How can we find the variance of S_x, S_y, and S_z?

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To find the variance of S_x, S_y, and S_z for the state |Psi> = a|up> + b|down>, the calculation involves determining the expectation values <S^2_x>, <S^2_y>, and <S^2_z>. The challenge arises in simplifying the second term of the variance formula, which includes numerous terms involving a*, a, b*, and b that do not cancel out. Participants are encouraged to share their approaches to calculating <S_x> and other related expectation values. Clarifying these calculations could help resolve the issues with the variance computation. Understanding these steps is crucial for accurately determining the variances in quantum mechanics.
man@SUT
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Given |Psi> = a|up>+b|down>, in principle it should not be so difficult but when I calculate

Delta S=Sqrt[<S^2_x,S^2_y, or S^2_z>-<S_x,S_y, or S_z>^2]

the second term gives the problem. Lots of many terms a*, a, b*, b which is not canceled involve.

Whoever knows the way to get rid of this problem, please let me know.
 
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Can you show what you've tried so far?
For example, how did you go about finding <S_x>?
 

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