We will now use a very general procedure to show that probability is
conserved under the dynamics given by the TDSE. By conservation of
probability we mean the requirement that the integral of 
will
remain properly normalized 
for all time t>0.
The procedure we are about to use is very general. It is used, for
instance, to show that energy and momentum are conserved in
electrodynamics under Maxwell's equations and that mass and momentum
are conserved under the equations of Newtonian fluid flow.
As we saw in the previous section the mathematical condition that
be conserved is that it obey the
continuity equation with no source term (
as
evolves according to the TDSE. To demonstrate
conservation all that we must now do is identify a probability
current density 
so that 
.
To identify this current density, we begin with the time derivative
in the equation of continuity (3) and then employ our
knowledge of the time derivatives of 
through the TDSE (2),
To complete the identification of 
, express the
hand side of the above expression as the divergence of some quantity.
This we do by noting the following identity of vector calculus, which
amounts to integration by parts,
Inserting this into our previous result gives
Thus, our final form for the probability current density is
For 
developing under the Schrödinger equation, this
satisfies the continuity equation with no source term,
Thus, probability is conserved and the rate of change of probability in a region V balances the total flux leaving the region,
and the total probability in all of space is conserved,
