Finding expectation value of two operators in a 3 state QM system

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The discussion focused on finding the expectation value of two operators in a three-state quantum mechanics system. The original thread was deleted but was restored using a cached version from Yahoo.com. Participants engaged in detailed discussions, and the original poster's questions were successfully addressed. Acknowledgment was given to @PeroK for their significant contributions to the thread. The thread has now been closed following the resolution of the inquiries.
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Homework Statement
Finding expectation value of two operators in a 3 state QM system
Relevant Equations
(see below)
A recent thread by @coolcantalope was accidentally deleted by a Mentor (I won't say which one...), so to restore it we had to use the cached version from Yahoo.com. Below are the posts and replies from that thread.

The cached 2-page thread can be found by searching on the thread title, and is here:

https://cc.bingj.com/cache.aspx?q="...lang=en-US&w=WqoRsTRSSoiCPSlyCzATk534wzOv9qXu

https://cc.bingj.com/cache.aspx?q="...lang=en-US&w=E37oxZc2R1_7fPuG11KuK7LqMbqr-EkC

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