What you are looking at

An electron, drawn as a wavepacket, moves right towards a barrier. When its energy is below the top of the barrier, a classical electron would always bounce back, yet part of the wave gets through. The bar under the plot counts how much ends up on each side and marks what the theory predicts for this packet.

Glossary

Tunnelling

A quantum particle getting through a barrier that it does not have the energy to climb. The wave leaks into the barrier, and if the barrier is thin some of it comes out the other side.

Why it matters It powers the scanning tunnelling microscope and the flash memory in phones.

Wavepacket

A short burst of wave that stands for one electron. Its shape tells you where the electron is likely to be found; here it is a Gaussian, a bell curve.

V(x)

The potential: the energy the electron would need to be at each position x. The grey filled shape is the barrier; everywhere else it is zero.

Barrier shape

Rectangle is one flat-topped wall. Double wall is two walls of width a with a 1.5 nm gap between them. Smooth bump is a bell-shaped hill whose width at half its height is a.

Resonant energy

With the double wall, the gap between the walls acts as a trap. At certain energies a wave with exactly that energy crosses both walls completely. Move E₀ to find them.

Why it matters Two walls can let through more than one wall alone.

Packet energy E₀

The electron's average energy of motion, from 0.3 to 2.5 eV. It is the dashed line on the plot, and the packet is drawn riding on it.

Packet width σ

How spread out the packet is in position, from 0.8 to 2 nm. A narrower packet holds a wider range of energies.

Why it matters That range is why Theory, this packet differs from Plane wave at E₀.

Barrier height V₀

The energy at the top of the barrier, from 0 to 3 eV. When E₀ is below V₀ a classical electron always bounces back.

Barrier width a

How thick the barrier is, from 0.1 to 1.5 nm: the rectangle's width, each wall's width, or the bump's width at half height.

Why it matters Below the top, the share that gets through drops very fast as the barrier gets thicker.

|ψ|²

The filled shape: how likely it is to find the electron at each position. The whole area always adds up to 1, one electron.

Re ψ

The thin wavy line: one part of the wave itself, which goes up and down. Higher energy means shorter ripples.

Transmitted

The share of the chance of finding the electron that is now right of the barrier, including what has left the box on the right. It grows during the run and then settles.

Reflected

The share that is left of the barrier, including what has left the box on the left. It reads Not yet until the packet has finished leaving the barrier.

Theory, this packet

The share the textbook formula predicts for this packet: the transmission of every energy in it, weighted by how much of the packet has that energy.

Why it matters When the run ends, Transmitted should land on this number.

Plane wave at E₀

The share that would get through for an endless wave with exactly the energy E₀ and no spread. A wide packet comes close to it.

Decay length 1/κ

Inside the barrier the wave shrinks as e^(−κx): every 1/κ it drops to about a third. It reads Over the top when E₀ is at or above V₀, because then the wave does not shrink.

Why it matters A barrier a few decay lengths thick lets almost nothing through.

Time

The electron's own time since release, in femtoseconds (fs), millionths of a billionth of a second. The moving view slows it so the packet takes about 6 seconds to cross.

Left, Right, Theory

The bar under the plot. The grey part on the left is what is left of the barrier, the coloured part on the right what is right of it, the faint middle what is still inside. The white mark is the theory.

eV

The electronvolt: the energy an electron gains crossing 1 volt. It is the energy scale on the left of the plot.

Position (nm)

Where along the line the electron is, in nanometres, millionths of a millimetre. The barrier starts at 0; the view shows 20 nm either side.

Schrödinger equation

The rule for how a quantum wave changes over time. Each run solves it for the electron on 4160 points, using a method called Crank–Nicolson.

Absorbing layers

Strips past both edges of the view that soak up the wave instead of bouncing it back. The solver counts what each one takes, so the three parts of the bar always add up to 1.