Consider the one-dimensional, time-dependent probability density
and the current density
where and .
Show by calculation that
Hint: Use the Schrödinger equation and its complex conjugate.
Calculate the partial derivative of the probability density with respect to .
The Schrödinger equation and its complex conjugate can be written as
Calculate the partial derivative of the current density with respect to .
It is clear from the above calculations that , therefore;
The conservation of probability in 3D space can be compactly expressed by the vector equation
where and are the wavefunction and its complex conjugate, is the probability density, and is the current density.