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I am reading The Principles of Quantum Mechanics by Dirac, in chapter 28 Heisenberg's form for the equations of motion, there is a statement about the classic analog about the Hamiltion form between classic mechanics of and quantum mechanics. My questions are:

  1. If classic analog means that the Hamiltonian operator is the function of $q$ and $p$ (position and mom), then what is the premise of this assumption?

  2. Is there any example of a Hamiltonian that couldn't be expressed as the function of $q$ and $p$?

  3. There is a footnote saying that under Curvilinear coordinates, this assumption is NOT right, so I guess that under Curvilinear coordinates, the classic Hamiltonian form and quantum Hamiltonian form are NOT the same, is there an example of this situation? And why would this happen?

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  • $\begingroup$ I don't really understand what do you mean by: "what is the premise of this assumption"; can you be more explicit on what you are asking? $\endgroup$
    – Noumeno
    Commented Jun 22, 2021 at 8:45
  • $\begingroup$ @Noumeno I wanna know if there is an example that Hamiltonian is not the function of only p and q, except when there are external fields. $\endgroup$
    – Jack
    Commented Jun 22, 2021 at 8:50
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    $\begingroup$ Arguably linked. $\endgroup$ Commented Jun 22, 2021 at 13:40

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Quantization/dequantization is a huge topic, so we will only try to address OP's specific questions:

  1. See e.g. this related Phys.SE post.

  2. The simplest example is perhaps spin/internal angular momentum operators $\hat{S}_x,\hat{S}_y,\hat{S}_z$.

  3. Dirac is referring to operator ordering ambiguities, see e.g. this, this & this related Phys.SE posts.

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  • $\begingroup$ So if I dequantize a quantum theory, an operator degenerate into a function, is there any geometry concepts analog of this process? $\endgroup$
    – Jack
    Commented Jun 22, 2021 at 11:10
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    $\begingroup$ @Jack there is an entire cottage industry of the limits of metaplectic geometry. Perhaps you'd like to brush up on deformation quantization and geometric quantization. It's subtle. $\endgroup$ Commented Jun 22, 2021 at 13:28
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    $\begingroup$ There are literally dozens of books in the market. Start by trawling in Wikipedia, and especially here, and explore the books in the literature. Might take a look at this one. $\endgroup$ Commented Jun 22, 2021 at 13:37

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