Canonical quantization

Definition

Let $\R^{2n}$ be the phase space with coordinates $(q_1, \dots, q_n, p_1, \dots, p_n)$ and Poisson bracket $\{f, g\} = \sum_j \left( \frac{\partial f}{\partial q_j}\frac{\partial g}{\partial p_j} - \frac{\partial f}{\partial p_j}\frac{\partial g}{\partial q_j} \right)$, and let $\hbar > 0$. Canonical quantization assigns:

1. the Hilbert space $L^2(\R^n)$ as the space of states;
2. to $q_j$ the position operator $(Q_j \psi)(x) = x_j \psi(x)$, and to $p_j$ the momentum operator $P_j \psi = -i\hbar\, \partial \psi / \partial x_j$, both self-adjoint and both mapping the Schwartz space $\mathcal{S}(\R^n)$ into itself;
3. to the constant function $1$ the identity operator $I$.

On $\mathcal{S}(\R^n)$ these operators satisfy the canonical commutation relations $$[Q_j, Q_k] = 0, \qquad [P_j, P_k] = 0, \qquad [Q_j, P_k] = i\hbar\, \delta_{jk}\, I,$$ which match $\{q_j, q_k\} = \{p_j, p_k\} = 0$ and $\{q_j, p_k\} = \delta_{jk}$ under the rule $\{f, g\} \mapsto \frac{1}{i\hbar}[\,\cdot\,, \cdot\,]$. A further observable $f(q, p)$ is quantized as an operator $f(Q, P)$ after the choice of an ordering rule.