Canonical quantization is a method of relating, or associating, a classical system
of the form
, where
is a manifold,
is the canonical symplectic form on
, with a (more complex) quantum system represented by
, where
is the
Hamiltonian operator. Some of the early formulations of quantum mechanics
used such quantization methods under the umbrella of the correspondence principle or postulate.
The latter states that a correspondence exists between certain classical and quantum operators, (such as the Hamiltonian operators) or algebras (such as Lie or Poisson (brackets)), with the classical ones being in the real (
) domain, and the quantum ones being in the complex (
) domain.
Whereas all classical Observables and States
are specified only by real numbers, the 'wave' amplitudes
in quantum theories
are represented by complex functions.
Let
be a set of Darboux coordinates on
. Then we may obtain from each coordinate function an operator
on the Hilbert space
, consisting of functions on
that are square-integrable with respect to some measure
, by the operator substitution rule:
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