Canonical quantization

Canonical quantization is a procedure for constructing a quantum theory from a classical Hamiltonian system by promoting canonical variables to operators with prescribed commutation relations.

Canonical commutation relations: Commutation relations stating that position and momentum operators satisfy [q, p] = iℏ for each canonical pair. They are the defining algebraic rule used to promote classical canonical variables.
Hamiltonian mechanics: A formulation of classical mechanics in terms of phase space, a Hamiltonian, and canonical equations of motion. Canonical quantization takes its variables and dynamics from this classical framework.
Path integral formulation: A quantum formulation that computes amplitudes by summing contributions from possible histories weighted by the action. It quantizes through histories rather than directly promoting canonical variables to operators.
Schrödinger equation: The equation governing the time evolution of a nonrelativistic quantum state. Canonical quantization of a particle’s Hamiltonian yields this equation of motion.
Canonical coordinates: Coordinates on phase space in which Hamilton’s equations take their standard symplectic form. These classical variables become the operator pairs in the quantization prescription.
Phase space: The space of positions and momenta that specifies a classical mechanical state. Canonical variables are coordinates on phase space before they become operators.
Geometric quantization: A mathematical framework that constructs quantum state spaces from symplectic manifolds using geometric structures. It gives a structured alternative when naive operator assignment is ambiguous.
Quantum field theory: A framework combining quantum principles with fields and, usually, special relativity. Canonical quantization extends the operator prescription from particle variables to field variables.
Operator ordering problem: The ambiguity that arises when classically commuting quantities become noncommuting operators. Different orderings can produce inequivalent quantum operators from the same classical expression.
Symplectic geometry: The study of spaces equipped with a closed, nondegenerate differential two-form. Its structure underlies canonical coordinates and the Poisson brackets used in quantization.

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