PlanetPhysics/Canonical Quantization

Canonical quantization is a method of relating, or associating, a classical system of the form , where is a manifold, is the canonical symplectic form on , with a (more complex) quantum system represented by , where is the Hamiltonian operator. Some of the early formulations of quantum mechanics used such quantization methods under the umbrella of the \htmladdnormallink{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 () domain, and the quantum ones being in the complex () 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 be a set of Darboux coordinates on . Then we may obtain from each coordinate function an operator on the Hilbert space , consisting of functions on that are square-integrable with respect to some measure , 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 is the "multiplication by " operator. Using this rule, we may obtain operators from a larger class of functions. For example,

  1. ,
  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 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 when , since , whereas . This is the operator ordering problem. One possible solution is to choose

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}}


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

which agrees with the Poisson bracket . \end{rmk}

\begin{ex} Let . The Hamiltonian function for a one-dimensional point particle with mass is

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