Editing Probability amplitude (section)

==Conservation of probabilities and the continuity equation==
{{main|Probability current}}
Intuitively, since a normalised wave function stays normalised while evolving according to the wave equation, there will be a relationship between the change in the probability density of the particle's position and the change in the amplitude at these positions.

Define the [[probability current]] (or flux) {{math|'''j'''}} as
:<math> \mathbf{j} = {\hbar \over m} {1 \over {2 i}} \left( \psi ^{*} \nabla \psi  - \psi \nabla \psi^{*} \right)  = {\hbar \over m} \operatorname{Im} \left( \psi ^{*} \nabla \psi \right),</math>
measured in units of (probability)/(area&nbsp;&times;&nbsp;time).

Then the current satisfies the equation
:<math> \nabla \cdot \mathbf{j} + { \partial \over \partial t} |\psi|^2 = 0.</math>
The probability density is <math>\rho=|\psi|^2</math>, this equation is exactly the [[continuity equation]], appearing in many situations in physics where we need to describe the local conservation of quantities. The best example is in classical electrodynamics, where {{math|'''j'''}} corresponds to current density corresponding to electric charge, and the density is the charge-density. The corresponding continuity equation describes the local [[Charge conservation|conservation of charges]].