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I have a density matrix $\rho$ where $$\rho = \frac{1}{4} \cdot \begin{pmatrix} 2 & 1 & 1\\ 1 & 1 & 0\\ 1 & 0 & 1 \end{pmatrix}$$

and the x component of a spin - 1 particle that is $$J_{x}=\frac{1}{\sqrt{2}} \cdot \begin{pmatrix} 0 & 1 & 0\\ 1 & 0 & 1\\ 0 & 1 & 0 \end{pmatrix}$$

How can I calculate the expectation value of the x component of the spin, $\langle J_{x} \rangle$, given the statistical ensemble described by the density matrix ?

And also, how to compute the standard deviation ?

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This reads like it is some kind of homework assignment, so I won't compute the full answer.

For a density matrix, the expectation value of any operator $\hat{O}$ is given by $\langle \hat{O} \rangle = \text{tr}(\rho\hat{O})$, and the standard deviation of that operator is $\Delta\hat{O} = \sqrt{\langle \hat{O}^2 \rangle - \langle \hat{O}\rangle^2}$

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