My book on quantum physics says that if two Hermitian operators commute then it emerges that they have common eigenfunctions. Is that true? If A,B hermitian commuting operators and Ψ a random wavefunction then: [A,B]Ψ=0 => ABΨ=BAΨ If we assume that Ψ is B`s eigenfunction: b*AΨ=BAΨ From this equation, how does it emerge for A to have common eigenfunctions with B? I agree that if Ψ was A`s eigenfunction then the equation would be satisfied, but in the more general case of a random Ψ it doesnt emerge that Ψ is also A`s eigenfunction! Another argument to support this is the following. If those operators commute, then their uncertainties satisfy the following inequality: ΔΑ*ΔΒ >=0 Ψ is B`s eigenfunction so: ΔB=0. Now the inequality is satisfied for ANY ΔΑ (including zero). So is the statement << if two hermitian operators commute then it emerges that they have common eigenfunctions. >> true or false?