Non Relativistic Quantum Mechanics

We describe matter with a wave function, $\psi(x)$ in a Hilbert space. The wave function evolves via the Schrodinger equation. Its absolute square gives the probability distribution under a measurement.

To construct the Schrodinger equation, one needs the Hamiltonian of the system. This comes out of classical mechanics.

(1)
\begin{align} \dot \psi = i\hbar \frac{d\psi}{dt} \end{align}