probability current

The probability current describes how probability flows from place to place as a wavefunction evolves in time. If the probability density |ψ|² is like the amount of a fluid at each point, the probability current is the rate and direction in which that fluid streams past. It is built from the wavefunction and its spatial variation, and it points the way the particle's likelihood is drifting.

The current matters because probability is not static. As ψ changes with time, regions where the particle is likely to be found shift, swell, and shrink. Wherever the density is rising, current must be flowing in; wherever it is falling, current must be flowing out. The current is exactly the link that makes this local bookkeeping consistent, tying the change of |ψ|² at a point to the flow across the boundary around it.

A revealing feature is that a real-valued wavefunction carries no current at all — the flow depends on how the phase of ψ varies through space. A steadily advancing phase signals a particle drifting in that direction, much as the marching ripples on a pond reveal which way the water's disturbance is travelling. This is one more place where the phase of the complex amplitude, invisible to |ψ|² alone, shows its physical teeth.