Editing Canonical commutation relation (section)

== Relation to classical mechanics ==
By contrast, in [[classical physics]], all observables commute and the [[commutator]] would be zero. However, an analogous relation exists, which is obtained by replacing the commutator with the [[Poisson bracket]] multiplied by <math>i\hbar</math>,
<math display="block">\{x,p\} = 1 \, .</math>

This observation led [[Paul Dirac|Dirac]] to propose that the quantum counterparts <math>\hat{f}</math>, <math>\hat{g}</math> of classical observables {{mvar|f}}, {{mvar|g}} satisfy
<math display="block">[\hat f,\hat g]= i\hbar\widehat{\{f,g\}} \, .</math>

In 1946, [[Hilbrand J. Groenewold|Hip Groenewold]] demonstrated that a ''general systematic correspondence'' between quantum commutators and Poisson brackets could not hold consistently.<ref name="groenewold">{{Cite journal | last1 = Groenewold | first1 = H. J. | title = On the principles of elementary quantum mechanics | doi = 10.1016/S0031-8914(46)80059-4 | journal = Physica | volume = 12 | issue = 7 | pages = 405–460 | year = 1946 |bibcode = 1946Phy....12..405G }}</ref><ref>{{harvnb|Hall|2013}} Theorem 13.13</ref>

However, he further appreciated that such a systematic correspondence does, in fact, exist between the quantum commutator and a ''[[Deformation theory|deformation]]'' of the Poisson bracket, today called the [[Moyal bracket]], and, in general, quantum operators and classical observables and distributions in [[phase space]]. He thus finally elucidated the consistent correspondence mechanism, the [[Wigner–Weyl transform]], that underlies an alternate equivalent mathematical representation of quantum mechanics known as [[Phase-space formulation|deformation quantization]].<ref name="groenewold"/><ref>{{Cite journal | last1 = Curtright | first1 = T. L. | last2 = Zachos | first2 = C. K. | doi = 10.1142/S2251158X12000069 | title = Quantum Mechanics in Phase Space | journal = Asia Pacific Physics Newsletter | volume = 01 | pages = 37–46 | year = 2012 | arxiv = 1104.5269 | s2cid = 119230734 }}</ref>

=== Derivation from Hamiltonian mechanics ===
According to the [[correspondence principle]], in certain limits the quantum equations of states must approach [[Poisson bracket#Hamilton's equations of motion|Hamilton's equations of motion]]. The latter state the following relation between the generalized coordinate ''q'' (e.g. position) and the generalized momentum ''p'':
<math display="block">\begin{cases}
  \dot{q} =  \frac{\partial H}{\partial p} = \{q, H\}; \\
  \dot{p} = -\frac{\partial H}{\partial q} = \{p, H\}.
\end{cases}</math>

In quantum mechanics the Hamiltonian <math>\hat{H}</math>, (generalized) coordinate <math>\hat{Q}</math> and (generalized) momentum <math>\hat{P}</math> are all linear operators.

The time derivative of a quantum state is represented by the operator <math>-i\hat{H}/\hbar</math> (by the [[Schrödinger equation]]). Equivalently, since in the Schrödinger picture the operators are not explicitly time-dependent, the operators can be seen to be evolving in time (for a contrary perspective where the operators are time dependent, see [[Heisenberg picture]]) according to their commutation relation with the Hamiltonian:
<math display="block">\frac {d\hat{Q}}{dt} = \frac {i}{\hbar} [\hat{H},\hat{Q}]</math>
<math display="block">\frac {d\hat{P}}{dt} = \frac {i}{\hbar} [\hat{H},\hat{P}] \,\, .</math>

In order for that to reconcile in the classical limit with Hamilton's equations of motion, <math> [\hat{H},\hat{Q}]</math> must depend entirely on the appearance of <math>\hat{P}</math> in the Hamiltonian and <math>[\hat{H},\hat{P}]</math> must depend entirely on the appearance of <math>\hat{Q}</math> in the Hamiltonian. Further, since the Hamiltonian operator depends on the (generalized) coordinate and momentum operators, it can be viewed as a functional, and we may write (using [[functional derivative]]s):
<math display="block">[\hat{H},\hat{Q}] = \frac {\delta \hat{H}}{\delta \hat{P}} \cdot [\hat{P},\hat{Q}]</math>
<math display="block">[\hat{H},\hat{P}] = \frac {\delta \hat{H}}{\delta \hat{Q}} \cdot [\hat{Q},\hat{P}] \, . </math>

In order to obtain the classical limit we must then have
<math display="block"> [\hat{Q},\hat{P}] = i \hbar ~ I.</math>