canonical quantization

Canonical quantizationPlanetmathPlanetmath is a method of relating, or associating, a classical system of the form (T*X,ω,H), where X is a manifold, ω is the canonical symplectic formMathworldPlanetmath on T*X, with a (more complex) quantum system represented by HC(X), where H is the Hamiltonian operatorPlanetmathPlanetmath ( 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 theoriesPlanetmathPlanetmath are represented by complex functions.

Let (xi,pi) be a set of Darboux coordinates on T*X. Then we may obtain from each coordinate function an operator on the Hilbert spaceMathworldPlanetmath =L2(X,μ), consisting of functions on X that are square-integrable with respect to some measure μ, by the operator substitution rule:

xix^i =xi, (1)
pip^i =-ixi, (2)

where xi is the “multiplication by xi” operator. Using this rule, we may obtain operators from a larger class of functions. For example,

  1. 1.


  2. 2.


  3. 3.

    if ij then xipjx^ip^j=-ixixj.


The substitution rule creates an ambiguity for the function xipj when i=j, since xipj=pjxi, whereas x^ip^jp^jx^i. This is the operator ordering problem. One possible solution is to choose


since this choice produces an operator that is self-adjoint and therefore corresponds to a physical observable. More generally, there is a construction known as Weyl quantization that uses Fourier transformsMathworldPlanetmath to extend the substitution rules (1)-(2) to a map

C(T*X) Op()
f f^.

This procedure is called “canonical” because it preserves the canonical Poisson brackets. In particular, we have that


which agrees with the Poisson bracket {xi,pj}=δji.

Example 1.

Let X=. The Hamiltonian function for a one-dimensional point particle with mass m is


where V(x) is the potential energy. Then, by operator substitution, we obtain the Hamiltonian operator

Title canonical quantization
Canonical name CanonicalQuantization
Date of creation 2013-03-22 15:53:34
Last modified on 2013-03-22 15:53:34
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 18
Author bci1 (20947)
Entry type Definition
Classification msc 81S10
Classification msc 53D50
Classification msc 46L65
Related topic Quantization
Related topic PoissonBracket
Related topic HamiltonianOperatorOfAQuantumSystem
Related topic SchrodingerOperator
Related topic AsymptoticMorphismsAndWignerWeylMoyalQuantizationProcedures
Defines operator substitution rule
Defines operator ordering problem