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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?

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

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ What about e.g. the Wikipedia article about the mean is unclear to you? Please be more specific about what you want to know. $\endgroup$ – ACuriousMind Aug 10 '16 at 10:31
  • $\begingroup$ I wanted to know if can I deduce something about operators based on its mean value. $\endgroup$ – bawq Aug 10 '16 at 11:40
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The expectation value of an operator in physics has the same meaning as the expectation value in measure theory: it gives the sum of the possible outcomes weighted with the corresponding probabilities.

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.

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