Conservation of probabilities and the continuity equation Probability amplitude

intuitively, since normalised wave function stays normalised while evolving according wave equation, there relationship between change in probability density of particle s position , change in amplitude @ these positions.

define probability current (or flux) j as

j

=


ℏ
m




1

2
i




(

ψ

∗


∇
ψ
−
ψ
∇

ψ

∗


)

=


ℏ
m


im
⁡

(

ψ

∗


∇
ψ
)

,


{\displaystyle \mathbf {j} ={\hbar \over m}{1 \over {2i}}\left(\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*}\right)={\hbar \over m}\operatorname {im} \left(\psi ^{*}\nabla \psi \right),}

measured in units of (probability)/(area × time).

then current satisfies equation

∇
⋅

j

+


∂

∂
t




|

ψ


|


2


=
0.


{\displaystyle \nabla \cdot \mathbf {j} +{\partial \over \partial t}|\psi |^{2}=0.}

the probability density



ρ
=

|

ψ


|


2




{\displaystyle \rho =|\psi |^{2}}

, equation continuity equation, appearing in many situations in physics need describe local conservation of quantities. best example in classical electrodynamics, j corresponds current density corresponding electric charge, , density charge-density. corresponding continuity equation describes local conservation of charges.