Dirac quantization

From DispersiveWiki

For a Hamiltonian system without constraints, Dirac quantization can be imposed through the following condition between Poisson brackets and quantum brackets

being and functions of the canonical variables and the hat is there to remember that, in the quantum case, one has operators acting on a Hilbert space. The definition of these functions for operators incurs into an ordering problem.

So, for a mechanical system with Hamiltonian having the following set of canonical equations describing the dynamics

one can postulate a corresponding quantum system with dynamical equations

The operatorial equations describing time evolution of the operators are now termed Heisenberg equations. In its more general form, Heisenberg equation for an operator is written, again using Dirac quantization on Poisson brackets, as