Editing Canonical commutation relation (section)

=== 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>