canonical quantization
Canonical quantization^{} is a method of relating, or associating, a classical system of the form $({T}^{*}X,\omega ,H)$, where $X$ is a manifold, $\omega $ is the canonical symplectic form^{} on ${T}^{*}X$, with a (more complex) quantum system represented by $H\in {C}^{\mathrm{\infty}}(X)$, where $H$ is the
Hamiltonian operator^{} (http://planetmath.org/HamiltonianOperatorOfAQuantumSystem). 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 ($\mathbb{R}$) domain, and the quantum ones being in the complex ($\u2102$) 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 $({x}^{i},{p}_{i})$ be a set of Darboux coordinates on ${T}^{*}X$. Then we may obtain from each coordinate function an operator on the Hilbert space^{} $\mathscr{H}={L}^{2}(X,\mu )$, consisting of functions on $X$ that are squareintegrable with respect to some measure $\mu $, by the operator substitution rule:
${x}^{i}\mapsto {\widehat{x}}^{i}$  $={x}^{i}\cdot ,$  (1)  
${p}_{i}\mapsto {\widehat{p}}_{i}$  $=i\mathrm{\hslash}{\displaystyle \frac{\partial}{\partial {x}^{i}}},$  (2) 
where ${x}^{i}\cdot $ is the “multiplication by ${x}^{i}$” operator. Using this rule, we may obtain operators from a larger class of functions. For example,

1.
${x}^{i}{x}^{j}\mapsto {\widehat{x}}^{i}{\widehat{x}}^{j}={x}^{i}{x}^{j}\cdot $,

2.
${p}_{i}{p}_{j}\mapsto {\widehat{p}}_{i}{\widehat{p}}_{j}={\mathrm{\hslash}}^{2}\frac{{\partial}^{2}}{\partial {x}^{i}{x}^{j}}$,

3.
if $i\ne j$ then ${x}^{i}{p}_{j}\mapsto {\widehat{x}}^{i}{\widehat{p}}_{j}=i\mathrm{\hslash}{x}^{i}\frac{\partial}{\partial {x}^{j}}$.
Remark.
The substitution rule creates an ambiguity for the function ${x}^{i}{p}_{j}$ when $i=j$, since ${x}^{i}{p}_{j}={p}_{j}{x}^{i}$, whereas ${\widehat{x}}^{i}{\widehat{p}}_{j}\ne {\widehat{p}}_{j}{\widehat{x}}^{i}$. This is the operator ordering problem. One possible solution is to choose
$${x}^{i}{p}_{j}\mapsto \frac{1}{2}\left({\widehat{x}}^{i}{\widehat{p}}_{j}+{\widehat{p}}_{j}{\widehat{x}}^{i}\right),$$ 
since this choice produces an operator that is selfadjoint and therefore corresponds to a physical observable. More generally, there is a construction known as Weyl quantization that uses Fourier transforms^{} to extend the substitution rules (1)(2) to a map
${C}^{\mathrm{\infty}}({T}^{*}X)$  $\to \mathrm{Op}(\mathscr{H})$  
$f$  $\mapsto \widehat{f}.$ 
Remark.
This procedure is called “canonical” because it preserves the canonical Poisson brackets. In particular, we have that
$$\frac{i}{\mathrm{\hslash}}[{\widehat{x}}^{i},{\widehat{p}}_{j}]:=\frac{i}{\mathrm{\hslash}}\left({\widehat{x}}^{i}{\widehat{p}}_{j}{\widehat{p}}_{j}{\widehat{x}}^{i}\right)={\delta}_{j}^{i},$$ 
which agrees with the Poisson bracket $\{{x}^{i},{p}_{j}\}={\delta}_{j}^{i}$.
Example 1.
Let $X=\mathbb{R}$. The Hamiltonian function for a onedimensional point particle with mass $m$ is
$$H=\frac{{p}^{2}}{2m}+V(x),$$ 
where $V(x)$ is the potential energy. Then, by operator substitution, we obtain the Hamiltonian operator
$$\widehat{H}=\frac{{\mathrm{\hslash}}^{2}}{2m}\frac{{d}^{2}}{d{x}^{2}}+V(x).$$ 
Title  canonical quantization 
Canonical name  CanonicalQuantization 
Date of creation  20130322 15:53:34 
Last modified on  20130322 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 