# 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 $H\in C^{\infty}(X)$, 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 ($\mathbb{C}$) 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 $\mathcal{H}=L^{2}(X,\mu)$, consisting of functions on $X$ that are square-integrable with respect to some measure $\mu$, by the operator substitution rule:

 $\displaystyle x^{i}\mapsto\hat{x}^{i}$ $\displaystyle=x^{i}\cdot,$ (1) $\displaystyle p_{i}\mapsto\hat{p}_{i}$ $\displaystyle=-i\hbar\frac{\partial}{\partial x^{i}},$ (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. 1.

$x^{i}x^{j}\mapsto\hat{x}^{i}\hat{x}^{j}=x^{i}x^{j}\cdot$,

2. 2.

$p_{i}p_{j}\mapsto\hat{p}_{i}\hat{p}_{j}=-\hbar^{2}\frac{\partial^{2}}{\partial x% ^{i}x^{j}}$,

3. 3.

if $i\neq j$ then $x^{i}p_{j}\mapsto\hat{x}^{i}\hat{p}_{j}=-i\hbar x^{i}\frac{\partial}{\partial x% ^{j}}$.

###### 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 $\hat{x}^{i}\hat{p}_{j}\neq\hat{p}_{j}\hat{x}^{i}$. This is the operator ordering problem. One possible solution is to choose

 $x^{i}p_{j}\mapsto\frac{1}{2}\left(\hat{x}^{i}\hat{p}_{j}+\hat{p}_{j}\hat{x}^{i% }\right),$

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 transforms to extend the substitution rules (1)-(2) to a map

 $\displaystyle C^{\infty}(T^{*}X)$ $\displaystyle\to\operatorname{Op}(\mathcal{H})$ $\displaystyle f$ $\displaystyle\mapsto\hat{f}.$
###### Remark.

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

 $\frac{-i}{\hbar}[\hat{x}^{i},\hat{p}_{j}]:=\frac{-i}{\hbar}\left(\hat{x}^{i}% \hat{p}_{j}-\hat{p}_{j}\hat{x}^{i}\right)=\delta^{i}_{j},$

which agrees with the Poisson bracket $\{x^{i},p_{j}\}=\delta^{i}_{j}$.

###### Example 1.

Let $X=\mathbb{R}$. The Hamiltonian function for a one-dimensional 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

 $\hat{H}=\frac{-\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}+V(x).$
 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