The Born Rule -- Axiom or Result?

Abstract

The Born rule is part of the collapse axiom in the standard version of quantum theory, as presented by standard textbooks on the subject. We show here that its signature quadratic dependence follows from a single additional physical assumption beyond the other axioms - namely, that the probability of a particular measurement outcome (the state $\phi_k$, say) is independent of the choice of observable to be measured, so long as one of its eigenstates corresponds to that outcome. We call this assumption ``observable independence.'' As a consequence, the Born rule cannot be completely eliminated from the list of axioms, but it can, in principle, be reduced to a more physical statement. 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.