Commutation and Eigenfunctions

[CanIExplore]

My first question is, does any operator commute with itself? If this is the case, is there a simple proof to show so? If not, what would be a counter-example or a "counter-proof"?

My second question has to do with the properties of an eigenvalue problem. If you have an operator Q such that \hat{Q}\psi=\lambda\psi, then is the following always true?

\hat{Q}^{2}\psi=\lambda^{2}\psi

Since \hat{Q}^{2}\psi=\hat{Q}\hat{Q}\psi=\hat{Q}\lambda\psi=\lambda\hat{Q}\psi=\lambda\lambda\psi=\lambda^{2}\psi.

And is there a more general way of saying this? Like,

\hat{Q}_{i}\hat{Q}_{j}\psi=\hat{Q}_{i}\lambda_{j}\psi=\lambda_{j}\hat{Q}_{i}\psi=\lambda_{i}\lambda_{j}\psi

Which should always be true even if the operators don't commute because the lambdas are just scalars right?

 

[sgd37]

Of course an operator commutes with itself [Q,Q] is always 0 from the definition of the commutator

For the last part you wouldn't be able to write it like that if the operators don't commute. Commuting operators have simultaneous eigenfunctions, non-commuting ones do not