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 ($ℝ$) domain, and the quantum ones being in the complex ($ℂ$) 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 $\mathscr{H}=L^2(X,\mu )$, consisting of functions on $X$ that are square-integrable with respect to some measure $\mu$, by the operator substitution rule:

$x^i\mapsto {\widehat{x}}^i$ $=x^i\cdot ,$ (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 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

$C^{\mathrm{\infty }}(T^{*}X)$ $\to \mathrm{Op}(\mathscr{H})$

$f$ $\mapsto \widehat{f}.$

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 $X=ℝ$. 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

$$\widehat{H}=\frac{-{\mathrm{\hslash }}^2}{2m}\frac{d^2}{d{x}^2}+V(x).$$

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