PlanetMath: canonical quantization

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canonical quantization

(Definition)

Canonical quantization is a method of turning a classical system of the form $(T^*X, \omega, H)$ , where is a manifold, $\omega$ is the canonical symplectic form on , and $H \in C^\infty(X)$ is the Hamiltonian, into a quantum system. The first formulations of the theory of quantum mechanics used this method, and it continues to be the most practical and accessible method of quantization.

Let be a set of Darboux coordinates on . Then we may obtain from each coordinate function an operator on the Hilbert space $\mathcal{H} = L^2(X, \mu)$ , consisting of functions on 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

” operator. Using this rule, we may obtain operators from a larger class of functions. For example,

$x^i x^j \mapsto \hat{x}^i \hat{x}^j = x^i x^j \cdot$ ,
$p_i p_j \mapsto \hat{p}_i \hat{p}_j = -\hbar^2 \frac{\partial ^2}{\partial x^i x^j}$ ,
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 1 The substitution rule creates an ambiguity for the function

when

, since

, 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

$\displaystyle x^i p_j \mapsto \frac{1}{2}\left(\hat{x}^i \hat{p}_j + \hat{p}_j \hat{x}^i\right),$

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

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

Remark 2 This procedure is called “canonical” because it preserves the canonical Poisson brackets. In particular, we have that

$\displaystyle \frac{-i}{\hbar}[\hat{x}^i, \hat{p}_j] := \frac{-i}{\hbar}\left(\hat{x}^i\hat{p}_j - \hat{p}_j\hat{x}^i\right) = \delta^i_j,$

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

$\displaystyle H = \frac{p^2}{2m} + V(x),$

where

is the potential energy. Then, by operator substitution, we obtain the Hamiltonian operator

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

"canonical quantization" is owned by plinko.

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See Also: quantization, Poisson bracket

Also defines:

operator substitution rule, operator ordering problem

Keywords:

quantization, quantum

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Cross-references: potential, mass, point, Poisson brackets, preserves, map, Fourier transforms, self-adjoint, solution, ordering, class, measure, Hilbert space, operator, function, coordinate, Darboux coordinates, quantization, theory, quantum system, Hamiltonian, symplectic form, canonical, manifold, classical system
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This is version 1 of canonical quantization, born on 2006-05-02.
Object id is 7894, canonical name is CanonicalQuantization.
Accessed 2095 times total.

Classification:

AMS MSC:	46L65 (Functional analysis :: Selfadjoint operator algebras :: Quantizations, deformations)
	53D50 (Differential geometry :: Symplectic geometry, contact geometry :: Geometric quantization)
	81S10 (Quantum theory :: General quantum mechanics and problems of quantization :: Geometry and quantization, symplectic methods)

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