Derivation of the Born rule from the unitarity of quantum evolution

Abstract

In order to make the quantum mechanics a closed theory one has to derive the Born rule from the first principles, like the Schroedinger equation, rather than postulate it. The Born rule was in certain sense derived in several articles, e.g. in [D. Deutsch, Proc. R. Soc. Lond. A455, 3129 (1999)] and [W. H. Zurek, Phys. Rev. Lett. 90, 120404 (2003)]. In this work some arguments of previous authors are simplified and made more "physical". It is shown how to derive the Born rule using the conservation of quantum state norm $\langle\Psi|\Psi\rangle$ that is the unitary evolution property determined by the Schroedinger equation. It is this property that makes the probability equal to the square of the amplitude modulus. We also present arguments in the spirit of the Many-World Interpretation to explain the origin of probabilistic behavior. Simply speaking, the randomness appears as a result of representing the wave function by using a detector of discrete nature that is found only in one state at a time, out of two or more possible states.