tunnelling underpins real phenomena such as nuclear fusion in the Sun, alpha decay, flash memory and the scanning tunnelling microscope.
What is quantum tunnelling?
Quantum tunnelling is the phenomenon where a particle has a non-zero probability of passing through a potential barrier even when its energy is lower than the barrier height. Classically this is impossible, but the particle's wavefunction does not stop abruptly at the barrier — it decays exponentially inside and emerges with reduced amplitude on the far side.
What does this simulation actually show?
It renders a Gaussian wave packet as a probability-density plot moving towards a centred rectangular barrier. As the launch progresses the packet strikes the barrier and divides into a reflected packet that travels back and a transmitted packet that continues to the right, with their heights scaled by R and T. An evanescent tail is drawn inside the barrier when E is below V₀.
What do the E, V₀ and Width controls do?
E sets the energy of the incoming particle (slider range 5 to 120), V₀ sets the barrier height (0 to 120), and Width sets the barrier thickness L (1 to 30). Lowering V₀, raising E or narrowing the barrier all increase transmission. Each change updates the live readout for T, R and E/V₀ immediately.
For E below V₀ the code uses the exact rectangular-barrier result T = 1 / (1 + V₀² sinh²(κL) / (4E(V₀−E))), with κ = √(V₀−E). For E above V₀ it uses the over-barrier analogue with sin in place of sinh, which produces resonance peaks at certain energies. The reflection coefficient is simply R = 1 − T.
Inside the classically forbidden region the wavefunction is not zero; it decays exponentially as exp(−κx) rather than vanishing. If the barrier is thin enough, a residual amplitude survives to the far edge, so there is a finite chance of finding the particle beyond the barrier. No energy is borrowed — the particle simply has a spread-out wave nature that allows leakage.
E/V₀ is the ratio of the particle's energy to the barrier height. When it is below 1 the case is classically forbidden and transmission relies entirely on tunnelling. When it is above 1 the particle has enough energy to pass over the barrier, though reflection can still occur and resonances appear at specific energies.
Width has a very strong, roughly exponential effect. For a thick barrier the approximate transmission scales like exp(−2κL), so doubling the width can reduce the chance of tunnelling by orders of magnitude. This sensitivity is exactly why the scanning tunnelling microscope can resolve individual atoms from tiny changes in tip-to-surface distance.
The transmission and reflection values come from the genuine analytic rectangular-barrier formulae, so the numbers are physically faithful. The moving packets, however, are a stylised illustration: the split into discrete reflected and transmitted lobes is animated for clarity rather than solved as a full time-dependent Schrödinger evolution, and the on-screen lengths are scaled to fit the display.
The evanescent wave is the exponentially decaying part of the wavefunction within the classically forbidden region. It carries no net probability current on its own but connects the incoming and outgoing waves. In the simulation it appears as a faint tail inside the barrier shading when E is below V₀, fading with distance into the barrier.
Tunnelling drives nuclear fusion in stars, where protons tunnel through their electrostatic repulsion, and it explains radioactive alpha decay. In technology it underlies the scanning tunnelling microscope, tunnel diodes, and the charge transfer that writes and erases flash memory, as well as posing a leakage problem in very small transistors.