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I am working through this quantum mechanics homework question and I am a little confused on what I am being asked to do. I have come to one of two possible answers but I don't think either is correct.

Say you have a ket $\psi$ I am asked to find the possible outcomes of measuring the observable values of the operators given in the problem.

I have already shown that they are Hermitian. But I am not sure if I am supposed to calculate the expected values of the operator $\Omega$. where the expecred value is found by $<\Omega> = <\psi|\Omega|\psi>$ I don't think this is correct because the problem asks for the outcomes of measuring the obersvables.

So then I thought maybe because I diagonalized $\Omega$ all I need to do is show that the observables are the eigenvalues of the operator? Which are just down the diagonal of $\Omega$. But again I am suspicious of this because then I am asked to find the probability of each of the observables and I am not sure how to do that with just knowing that they are observable. Don't I need like an observation state ket or something like that? Something that would go in the equation: $P(\omega)= |<\psi|\omega>|^2$ I tried to find some stuff online but some of the online solutions seem to do the reverse order for $\psi$ and $\omega$ to find the probability.

Which of these methods is correct to measure the observables? Or am I completely off?

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  • $\begingroup$ Hint: try to relate $\omega$ to your operator $\endgroup$ – Superfast Jellyfish Mar 1 at 21:27
  • $\begingroup$ Relating it to the operator would be like finding the column vector corresponding to the eigenvector of that eigenvalue??? Like: $|\omega>=$ (Eigenvector)? $\endgroup$ – user255496 Mar 1 at 21:29
  • $\begingroup$ Yup. That’s right. $\endgroup$ – Superfast Jellyfish Mar 1 at 21:32
  • $\begingroup$ To make sure I am understanding correctly: Do I just do the inner product of the $\psi$ and the $|\omega>$? So the possible outcomes of $\Omega$ would be $<\omega_1|\psi>$ and then the same thing for $|\omega_2>$ and $|\omega_3>$? The result of those three inner products would be the posible outcomes of measuring the observables? $\endgroup$ – user255496 Mar 1 at 21:37
  • $\begingroup$ Outcomes would always be an eigenstate (of that operator) with the observable being the corresponding eigenvalue. The probability of each of them is given by your $P$ expression. Remember that inner products are just (complex) numbers. $\endgroup$ – Superfast Jellyfish Mar 1 at 21:39

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