As said, the mathematical tools of quantum mechanics are probability algorithms. They serve a single purpose, which is to assign probabilities to the possible outcomes of measurements, and they are governed by two overarching rules.
Suppose that you want to calculate the probability of a particular outcome of a measurement M2 (performed at the time t2), given the outcome of a measurement M1 (performed at an earlier time t1). Here is what you have to do:
Choose any sequence of measurements that may be made in the meantime.
Refer to a possible sequence of outcomes as an alternative.
Assign to each alternative a complex number and refer to it as its amplitude.
Apply either of these rules:
- Rule A: If the intermediate measurements are made (or if it is possible to find out what their outcomes would have been if they had been made), first square the magnitudes of the amplitudes of the alternatives and then add the results.
- Rule B: If the intermediate measurements are not made (and if it is impossible to find out what their outcomes would have been if they had been made), first add the amplitudes of the alternatives and then square the magnitude of the result.
To get the idea, let’s apply these rules to some experimental situations.