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arXiv:quant-ph/9912015 · Journal of Mathematical Physics

Derivation of the wave function collapse in the context of Nelson’s stochastic mechanics

Michele Pavon

Published Nov 1, 1999• 14 citations
arXivHTMLPDF

Abstract

The von Neumann collapse of the quantum mechanical wave function after a position measurement is derived by a purely probabilistic mechanism in the context of Nelson’s stochastic mechanics.
DOI
10.1063/1.533046

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Elaborating on M. Pavon, J.Math. Phys. 40 (1999), 5565-5577, we develop a simplified version of a variational principle within Nelson stochastic mechanics that produces the von Neumann wave packet reduction after a position measurement. This stochastic control problem parallels, with a different kinematics, the problem of the Schroedinger bridge. This gives a profound meaning to what was observed by Schroedinger in 1931 concerning Schroedinger bridges: "Merkwuerdige Analogien zur Quantenmechanik, die mir sehr des Hindenkens wert erscheinen".
In quantum mechanics and quantum information science, state evolution occurs via two fundamental mechanisms: deterministic unitary gate transformations and projective (von Neumann) measurements. This paper formalizes the three mathematical and physical pillars that differentiate projective measurements from unitary operations: stochastic behavior governed by the Born rule, non-linearity introduced via wave function collapse normalization, and operational irreversibility. It provides rigorous mathematical derivations for state update rules, explores the role of eigenvalues and observables, and demonstrates the non-linearity of projective measurements through concrete mathematical examples.