Canonical quantization promotes phase-space variables to operators with $[\hat q_i,\hat p_j]=i\hbar\delta_{ij}$. Classical observables become operators on a Hilbert space, and the classical Poisson algebra is deformed into a noncommutative quantum algebra. This is conceptually powerful but must be applied carefully because operator ordering can matter.

A quantum state is represented by a normalized vector $|\psi\rangle$ in a complex Hilbert space, while observables are Hermitian operators. The spectral theorem explains why measurement outcomes are eigenvalues and why the Born rule gives probabilities from $|\langle a|\psi\rangle|^2$.

For two observables $A$ and $B$, the Robertson inequality gives $\Delta A\,\Delta B \ge \tfrac{1}{2}|\langle[A,B]\rangle|$. The familiar position-momentum relation $\Delta x\,\Delta p \ge \hbar/2$ reflects the noncommutative structure of quantum observables, not experimental imperfection alone.

The path integral expresses the transition amplitude as $\int \mathcal{D}q\,e^{iS[q]/\hbar}$. In the semiclassical limit, stationary-phase contributions dominate, connecting quantum behavior to classical trajectories. Path integrals are especially useful in field theory, statistical mechanics, and gauge theory quantization.