 Solution

The postulates of quantum mechanics assign a pure state to a normalized ray in a complex Hilbert space, and an observable to a self-adjoint operator. A spectral outcome projector $P_a$ has Born rule probability $p_a=⟨\psi |P_a|\psi ⟩$. An ideal projective measurement updates the state to $P_a|\psi ⟩/\sqrt{p_a}$ when $p_a>0$; degeneracy requires the full outcome projector.

Between measurements, an isolated system obeys $i\hslash {\partial }_t|\psi ⟩=H|\psi ⟩$, giving unitary evolution. Composite systems use tensor products; identical particles additionally obey the appropriate exchange symmetry. Mixed preparations use a density matrix, with $p_a=\mathrm{tr}(\rho P_a)$. These postulates require a specified Hamiltonian and measurement model; they do not assign those automatically.^[ref]