Indexing the archive…
Your Universe of Digital Possibilities
The square |ψ|² — the cloud every textbook draws — is only where the particle might be. The other half is the phase: the part that interferes, that tunnels, that flows. Paint phase as colour and density as light and the hidden half appears — and with it, the points where probability vanishes and the colour wheel closes into a vortex. Pick a scenario, or drag the field to aim a packet anywhere.
The wavefunction’s whole future, set by its curvature and the potential V. It is complex — amplitude and phase — and only the pair together is the state.
The cloud is the square of the wave. Squaring discards the phase — which is exactly the half this instrument paints back in as colour.
Probability actually flows, and its current is set by the gradient of the phase φ. Flat phase, no flow; a winding phase is a current going somewhere.
Trapped at a node, the phase has nowhere to point — so it winds a whole turn around the point. A quantised vortex, its charge always an integer. The ones the rings mark.
We solve it exactly per step by splitting: a potential kick in real space, a kinetic drift in Fourier space (where −½∇² is just ×|k|²/2). Unitary — |ψ|² is conserved.
A free packet doesn’t hold together — the faster components outrun the slow, so σ grows with time. The spread you watch is the price of ever having localised it.
Everything strange about the quantum world is hiding in that second half. Interference is two phases adding or cancelling; tunnelling is a phase that keeps going where amplitude can’t; the vortices threading the dark fringes are the same objects that, in a superfluid or a Bose–Einstein condensate, lock into rigid lattices and refuse to spin like an ordinary fluid. A probability map is a photograph with the motion taken out. Put the phase back, and the wave starts to move.