The Schrödinger equation conserves total probability through a local continuity law. For a wavefunction $\psi(\mathbf r,t)$, the probability density is $$\rho=|\psi|^2.$$ Define the probability current density $$\mathbf j=\frac{\hbar}{2mi}\left(\psi^\nabla\psi-\psi\nabla\psi^\right).$$ For the ordinary Schrödinger equation with a real potential, $$\frac{\partial\rho}{\partial t}+\nabla\cdot\mathbf j=0.$$

This is a continuity equation. Integrating over a volume $V$ gives $$\frac{d}{dt}\int_V\rho,dV =-\oint_{\partial V}\mathbf j\cdot d\mathbf A.$$ Probability inside the region changes only because probability flux crosses its boundary.

For a one-dimensional plane wave $$\psi=Ae^{i(kx-\omega t)},$$ the density is $|A|^2$ and the current is $$j=\frac{\hbar k}{m}|A|^2.$$ A wave with negative $k$ carries current in the opposite direction.

This distinction matters in scattering. Reflection and transmission probabilities are determined by ratios of outgoing to incoming probability currents, not merely by ratios of wavefunction amplitudes when the particle speeds differ between regions.

A stationary bound state can have time-independent density even though its phase evolves; in many simple one-dimensional bound states the net probability current is zero.

Probability current gives wavefunction evolution a local conservation interpretation. It connects the Born probability density to transport, tunneling and scattering without imagining particles following definite hidden trajectories between measurements.