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

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SUMMARY

The discussion focuses on calculating the variance of spin components S_x, S_y, and S_z for a quantum state |Psi> = a|up> + b|down>. The primary challenge arises in evaluating the expression Delta S = Sqrt[ - ^2], particularly in simplifying the second term, which contains numerous non-canceling terms involving a*, a, b*, and b. Participants are encouraged to share their approaches to calculating to facilitate problem-solving.

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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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