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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.

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    $\begingroup$ You're right. I guess there's nothing to discuss :) $\endgroup$ – DanielSank Feb 21 '16 at 0:06
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The simplest version refers to a single particle. It may be more obvious in integral form: apply the Stokes theorem. The the equation states that the time derivative of the probability of the particle being measured in V is equal to the rate at which probability flows into V.

So your version is close.

See https://en.wikipedia.org/wiki/Probability_current#Continuity_equation_for_quantum_mechanics

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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.

In classical physics, thermodynamics says that energy can not enter a region unless it comes in through the boundaries. In fluid dynamics this is extremely intuitive, since the fluid is the only type of energy, saying energy can not show up in a region without coming in through the borders is the same as saying fluid can not show up in a region without entering through the borders.

This picture is totally analagous to the situation in quantum mechanics, except the interpretation of the fluid is more interesting. The continuity equation in quantum mechanics says that the actual rule is that the probability of finding a particle in some region can not change unless the probability of finding it in adjacent regions has changed. So in that way, you can think of probability as flowing like a liquid obeying the continuity equation.

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.

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