Prove that spin in arbitary direction is hbar/2

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SUMMARY

The discussion centers on the calculation of spin in an arbitrary direction, specifically addressing the eigenvalues of the spin matrix denoted as S_{z'}. Participants emphasize the importance of focusing on the eigenvalues of the constructed spin matrix rather than those of S_{z'}^2. Clarification is provided regarding the distinction between total spin angular momentum, represented as S^2 = S_x^2 + S_y^2 + S_z^2, and the component of spin angular momentum associated with an arbitrary direction.

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the question is in the jpeg file.
 

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How come you don't just solve the eigenvalues for the matrix that you have denoted by [tex]S_{z'}[/tex]? We are not interested in the eigenvalues of [tex]S_{z'}^2[/tex], but just on the eigenvalues of the spin matrix you have constructed in the fourth line of your attachment. I think you have got a bit messed up with the eigenvalues related to the total spin angular momentum [tex]S^2=S_x^2+S_y^2+S_z^2[/tex] and the component of spin angular momentum related to an arbitrary direction.

I hope this made any sense to you, it's my first post!
 
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