# Find the constant in the following matrix

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.

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!