The Born rule is part of the collapse axioms in the standard version of quantum theory, as presented by most textbooks on the subject. We show here that its signature quadratic dependence on the initial wavefunction's projection onto the measured outcome state follows from a single additional assumption beyond the other axioms. We give two examples of such an assumption, with a separate derivation for each, and we discuss their relationship with existing derivations. Our presentation is suitable for advanced undergraduates or graduate students who have taken a standard course in quantum theory. It does not depend on any particular interpretation of the theory.

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Formally, the system is described by a ray in H—a vector without a defined magnitude or phase. We fix the magnitude by normalizing to unity, ψ | ψ = 1.
3.
The state update rule is implicit in the quantum measurement formalism of von Neumann,4 although it is an idealization. Strictly, it is valid only if the measurement is ideal,5 that is, minimally disturbing and repeatable on the same system. As a common counterexample, measurement sometimes destroys the system, as in a photon polarization measurement.
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10.
Note that, unlike the example with J = 1, the squares of the Si are proportional to the identity, S z 2 = S x 2 = S y 2 = 2 / 4, and do not provide new observables.
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19.
See ‘Assertion’ on p. 4 of Ref. 18.
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[PubMed]
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