# Quantum Measuring Sequence

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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