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For a Hamiltonian system without constraints, Dirac quantization can be imposed through the following condition between Poisson brackets and quantum brackets
![{\displaystyle \{f_{1},f_{2}\}\rightarrow -{\frac {i}{\hbar }}[{\hat {f}}_{1},{\hat {f}}_{2}]}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/e701b1a5cdd7359199930a2882f05da5194ae73d)
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
![{\displaystyle [{\hat {q}}_{i},{\hat {p}}_{k}]=i\hbar \delta _{ik},\ [{\hat {q}}_{i},{\hat {q}}_{k}]=0,\ [{\hat {p}}_{i},{\hat {p}}_{k}]=0}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/48c2a6962f95023b60ab59aa297b9ca23c7f54ef)
![{\displaystyle \partial _{t}{\hat {p}}_{i}=-{\frac {i}{\hbar }}[{\hat {p}}_{i},{\hat {H}}],\ \partial _{t}{\hat {q}}_{i}=-{\frac {i}{\hbar }}[{\hat {q}}_{i},{\hat {H}}].}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/df5a2a797fba7c2b58d90cfe8a327946209c4c39)
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
![{\displaystyle d_{t}{\hat {A}}=\partial _{t}{\hat {A}}-{\frac {i}{\hbar }}[{\hat {A}},{\hat {H}}].}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/7bc6fab6f02e8b950563cbafca1f0b5116557404)
