It is stated that commuting Hermitian (linear + self-adjoint) operators have 'simultaneous eigenstates'? Which of the following is most correct, and why?
2 commuting operators share AN eigenstate
2 commuting operators share SOME eigenstates
2 commuting operators share THE SET of all possible eigenstates of the operator
My intuition would be that 2 commuting operators have to share the EXACT SAME FULL SET of all possible eigenstates, but the Quantum Mechanics textbook I am reading from is not sufficiently specific. If I am correct, any relatively accessible mathematical proof, or at least conceptual explanation, would be much appreciated.