Probability current

For a nonrelativistic particle of mass $m$, the probability current is

For a wavefunction satisfying the Schrodinger equation with a real potential,This continuity equation expresses local conservation of probability.

If the probability current of a normalizable wavefunction vanishes at spatial infinity, integrating the probability continuity equation over all space shows that the total probability $\int |\psi {|}^2$ is constant in time.