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As far as I understood, when we want to quantize a system, the procedure can be the following (but probably not the most general one):

We start by writing down the Lagrangian of the system.

We compute the momentum associated to the generalized coordinates and we can then write down the Hamiltonian.

We promote the canonical coordinates $(q,p)$ into operators by imposing: $[\widehat{q},\widehat{p}]=i \hbar$

My questions:

  • Does such procedure always work? It looks like a "magic" recipe for me. My concern is more about non-relativistic Q.M.

  • If we have equations of motion, whatever the Lagrangian we find that will produce the good classical equation of motion will be okay to start with for this quantization procedure ? Or it is more subtle.

  • Are there good reasons to understand why quantization "works" or it should be taken as a postulate?

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I understand the "magic recipe" impression:) It is better to introduce quantization as an answer to a question. The latter is: given a classical Hamiltonian dynamical system, how to construct a family of quantum systems indexed by $\hbar$ such that, when taking the $\hbar\rightarrow 0$ limit, one recovers the given classical system. The magic recipe (correspondence principle) is one clever way (among many approaches) of answering the question. It works well for integrable systems. For chaotic systems, there is a lot which is not understood; see, e.g., the Bohigas-Giannoni-Schmit conjecture.

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    $\begingroup$ Thank you for your answer. I don't really understand this fact when we say $\hbar \rightarrow 0$ we find the classical limit. The full structure of the theory is very different after quantization, whatever the value of $\hbar$ you put (even $0$). You started with number you ended up with operators. Maybe they will commute in this limit but for me it doesn't correspond to the classical limit. At least I don't understand why properly $\endgroup$
    – StarBucK
    Feb 6, 2020 at 15:41
  • $\begingroup$ The kind of limits are commutators giving Poisson brackets of classical observables. I can't explain to you in just a paragraph or so, a whole big field of knowledge. Look up: "semiclassical analysis", and "deformation quantization". $\endgroup$ Feb 6, 2020 at 15:44
  • $\begingroup$ Allright thanks. And for the other questions like for integrable system lets say, if I find any Lagrangian that leads to the good equation of motion, then I can use this recipe without risks to quantize properly the theory ? $\endgroup$
    – StarBucK
    Feb 6, 2020 at 15:50
  • $\begingroup$ @StarBucK: That's a good question. I don't know the answer off the bat. I think you should ask it as a separate post. Somewhat related: I know that a good part of semi-classical analysis is about canonical transformations on the classical side and their relation to quantum symmetries given by unitaries (Fourier integral operators). Your new question has a similar flavor. There may be results saying that if you change the Lagrangian like this, then it amounts to a unitary transformation for the quantum systems. $\endgroup$ Feb 6, 2020 at 18:07
  • $\begingroup$ A relevant reference for this kind of questions is "Quantization of Gauge Systems" by Henneaux and Teitelboim. Another book which I'm sure you will find interesting is "From c-numbers to q-Numbers" by Darrigol. $\endgroup$ Feb 6, 2020 at 18:12

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