Deriving Born Rule Probabilities via the Method of Arbitrary Functions

The goal of this paper is to apply the collection of mathematical tools known as the method of arbitrary functions to analyze how the Born rule probabilities arise from quantum dynamics. We argue that in a toy model of quantum measurement the Born rule probabilities can be derived from the unitary Schrödinger dynamics when certain dynamical parameters are treated as random variables with initial probability distributions. Specifically, we study the perturbed double well model, in which the perturbation is treated as a random variable, and we show that for arbitrary initial distributions within a certain class, the dynamics yield the Born rule probabilities in the joint limits given by long times and small values of Planck’s constant (the classical limit). Our results establish the Born rule as a type of universal limiting behavior that is independent of the precise initial dynamical parameters. We suggest that this provides a proof of concept for an explanatory schema for the Born rule, relying only on Schrödinger dynamics, which could generate a full explanation when supplemented with further interpretive details.