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I am having a little trouble with the concept of finding out the maximal set of commuting observables. Suppose I have n commuting operators. Then the wavefunction I use must have n parameters also. For instance, [itex] L_{3}, L^{2}[/itex] and [itex] H[/itex] where H is the Hamiltonian and L is the angular momentum operator all commute. So the common eigenstates have three variables inside i.e [itex]|n l m>[/itex].

What I can't figure out is how to ensure that these operators are "distinct" from each other. For example, if I take a one of the above operators and multiply it by a scalar I will get a different operator, say something like [itex]2H[/itex]. Now I have four commuting operators instead of three. Obviously, this is not going to give me another independent variable inside the wavefunction. This was a trivial example but in general, how should one ensure that the commuting operators that one has are all distinct?

My best guess so far is to check that the matrix representations form a linearly independent set but is there a) a weaker condition and b) a way to do this without resorting to matrix representation?

Thank you.