Problem

Let there be two operators and that represent their respective observables and . has two eigenvalues and , each corresponding to respective normalized eigenstates:   also has two eigenvalues and , which correspond respectively to normalized eigenstates and .

You make an initial measurement of , recording a value of . You then measure , then again. What is the probability that you record again?

Solution

If is measured and ​ is obtained, the state of the system after the measurement is . Then, is now measured with outcomes ​ of probability and ​ of probability .

Now is measured again, we can express and as:  The probability of obtaining ​ if we knew the outcome of the measurement of as and the system is in state is​ . The probability of obtaining if we knew the outcome of the measurement of as ​ and the system is in state is ​. The total probability of obtaining $\displaystyle a1​$ in the series of measurement is: = Probability of obtaining after obtaining + Probability of obtaining after obtaining    