Canonical quantization

Summary

Canonical quantization turns a classical Hamiltonian system into a quantum one. The canonical coordinates $q_j$ and momenta $p_j$ of phase space become self-adjoint operators $Q_j$ and $P_j$ on a Hilbert space. The Poisson bracket $\{q_j, p_k\} = \delta_{jk}$ becomes the commutator $[Q_j, P_k] = i\hbar\, \delta_{jk}$. Other observables, such as the Hamiltonian, are then replaced by functions of $Q$ and $P$, which needs an ordering rule such as Weyl quantization. Groenewold's theorem shows that no rule can turn all Poisson brackets into commutators exactly.