1. Observable measurement refers to mechanical quantities in physics, and all mechanical quantities can be expressed by self-adjoint operators, that is, observable measurements all correspond to self-adjoint operators, so there is no problem.
2. There must be an identical eigenfunction system between two reciprocal observable values, especially here, instead of one or several eigenfunctions. Facts have proved that:
If [A, B] = 0 and Af=Anf, we can know that A(Bf)=B(Af)=BAnf=AnBf under nondegenerate conditions, that is to say, Bf is also the eigenstate of A, and the eigenvalue is an. Because An is nondegenerate, Bf and F can only differ by a constant factor at most, which is denoted as Bn, that is, Bf=Bnf.
In the case of degeneration, it can be proved that it is not very convenient to write by recombination and using the conditions of non-mediocre solutions of odd linear equations. Please call me if you don't understand.
3. Because the two operators are commutative, that is to say, there will be * * * identical eigenfunction systems. If two operators are measured in this space, there will be a certain value, that is, A and B can be measured accurately at the same time. If the selected eigenfunction system is not composed of * * * identical eigenfunction systems, an uncertain relationship will arise.