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Question That image shows the angular momentum operators. I need to

Question : Question That image shows the angular momentum operators. I need to

Question

That image shows the angular momentum operators. I need to know how to write the third one, Lz, in terms of the following comlex coordinates:

But I am not sure how to do that, I have tried using a test function but I still can't get it to match up. Every time I try something I have extra terms and we did not get any hints about where to start... help would be great!

 

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