Let there be two operators \(\hat{A}\) and \(\hat{B}\) that represent their respective observables \(A\) and \(B\).

\(\hat{A}\) has two eigenvalues \({a}_{1}\) and \({a}_{2}\), each corresponding to respective normalized eigenstates:

\[{\psi}_{1} = \frac{1}{5}(3{\phi}_{1} + 4{\phi}_{2})\]

\[{\psi}_{2} = \frac{1}{5}(4{\phi}_{1} - 3{\phi}_{2}).\]

\(\hat{B}\) also has two eigenvalues \({b}_{1}\) and \({b}_{2}\), which correspond respectively to normalized eigenstates \({\phi}_{1}\) and \({\phi}_{2}\).

You make an initial measurement of \(A\), recording a value of \(a_1\). You then measure \(B\), then \(A\) again. What is the probability that you record \(a_1\) again?

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