A general theory of quantum measurements

TL;DR

The paper proposes a universal energy-eigenvalue framework for quantum measurements that couples a sub-system to its surroundings through the full system Hamiltonian H. It develops existence, completeness, and perturbative methods for solving the resulting eigenproblem, and applies the approach to the hydrogen atom to obtain a new relativistic Hamiltonian and a small but meaningful finite-proton-mass correction to the fine structure. It then links measurement outcomes to photon emission/absorption processes, deriving photon-localization models, transition-rate formulas, and a sub-system selection rule that maximizes the transition rate, with a detailed numerical case in ZnS showing how hundreds of photons can mediate a high-resolution position signal. Overall, the work claims to establish a general, self-consistent mechanism that determines the sub-system, the preferred basis, and the transition rates across quantum-measurement scenarios, potentially addressing foundational questions about when and how measurements occur. The formalism is mathematically explicit, bridging microscopic dynamics with measurable outcomes through energy-based measurement events.

Abstract

A universal energy eigenvalue equation is proposed in this paper. It is proven that the unique set of eigenfunctions or preferred basis exists for any non-isolated sub-system. Applying the new eigenvalue equation to the relative motion of a hydrogen atom together with the derived relativistic Hamiltonian to quantify the impact of finite proton mass to the fine structure, correction to the fine structure is obtained as a result of the entanglement of the relative motion and the center-of-mass motion, which can be used to verify the correctness of the proposed eigenvalue equation. Applying the equation to the measurement of electron double-slit interference, it is analyzed that the photon packets with Lorentzian spectral lineshape, the domain and state density of the sub-system, and the energy of the incident electron together determines the spontaneous emission rate of the incident electron. Photons generated in this process excite electrons from the valence band to the conduction band of the detector. Corresponding to any emitted or absorbed photon, the sub-system is found to be uniquely determined by maximizing the transition rate. This new principle is valid for atoms too. Closed-form expressions are obtained for the transition rates and example numerical results show good correlation between calculation and the position measurement experiment. The discovered common mechanisms that determine the sub-systems, the preferred bases, and transition rates form the foundation of a new, general, and consistent theory of quantum measurement.