continuity equation
The continuity equation is the precise statement that probability is conserved: it can flow from place to place, but it is never created or destroyed. It ties together two ideas already met — the probability density |ψ|², which says how much probability sits at each point, and the probability current, which says how it streams. The equation declares that the rate at which density grows at a point exactly matches the net current flowing inward there.
The same shape of equation appears all over physics, for electric charge, for mass in a fluid, for energy. In every case it expresses a local conservation law: stuff does not teleport, it must move continuously through the space in between. For quantum mechanics the 'stuff' is probability, and the equation is what guarantees that if a wavefunction starts properly normalized, with total probability one, it stays normalized for all time as it evolves.
This is more than tidy bookkeeping. The continuity equation is a direct consequence of the Schrödinger equation, and it shows that the theory is self-consistent: the strange, spreading, interfering wavefunction can still never lose or duplicate the particle it describes. The particle's total presence in the universe is conserved, even as its probability cloud sloshes and reshapes from moment to moment.
Density rises at a point only as fast as current flows in — probability is locally conserved.
Conservation of probability holds only for the standard, smooth Schrödinger evolution. Measurement, in most accounts, makes ψ jump discontinuously, a process outside this equation.