I have only taken a basic quantum mechanics course (this book, so you know where I'm coming from), but I've been wondering about something.
If we set up a quantum system in a known state and take a measurement of two incompatible observables, we will get two real numbers. If we repeat this experiment multiple times, then we will get two lists of real numbers (each list corresponding to the measurements of one of the observables). Quantum mechanics allows us to predict the average and standard deviation of these numbers, but it does not allow us to predict the individual numbers.
If I understand correctly, this is a fundamental limit of the theory. The data is essentially random. Is it correct to say that most scientists believe that no theory will ever allow the prediction of these individual numbers? Why do they think that?
And secondly, is there any other property of those numbers that quantum mechanics predicts that I am missing (other than mean and standard deviation)?
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