# Classic analog of quantum mechanics when dealing with Hamiltonian operator

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?

• 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? Commented Jun 22, 2021 at 8:45
• @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.
– Jack
Commented Jun 22, 2021 at 8:50
• Arguably linked. Commented Jun 22, 2021 at 13:40

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.

• So if I dequantize a quantum theory, an operator degenerate into a function, is there any geometry concepts analog of this process?
– Jack
Commented Jun 22, 2021 at 11:10
• @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. Commented Jun 22, 2021 at 13:28
• 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. Commented Jun 22, 2021 at 13:37