Schroedinger Equation III

Probability Current Density: In general for 3 space dimensions the probability density
P(r, t)=ψ*ψ dτ where dτ is a volume element. Integrating P over all space gives us the total probability N ( if not normalized to 1).

Differentiating the integrated probability density w.r.t the time t and replacing the time derivatives of ψ* and ψ from the Schroedinger equation and its complex conjugate we have

∂/∂ t ∫_V P dτ= -i /2m ∫_V [ψ∇² ψ* - ψ*∇²ψ]

using the Green theorem on the RHS we can write the integrand as divergence of a vector field ∇.S where S= -/m Im (ψ∇&psi*). Thus we have the equation

∂/∂ t P = ∇.S which is a equation of continuity for the
probability and indicates that probability is conserved. Correspondingly S is called the Probability Current Density ( in analogy with the current density in EM theory or the velocity current in fluid mechanics). So we see that Probability
can flow. In particular for free particle we see that S=P v where
v is the velocity of the particle. This is just like J=ρ v in EM theory.

As an exercise do this for 1 dimensions Ref Gasiorowicz ( in 1 dim there is no Green Theorem and one will have to use total derivatives to reduce the integral).