Dirac quantization: Difference between revisions

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


$$\{f_1,f_2\}\rightarrow -\frac{i}{\hbar}[\hat f_1,\hat f_2]$$
<math>\{f_1,f_2\}\rightarrow -\frac{i}{\hbar}[\hat f_1,\hat f_2]</math>


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 incur into an ordering problem.
being <math>f_1</math> and <math>f_2</math> 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
So, for a mechanical system with Hamiltonian <math>H</math> having the following set of canonical equations describing the dynamics


$$\{q_i,p_k\}=\delta_{ik},\ \{q_i,q_k\}=0,\ \{p_i,p_k\}=0$$
<math>\{q_i,p_k\}=\delta_{ik},\ \{q_i,q_k\}=0,\ \{p_i,p_k\}=0</math>


$$\partial_t p_i=\{p_i,H\}, \partial_t q_i=\{q_i,H\},$$
<math>\partial_t p_i=\{p_i,H\},\ \partial_t q_i=\{q_i,H\},</math>


one can postulate a corresponding quantum system with dynamical equations
one can postulate a corresponding quantum system with dynamical equations


$$[\hat q_i,\hat p_k]=i\hbar\delta_{ik},\ [\hat q_i,\hat q_k]=0,\ [\hat p_i,\hat p_k]=0$$
<math>[\hat q_i,\hat p_k]=i\hbar\delta_{ik},\ [\hat q_i,\hat q_k]=0,\ [\hat p_i,\hat p_k]=0</math>
 
<math>\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].</math>


$$\partial_t \hat p_i=-\frac{i}{\hbar}[\hat p_i,\hat H], \partial_t \hat q_i=-\frac{i}{\hbar}[\hat q_i,\hat H].$$
The operatorial equations describing time evolution of the operators are now termed '''Heisenberg equations'''. In its more general form, Heisenberg equation for an operator <math>A</math> is written, again using Dirac quantization on Poisson brackets, as


The operatorial equations describing time evolution of the operators are now termed '''Heisenberg equations'''.
<math> d_t\hat A=\partial_t\hat A-\frac{i}{\hbar}[\hat A,\hat H].</math>


[[Category:methods]]
[[Category:methods]]

Latest revision as of 15:56, 1 July 2018

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