Schrödinger's equation is not an equation describing some probability density. It is a fully deterministic equation that takes for input a quantum state |$\Psi$> and tells you how this state evolves with time. If you know the state |$\Psi (t)$> at a certain time $t$ and the Hamiltonian of the system, then you know exactly (at least formally) what the state |$\Psi (t')$> will be at a time $t' > t$.
The fact that the quantity <$\Psi$| $\Psi$> describes the probability density to find a particle at some point $x$ in real space (or equivalently at some impulsion $p$ in reciprocal space, depending on the representation you're using) tells you a different story. It is linked to the fact that the most information you can know in the quantum framework is the wavefunction |$\Psi$>, contrary to classical mechanics where you can track the trajectory of a particle (thus know its impulsion $p$ and position $x$ at every moment).
The "uncertainty" you have in quantum mechanics does not mean we lack some information on the system or that the evolution is non-deterministic. It just means that quantum determinism is different than classical determinism.