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

[man@SUT]

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.

 

[Galileo]

Can you show what you've tried so far?
For example, how did you go about finding <S_x>?