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Nov 30, 2020 at 20:07 vote accept Noumeno
Nov 30, 2020 at 20:07 comment added Noumeno Wonderful. For an eventual future reader I think this link may be helpful: it expands on spherical basis vectors and how they work.
Nov 30, 2020 at 16:47 history edited JEB CC BY-SA 4.0
added an addendum regarding cartesian reps and spherical reps per OP's comment.
Nov 30, 2020 at 16:04 comment added JEB That's why I mention coordinates. $\vec x$ depends on theta and phi. I will add an addendum that will convince you...because it is what convince me.
Nov 30, 2020 at 15:12 comment added Noumeno I was following you until you re-written my equation (1). I feel like you breeze through it without properly showing why we can say that the equation is true, and this was one of the main points of my question. Don't get me wrong your answer is useful but I still have some perplexities, like: Y is dependent upon $\theta$ and $\phi$, but in the left hand side of the equation there is noting dependent on $\theta$ and $\phi$, so where does the dependence come from? I would expect something like $\langle \theta \ \phi| l \ m\rangle$. I think that you could expend on the derivation of equation (1).
Nov 30, 2020 at 2:32 history edited JEB CC BY-SA 4.0
corrected error about theta phi pure state.
Nov 29, 2020 at 21:04 history answered JEB CC BY-SA 4.0