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canonical quantization (Definition)

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. 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. $x^i x^j \mapsto \hat{x}^i \hat{x}^j = x^i x^j \cdot$,
  2. $p_i p_j \mapsto \hat{p}_i \hat{p}_j = -\hbar^2 \frac{\partial ^2}{\partial x^i x^j}$,
  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 1   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
$\displaystyle x^i p_j \mapsto \frac{1}{2}\left(\hat{x}^i \hat{p}_j + \hat{p}_j \hat{x}^i\right),$    
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 $m$ is
$\displaystyle H = \frac{p^2}{2m} + V(x),$    
where $V(x)$ is the potential energy. Then, by operator substitution, we obtain the Hamiltonian operator
$\displaystyle \hat{H} = \frac{-\hbar^2}{2m} \frac{d^2}{dx^2} + V(x).$    



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Also defines:  quantization methods
Keywords:  canonical, quantum theory, quantization

Cross-references: energy, mass, point particle, Fourier transforms, quantization, observable, Hilbert space, operator, functions, quantum theories, wave amplitudes, Observables and States, domain, Hamiltonian operators, quantum operators, correspondence principle, quantum mechanics, manifold, system
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This is version 2 of canonical quantization, born on 2010-06-04, modified 2010-06-04.
Object id is 867, canonical name is CanonicalQuantization.
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Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 02.70.-cxx (Computational techniques )
 02.90.+p (Other topics in mathematical methods in physics )
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