Local charge conservation in quantum mechanics The description of charged particles in electrodynamics obeys the continuity equation,
$$ \nabla \cdot \textbf{J} = -\frac{\partial \rho}{\partial t} $$
With the common physical understanding of this equation being that it describes local charge conservation.
My question is whether the quantum mechanical description of charged particles (like an electron) is consistent with this. Since in quantum mechanics we can't associate a well defined trajectory with the charged particle I can't see how local charge conservation is incorporated in quantum mechanics.
 A: If you multiply the well known continuity equation with -e you get the expression of charge conservation. I am assuming an electronic wave function.
A: Particles in quantum mechanics obey a similar continuity equation for probability. This is necessary for probability conservation. Whenever the probability for a particle being in a particular region increases, the probability for finding the particle in the rest of space must decrease. The total probability of finding the particle must always be 1 (or 100% if you prefer). If the particle has a charge associated with it, we consider the probability distribution of charge. This is found by just multiplying the spacial probability distribution by the charge of the particle. This charge probability distribution does indeed obey the continuity equation.
Too much information:
In quantum field theory, local charge conservation plays a very important role. Noether's theorem associates every conservation law with a symmetry. The symmetry associated with charge conservation is gauge invariance. (Gauge invariance means we can use multiple scalar and vector potential functions for the same physical situation. For example, you can add any constant to the electric potential function $V(\mathbf{r})$ without changing $\mathbf{E}=-\nabla V$.) Generalizations of the gauge invariance of electromagnetism allow us to construct particles with more interesting conserved charges, like the color charge associated with the strong force.
A: Local charge conservation is a must for global charge conservation due to special relativity.
Nothing in the framework of quantum mechanics violates the charge continuity equation.
Just because it is not possible to tell the exact position and momentum of a particle simultaneously ( that is its trajectory) doesn't mean this equation is violated. Wherever  the charge be and however it moves it always respects the continuity equation.
