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A general question..

In analytical mechanics, we take a given hamiltonian and re-write it in term of generalzed coordinates. In a way- we recode the hamiltonian to concern only the "essence" of the problem.

However, it seems to me, that in QM we do the opposite- we look for operators that commute with the hamiltonian and recode the hamiltonian in terms of these operators.

But.. isn't using conserved quantities the exact opposite of using with generalzed coordinates (idea-wise)?

And more specifically - Given any hamiltonian , I can simply choose an operator that has nothing to do with the problem (say.. choose the spin operator in a spinless problem). ofcourse this operator will commute with the hamiltonian, but it won't help in solving the problem (ie finding the energy values). This I dont understand- why do some commuting operators help in solving the problem, and some don't? I mean, formally, what is the differece btwn those who do help us, and those who don't?

Thanks!