Dirac quantization

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

${\displaystyle \{f_{1},f_{2}\}\rightarrow -{\frac {i}{\hbar }}[{\hat {f}}_{1},{\hat {f}}_{2}]}$

being ${\displaystyle f_{1}}$ and ${\displaystyle f_{2}}$ 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 ${\displaystyle H}$ having the following set of canonical equations describing the dynamics

${\displaystyle \{q_{i},p_{k}\}=\delta _{ik},\ \{q_{i},q_{k}\}=0,\ \{p_{i},p_{k}\}=0}$

${\displaystyle \partial _{t}p_{i}=\{p_{i},H\},\ \partial _{t}q_{i}=\{q_{i},H\},}$

one can postulate a corresponding quantum system with dynamical equations

${\displaystyle [{\hat {q}}_{i},{\hat {p}}_{k}]=i\hbar \delta _{ik},\ [{\hat {q}}_{i},{\hat {q}}_{k}]=0,\ [{\hat {p}}_{i},{\hat {p}}_{k}]=0}$

${\displaystyle \partial _{t}{\hat {p}}_{i}=-{\frac {i}{\hbar }}[{\hat {p}}_{i},{\hat {H}}],\ \partial _{t}{\hat {q}}_{i}=-{\frac {i}{\hbar }}[{\hat {q}}_{i},{\hat {H}}].}$

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

${\displaystyle d_{t}{\hat {A}}=\partial _{t}{\hat {A}}-{\frac {i}{\hbar }}[{\hat {A}},{\hat {H}}].}$