Can anyone explain it in simple words? I tried to read about the Born rule on Wikipedia but it seems it is just describing the rule. Is there any proof that is suitable for person with no solid math background?
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$\begingroup$ Same (nearly same) question (s ) has (have) been asked now & then at Phys.SE: Born rule and unitary evolution ; Born's Rule, What is the Reason?; Born Interpretation of Wave Function. So, don't think there is a need of same question here. $\endgroup$– user5764Dec 3, 2015 at 12:20
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First, I would not call the Born rule a law as it is done in the Wikipedia article linked by OP in the question. It is, of course, the question of the definitions, but I would call the Born rule rather a postulate. A law in physics is a principle (usually universal) deduced from observations, while a postulate is an assumption, a statement that we can not prove or derive but which is assumed to be right and tested to be consistent with experimental observations.
The Born rule is a postulate, thus, there is no any proof of it; it is assumed to be right. The only thing we can do is to justify a postulate: we can make experimentally verifiable predictions on the basis of it and test them. And it happens so that so far the experimental outcomes are always in full agreement with the predictions done using the Born rule. Thus, we know it provides a correct description of physical systems.
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$\begingroup$ The Born postulate, as you call it, is also widely called the "Born interpretation". Is one preferred over the other? $\endgroup$– YodaDec 3, 2015 at 11:54
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2$\begingroup$ @Goat'sMilk, I wouldn't call it an interpretation, since the word "interpretation" has a well-defined meaning in quantum theory: a set of statements attempting to explain the philosophical (primarily, ontological & epistemological) meaning of quantum theory beyond the rigorous physical theory. Born rule, on the other hand, is a part of the quantum theory itself, it is one of the fundamental principles of it, one of the key components of its mathematical formalism. Thus, I think one would better avoid calling it an interpretation. $\endgroup$– WildcatDec 3, 2015 at 12:16
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$\begingroup$ @Wildcat There are different interpretations of QM, and they can have different nuances about what the wave-function mean and how is it tied to physical reality (as the wave function itself is not physical). In that sense, the "interpretation" is a much better term than "postulation". $\endgroup$– GregOct 13, 2019 at 16:52