I was studying about what does it mean canonical quantization treatment. But now I have the next question.
Why if we establish canonical the commutation relations $$\left[q,p\right]=i\hbar,\quad \left[q,q\right]=\left[p,p\right]=0, $$ we can say that the system is quantized?
I mean, what does imply these relations in the system in order to obtain a discrete set of states? I know where the idea comes from, a generalization of Poisson brackets, so I would like to know if there is another way to see that the system is quantized just by imposing these relations?

  • $\begingroup$ So your point is since Poisson brackets obey similar relations, how come aren't classical energies quantized? It's part of the setup. You are familiar with ladder operators leading to the discrete spectrum for the oscillator? (also see this and the Royer review cited.) $\endgroup$ Mar 15 at 22:53

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