The Origin of the Born Rule from Spacetime Averaging

TL;DR

This paper proposes that the Born rule in quantum mechanics can be derived as an approximation resulting from spacetime averaging of energy fluctuations, supported by analysis of a particle in a potential well.

Contribution

It introduces a novel approach linking the Born rule to spacetime averaging of energy, providing a potential foundational explanation for quantum probabilities.

Findings

Energy expectation from spacetime averaging closely matches Born rule predictions

Differences between averaged energy and Born rule are within experimental error

Supports the idea that the Born rule is an emergent, approximate phenomenon

Abstract

The Born rule postulates that the probability of measurement in quantum mechanics is related to the squared modulus of the wave function $\psi$. We rearrange the equation for energy eigenfunctions to define the energy as the real part of $\hat{E}\psi/\psi$. For an eigenstate, this definition gives a constant energy eigenvalue. For a general wave function, the energy fluctuates in space and time. We consider a particle in a one-dimensional square well potential in a superposition of two states and average the energy over space and time. We show that, for most cases, such an energy expectation value differs by only a few percent from that calculated using the Born rule. This difference is consistent with experimental tests of the expectation value and suggests that the Born rule may be an approximation of spacetime averaging.