Equalities for Paulimatrix averages

[Funzies]

I've come across this a few times:

$$<S_x^2> = <S_y^2>=<S_z^2>=\hbar^2/4$$

But I can't seem to understand why this holds, as <S_x>, <S_y> and <S_z> sometimes give really strange values for a random spinor, with no correlation at all.

Can anyone explain this to me? Thanks!

 

[arkajad]

If $S_x=\frac{\hbar}{2}\sigma_x$, then $S_x^2=\hbar^2/4$, and the same for other components, thus your formula holds even without expectation values.

You should not mix $<S_x>^2$ with $<S_x^2>$. These are different things. If you jump to the left (-1) and to the right (+1) then the average can be zero, but the average of the squares will be >0.