What Are the Properties of Commutating Operators in Math?

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Help me to be sure I got this right.
Two commutating operators:
1)have common eigenstates or the same?
2)can be observe at the same time.

thanks
 
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I would say that you got it right, but would you please write down how you got this result..
 
Thread 'Help with Time-Independent Perturbation Theory "Good" States Proof'

Thread 'Help with Time-Independent Perturbation Theory "Good" States Proof'

(Disclaimer: this is not a HW question. I am self-studying, and this felt like the type of question I've seen in this forum. If there is somewhere better for me to share this doubt, please let me know and I'll transfer it right away.) I am currently reviewing Chapter 7 of Introduction to QM by Griffiths. I have been stuck for an hour or so trying to understand the last paragraph of this proof (pls check the attached file). It claims that we can express Ψ_{γ}(0) as a linear combination of...
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