# What does the continuity equation for probability in quantum mechanics mean?

In quantum mechanics, the continuity equation $-{d\rho}/{dt}=\nabla\cdot{J}$ holds for a probability density $\rho$ and probability current $J$. But what does it mean, from a physical point of view?

I imagine it means that a particle can not appear or disappear in a given volume $V$, there must be a "particle flux" in the walls of $V$ for particles entering or leaving $V$.

If I´m wrong, please tell me.

• You're right. I guess there's nothing to discuss :) – DanielSank Feb 21 '16 at 0:06

If we think about the wave function for a single particle in position space as a sort of fluid sitting in space, then the density of that fluid in some region is related to the probability of finding the particle there. i.e. consider $\psi(\vec{x})$ to be a sort of complex density over the space of values which $\vec{x}$ can take. A conventional density is a function $\rho(\vec{x})$ which assigns a real valued number to each point in space, in QM we extend that idea to complex numbers.
The way that density flows along a direction $x_i$ is characterized by $\frac{\partial}{\partial x_i} \psi(\vec{x}),$ which you might recognize as acting on the state with the momentum operator. This again could be seen as intuitive in some sense since the momentum of particle tells us the direction in which it is moving. The quantum picture does not have point particles though, it has extended objects called wave functions. The generalization ends up being the flow of the wave function.