# What information does the mean value of a matrix give us? [closed]

Last time I found that question and even I spent a lot of time I didn't find any answer.

By mean value of matrix, I mean mean value of some operator. E.g. Pauli matrix. Can someone explain what the mean value of $$\sigma_1 =\{\frac{1}{2}, \frac{1}{2}\}$$ tells us?

## closed as unclear what you're asking by Norbert Schuch, heather, Wolpertinger, sammy gerbil, John RennieAug 14 '16 at 19:26

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• What about e.g. the Wikipedia article about the mean is unclear to you? Please be more specific about what you want to know. – ACuriousMind Aug 10 '16 at 10:31
• I wanted to know if can I deduce something about operators based on its mean value. – bawq Aug 10 '16 at 11:40

In particular let $\hat{A}$ be a self-adjoint operator in quantum mechanics whose set of eigenvectors $|a\rangle$ such that $\hat{A}|a\rangle = a\,|a\rangle$ is a basis for the entire Hilbert space (i. e. an observable) and let $|\psi\rangle$ be a state upon which one wants to calculate the expectation value of $\hat{A}$. Given the above one can expand $$|\psi\rangle = \sum_{a}c_a|a\rangle$$ and thus $$\langle\psi|\hat{A}|\psi\rangle = \sum_{a'}\sum_a c_{a'}^* c_a \langle a'|\hat{A}|a\rangle = \sum_{a'}\sum_a c_{a'}^* c_a a' \delta_{a'a} = \sum_a a\,|c_a|^2$$ where $|c_a|^2$ is the probability that a measurement of the observable $\hat{A}$ gives back the value $a$ onto the state at hand.