= Postulates of quantum mechanics
For an isolated quantum system, pure states are normalized rays in a complex <Hilbert space>, observables are <self-adjoint operators>, and their probabilities obey the <Born rule>. A spectral projector $P_a$ has outcome probability $\langle\psi|P_a|\psi\rangle$; an ideal repeatable <projective measurement> has conditional state $P_a\psi/\sqrt{\langle\psi|P_a|\psi\rangle}$. The <Schrödinger equation> specifies continuous unitary evolution between such ideal measurements, given a <Hamiltonian operator>. General measurements need not have the projective state-update rule.
Composite systems use the <tensor product> of their state spaces. Identical particles additionally obey the bosonic or fermionic exchange symmetry appropriate to their species. The <photoelectric effect>, <Davisson–Germer experiment> and <double-slit experiment> motivate energy quanta, matter-wave phases and probability interference, respectively; these experiments constrain the framework but do not separately prove every abstract postulate or specify every system's Hamiltonian.
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