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An atom is prepared in the angular momentum state $$C\left(\begin{array}{c}1 \\ 2\end{array}\right)$$Here $C$ is a constant. This has benn written in the $S_z$ basis.

a)Find C

b)Work out $\langle S_y\rangle$ using matrices

c)Calculate the variance $\sigma_{S_{y}}^2$

I've calculated C to be $\frac{1}{\sqrt5}$ by normalization, and my $\langle S_y\rangle$ comes out to be zero while the variance is $\frac{\hbar^2}{4}$. Another part of the question asks for variances $\sigma_{S_{x}}^2$ (which I've calculated to be $\frac{9\hbar^2}{100}$) and $\sigma_{S_{z}}^2$(which I've calculated to be $\frac{4\hbar^2}{25}$). We are then asked whether the results are consistent with the uncertainty principle. Can anyone let me know how to show that the results are consistent?Any help would be appreciated.

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Your constant $C$ is right and your expectation value $<S_y> = 0$ is also right.

The variance is given by $<\sigma_{S_y}^2> = \frac{\hbar^2}{4}$; right!

Very good!

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  • $\begingroup$ Thank you.Can you please take a look at the edited version? $\endgroup$ – Paradox 101 Mar 2 '15 at 19:59

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