Continuity Equation For Quantum Mechanics
The definition of probability current and Schrödinger's equation can be used to derive the continuity equation, which has exactly the same forms as those for hydrodynamics and electromagnetism:
where the probability density is defined as
- .
If one were to integrate both sides of the continuity equation with respect to volume, so that
then the divergence theorem implies the continuity equation is equivalent to the integral equation
where the V is any volume and S is the boundary of V. This is the conservation law for probability in quantum mechanics.
In particular, if Ψ is a wavefunction describing a single particle, the integral in the first term of the preceding equation (without the time derivative) is the probability of obtaining a value within V when the position of the particle is measured. The second term is then the rate at which probability is flowing out of the volume V. Altogether the equation states that the time derivative of the chance of the probability of the particle being measured in V is equal to the rate at which probability flows into V.
Read more about this topic: Probability Current, Motivation
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