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There exists a physical operator F whose eigenvalues are a or b. Normalised eigenstate is A and B for each. The average value of F was f when measured from a certain system. Describe the eigenstate of this system in terms of A and B.

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I am just a physics student so I could be wrong, just trying to help you: using Dirac bracket notation you can write your system state $ |psi \rangle $ as a superposition of the eigenstates, with complex coeficients α and β, $$| \psi \rangle =\alpha |a\rangle + \beta |b\rangle, $$ and you can write the expected value as: $$\langle \psi | \hat F | \psi \rangle =f.$$ Knowing that $\hat F |a\rangle =a|a\rangle,\ \hat F |b\rangle =b|b\rangle$ I think that you should be able to finish.

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