Local charge conservation in quantum mechanics

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.