PlanetPhysics/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 \htmladdnormallink{correspondence principle {http://planetphysics.us/encyclopedia/PrincipleOfCorrespondingStates.html} 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:

Failed to parse (unknown function "\pdiff"): {\displaystyle \begin{matrix} x^i \mapsto \hat{x}^i &= x^i \cdot, (1)\\ p_i \mapsto \hat{p}_i &= -i \hbar \pdiff{}{x^i} (2), \end{matrix}}

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. Failed to parse (unknown function "\pdiff"): {\displaystyle p_i p_j \mapsto \hat{p}_i \hat{p}_j = -\hbar^2 \pdiff{^2}{x^i x^j}} ,
3. if $i\neq j$ then Failed to parse (unknown function "\pdiff"): {\displaystyle x^i p_j \mapsto \hat{x}^i \hat{p}_j = -i \hbar x^i \pdiff{}{x^j}} .

\begin{rmk} 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 (2)-(3) to a map

Failed to parse (unknown function "\Op"): {\displaystyle \begin{matrix} C^\infty(T^*X) &\to \Op (\mathcal{H}) \\ f &\mapsto \hat{f}. \end{matrix}}

\end{rmk}

\begin{rmk} This procedure is called "canonical" because it preserves the canonical Poisson brackets. In particular, we have that

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

which agrees with the Poisson bracket $\{x^{i},p_{j}\}=\delta _{j}^{i}$. \end{rmk}

\begin{ex} 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

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

\end{ex}