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Different authors get to those differently: some start from translator operators, some relate those to Poisson brackets, etc...

My objective is to get a good intuitive feeling of what commuting and not commuting observables imply about the physical system? What are good physical examples to justify this?

What other ways have people used to link ,for instance, uncertainty relation and commutativity?

Insightful, less mathematical answers (IF AT ALL POSSIBLE) are greatly appreciated as I already have plenty of access to the mathematical foundation.

Thanks,

sokrates