# Operators depending on the same independent variable but commuting between them [closed]

As far as I understood in quantum mechanics two operators can commute even though they are not functionally independent, which means that they can depend on the same independent variable.

On the other hand I am not 100% sure about this, I've never seen two operators that depend on the same variable commuting between them. If my statement is correct, do you have any example of such operators?

• What do you mean exactly here? Can you give examples? Sep 12, 2023 at 11:45
• Presumably OP means e.g. $[AA,AAA]=0$. Sep 12, 2023 at 11:57

To prove the "hard part" of this, a simple argument would be the following: two commuting self-adjoint operators share a common eigenbasis $$(v_i)_{i \in I}$$ for some set $$I$$, choose any family $$(\lambda_i)_{i \in I}$$ of pairwise distinct real numbers. Now take the two Lagrange polynomials that take the $$\lambda_i$$'s to the eigenvalues of the two operators and you're good.