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Understanding Spin in QM

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Summary: please help me understand the following questions from Physics GRE test Thank you very much! To be honest i really hate this formalism. Memorizing such things is pain. but like everything it is what it is

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You will have to tell us more. Which parts don’t you understand? How far have you gotten trying to solve these problems yourself?
 
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You will have to tell us more. Which parts don’t you understand? How far have you gotten trying to solve these problems yourself?
Only problem 2. I understand some part of it
I understand that for the ground state l should be 0 in total, and we can use j = l + s to do the selection. But would you mind sharing your steps so that i can learn from it?

Thank you very much!
 
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Question 2 is about addition of angular momentum. State (1) is the singlet state which contributes zero spin for the nucleus, it isn’t a possibility since you can’t get spin 3/2 by combining a spin 0 state with a spin 1/2 state. The next three states are the triplet spin 1 states which can give spin 3/2 when combined with the electron, however the last state is ruled out by the magnetic quantum number.
 

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