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 theHamiltonian 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 theclassical 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 quantumtheories 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

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