= Solution
One consistent five-postulate formulation is the following; different textbooks group the same content differently.
* \b[State.] A pure state is a normalized vector $|\psi\rangle$ in a complex <Hilbert space>, with $\langle\psi|\psi\rangle=1$; vectors differing only by an overall phase represent the same state. For a spinless particle in position representation, $\psi(\mathbf r)=\langle\mathbf r|\psi\rangle$ is its wavefunction.
* \b[Observables.] A physical observable A is represented by a <self-adjoint operator> $\hat A$. In a Cartesian position representation, $\hat x=x$ and $\hat p_x=-i\hbar\partial_x$, with $\hbar=h/(2\pi)$.
* \b[Outcomes and probabilities.] Allowed ideal measurement outcomes are spectral values $a$ of $\hat A$. The <Born rule> gives $P(a)=\langle\psi|P_a|\psi\rangle$, where $P_a$ projects onto the corresponding eigenspace; continuous spectra require the analogous spectral probability measure. The expectation is $\langle\hat A\rangle=\langle\psi|\hat A|\psi\rangle$.
* \b[Ideal measurement update.] Conditional on a nonzero-probability outcome $a$, an ideal projective measurement leaves $P_a|\psi\rangle/\sqrt{\langle\psi|P_a|\psi\rangle}$. For a nondegenerate result this is its eigenstate. This is the ideal projection rule, not a description of every general measurement apparatus.
* \b[Time evolution.] Between measurements a closed system evolves by the <Schrödinger equation>, $i\hbar\partial_t|\psi(t)\rangle=\hat H|\psi(t)\rangle$, with <Hamiltonian> $\hat H$. For time-independent $\hat H$, this gives unitary evolution $e^{-i\hat Ht/\hbar}$.
The position-dependent probability density is $|\psi|^2$, and its flux is the <probability current>. These distinguish the wave amplitude from the measurable particle flow in the following scattering problem.
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