Dirac quantization

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

$$\{f_1,f_2\}\rightarrow -\frac{i}{\hbar}[\hat f_1,\hat f_2]$$

being $f_1$ and $f_2$ 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 $H$ having the following set of canonical equations describing the dynamics

$$\{q_i,p_k\}=\delta_{ik},\ \{q_i,q_k\}=0,\ \{p_i,p_k\}=0$$

$$\partial_t p_i=\{p_i,H\}, \partial_t q_i=\{q_i,H\},$$

one can postulate a corresponding quantum system with dynamical equations

$$[\hat q_i,\hat p_k]=i\hbar\delta_{ik},\ [\hat q_i,\hat q_k]=0,\ [\hat p_i,\hat p_k]=0$$

$$\partial_t \hat p_i=-\frac{i}{\hbar}[\hat p_i,\hat H], \partial_t \hat q_i=-\frac{i}{\hbar}[\hat q_i,\hat H].$$

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

$$ d_t\hat A=\partial_t\hat A-\frac{i}{\hbar}[\hat A,\hat H].$$