Physical interpretation of $\frac{\partial \rho}{\partial t} + \frac{\partial j}{\partial x}=0$

Question

What is the physical interpretation of $\frac{\partial \rho}{\partial t} + \frac{\partial j}{\partial x}=0$? Here $\rho$ is the probability density and $j$ is the probability current of a particle of mass $m$ moving in one-dimension.

Answer 1

Roughly speaking, the one-dimensional equation of continuity says that if the probability current going in is different from the probability current going out, then the probability density will change.

So, for instance, if there is more (of whatever) going in than going out, the probability will increase at that position.