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I feel like there was a name for this sleight of hand approach and I've been unsuccessfully trying to google it for a while.

I think Heisenberg introduced it and it's basically "putting hats on all variables in classical equations", like the Hamiltonian, for example.

By what name is this known?

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Canonical quantisation was introduced by Dirac in 1926

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Canonical quantization mentioned by @CharlesFrancis is probably the most precise answer. Colloquially one would often use term correspondence principle, also known as Bohr's correspondence principle, and the Ehrenfest theorem. However in reality these have more to do with the system's behavior than with the actual quantization procedure.

Note in this context that some phenomena, such as spin, have no classical analogs.

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