Quantum Mechanics and rotations

Let me warn you up front that I cannot yet reconcile Feynman’s answers with McMahon’s recipe. Well, this is supposed to be about the doing of math, not just about math itself.

For Feynman, i’m in “Lectures on Physics”, volume III, Quantum Mechanics. For McMahon, i’m using “Quantum Mechanics Demystified.” References to Schiff are to his “Quantum Mechanics”.

We have looked at the following here: if I know that a spin-1 particle is in the Jz state |+>, i.e.

and I wanted to know: what are possible values of Jx (the x-component of angular momentum)? and with what probabilities would they occur? What we looked at was the relationship between Jz and Jx in a given coordinate system

I want to consider a different question; I want to rotate the coordinate system.