Wave Packets and Scattering

This is a sandbox rather than a demonstration. Pick an obstacle, choose how much energy to throw at it, and press play. Three things are worth watching, in this order: the packet getting wider even when nothing is in its way, the moment it meets the obstacle and becomes two packets, and the current strip underneath, where the reflected part shows up as flow that has changed direction.

You can also draw. Turn on Draw the potential and drag a shape onto the canvas — or focus the canvas and use the arrow keys, left and right to move the cursor, up and down to raise and lower the ground under it. Anything you draw is solved exactly the same way as the presets.

Choose Free, with nothing in the way, and the packet still changes: it flattens and widens as it goes. A packet is a bundle of momenta, and a bundle of momenta is a bundle of speeds, so the fast components run ahead of the slow ones and the parcel comes apart. The width follows σ(t) = σ₀√(1 + (ħt/2mσ₀²)²), which the Free-spread σ(t) readout is tracking live.

Now make σ small. The packet you localised hardest is the one that disintegrates fastest, because pinning down position costs you a wide spread in momentum. This is the uncertainty principle showing up not as an inequality on a page but as a rate: localise an electron to an atom’s width and it smears across a millimetre in under a microsecond.

Press Reflect above it. The dashed energy line now sits well above the top of the amber block — this particle has more than enough energy to walk over the obstacle, and classically it would, without slowing down. Close to a fifth of it comes back. (It exceeds the Plane-wave T figure beside it, and should: that number is for a single sharp energy, while a real packet brings a spread of them, and the slower components in the spread reflect much more readily than the average.)

Nothing was blocking it. What reflects a quantum particle is not a wall but a change: the wavelength on top of the barrier is longer than the wavelength outside it, and any abrupt change in wavelength scatters a wave, exactly as light partially reflects off a pane of glass it is perfectly able to pass through. Widen the barrier slowly and the reflection oscillates, dropping to zero whenever the barrier happens to be a whole number of half-wavelengths wide and the two internal reflections cancel. That is an anti-reflective coating, and it is also how a resonant tunnelling diode works — try the Double barrier and hunt for the energy where the pair becomes transparent.

The strip under the main panel is j(x), the probability current: how much probability is flowing past each point, and which way. Sky is rightward, rose is leftward. Before the collision it is a single sky-coloured hump riding along with the packet. Afterwards there are two, of opposite colour, moving apart.

It is worth watching inside a barrier that the particle is tunnelling through. The density there decays exponentially, but the current does not vanish — it is small, flat and steady, because in steady state whatever flows in one side has to flow out the other. Probability is conserved, and j is the thing that does the conserving: ∂|ψ|²/∂t + ∂j/∂x = 0 is the same statement as charge conservation in a wire.

Press Bounce off a hole. The obstacle is now a well — a place where the particle would speed up, not slow down — and some of it still comes back. Once you accept that reflection is caused by a change in wavelength rather than by an obstruction, a downward step is as good a mirror as an upward one.

This is why neutrons reflect off the surface of a material, why an abrupt change in impedance sends a signal echoing back down a cable, and why the sudden drop in potential at the edge of a nucleus produces the resonances that make some isotopes swallow neutrons far more eagerly than their size alone would suggest.