mjhjmtu

In the lecture notes accompanying an introductory course in relativistic quantum mechanics, the Klein-Gordon probability density and current are defined as:
$$
begineqnarray
P & = & dfracihbar2mc^2left(Phi^*dfracpartialPhipartial t-PhidfracpartialPhi^*partial tright) \
vecj &=& dfrachbar2mileft(Phi^*vecnablaPhi-PhivecnablaPhi^*right)
endeqnarray
$$
together with the statement that:

One can show that in the non-relativistic limit, the known expressions for the probability density and current are recovered.

The 'known' expressions are:
$$
begineqnarray
rho &=& Psi^*Psi \
vecj &=& dfrachbar2mileft(Psi^*vecnablaPsi-PsivecnablaPsi^*right)
endeqnarray
$$

When taking a 'non-relativistic limit', I am used to taking the limit $c to infty$, which does give the right result for $vecj$, but for the density produces $P=0$. How should one then take said limit to recover the non-relativistic equations?

The trick is to make the approach for the relativistic Klein-Gordon wave function
$$ Phi(vecr,t) = Psi(vecr,t) e^-fracimc^2thbar $$
(The physical reasoning behind this approach is:
The fast oscillating exponential is the solution for the particle at rest.
And compared to that, $Psi$ will give only slow variations.)

From that you find its derivatives
$$
begineqnarray
fracpartialPhipartial t &=& left( fracpartialPsipartial t - fracimc^2hbarPsi right) e^-fracimc^2thbar \
vecnablaPhi &=& vecnablaPsi e^-fracimc^2thbar
endeqnarray
$$

Plug these into the definitions of the Klein-Gordon probability density and current ($P$ and $vecj$) and you get
$$
begineqnarray
P &=& Psi^* Psi + fracihbar2mc^2 left( Psi^*fracpartialPsipartial t - Psi fracpartialPsi^*partial t right) \
vecj &=& dfrachbar2mileft(Psi^*vecnablaPsi-PsivecnablaPsi^*right)
endeqnarray
$$

Now you can do the limit $c to infty$ and get the 'known' expressions of the non-relativistic Schrödinger probability density and current.

You can substitute $Phi = e^-mc^2t/hbar Psi$ and then neglect the second order time derivative of $Psi$. Drop the constant $mc^2$ and you will have recovered the Schrödinger equation.