Derivation of the Born rule from the unitarity of quantum evolution

TL;DR

This paper derives the Born rule from the unitarity of quantum evolution, showing that probability arises naturally from the conservation of the wave function's norm, with insights from the Many-Worlds Interpretation.

Contribution

It provides a simplified, more physical derivation of the Born rule directly from the Schrödinger equation's unitarity, avoiding additional postulates.

Findings

Born rule derived from unitarity of quantum evolution

Probability equals the square of the wave function amplitude

Wave function collapse explained via detector discreteness

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.