According to the topic of deformation quantization, the first few entries in the dictionary between

$$\tag{0} \text{Quantum Mechanics}\quad\longleftrightarrow\quad\text{Classical Mechanics}$$

read

$$\tag{1} \text{Operator}\quad\hat{f}\quad\longleftrightarrow\quad\text{Function/Symbol}\quad f,$$

$$\tag{2} \text{Composition}\quad\hat{f}\circ\hat{g} \quad\longleftrightarrow\quad\text{Star product}\quad f\star g ,$$
and

$$\tag{3} \text{Commutator}\quad [\hat{f},\hat{g}] \quad\longleftrightarrow\quad\text{Poisson bracket}\quad i\hbar\{f,g\}_{PB} + {\cal O}(\hbar^2). $$

Note that the correspondence (0) depends on which symbols one uses, e.g. Weyl symbols, and that there could in general be higher-order quantum corrections ${\cal O}(\hbar^2)$ in the identification (3).

*Example 1:* (Fundamental CCR)
$$\tag{4} [\hat{q},\hat{p}]~=~i\hbar{\bf 1}\quad\longleftrightarrow\quad
\{q,p\}_{PB}~=~1. $$

*Example 2:*

$$\tag{5} [\hat{q}^2,\hat{p}^2]~=~4[\hat{q},\hat{p}] (\hat{q}\hat{p})_W\quad\longleftrightarrow\quad
\{q^2,p^2\}_{PB}~=~4\{q,p\}_{PB} qp, $$
where $(\ldots)_W$ stands for Weyl-symmetrization of operators. See also e.g. this Phys.SE post.

*Example 3:*

$$\tag{6} [\hat{q}^3,\hat{p}^3]~=~9[\hat{q},\hat{p}] (\hat{q}^2\hat{p}^2)_W+\frac{3}{2}[\hat{q},\hat{p}]^3\quad\longleftrightarrow\quad
\{q^3,p^3\}_{PB}~=~9\{q,p\}_{PB} q^2p^2. $$
Note that there are higher-order quantum corrections ${\cal O}(\hbar^3)$ in eq. (6) even after Weyl-symmetrization.

This post imported from StackExchange Physics at 2017-03-13 12:21 (UTC), posted by SE-user Qmechanic