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# 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. 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.

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

$\displaystyle C^{\infty}(T^{*}X)$ | $\displaystyle\to\Op(\mathcal{H})$ | ||

$\displaystyle f$ | $\displaystyle\mapsto\hat{f}.$ |

###### 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

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

## Mathematics Subject Classification

81S10*no label found*53D50

*no label found*46L65

*no label found*

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