**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 **Failed to parse (syntax error): {\displaystyle a1 }**
in the series of measurement is:

- = Probability of obtaining after obtaining + Probability of obtaining after obtaining