[Paper Review] Quantum Indeterminism and First Passage Random Walks in Hilbert Space

Fariel Shafee|arXiv (Cornell University)|Feb 18, 2005

Quantum Mechanics and Applications14 references3 citations

TL;DR

This paper proposes a novel model for quantum measurement using first passage random walks in Hilbert space to dynamically explain wave function collapse and stochastic outcome selection. By coupling the quantum system to a mesoscopic detector and leveraging environmental degrees of freedom, the model induces coherent amplification of one eigenstate through constrained random walks, while other superposed states are dissipated, preserving quantum weights and decoherence without postulating modified dynamics or explicit collapse mechanisms.

ABSTRACT

We propose a new model for a measurement of a characteristic of a microscopic quantum state by a large system that selects stochastically the different eigenstates with appropriate quantum weights. Unlike previous works which formulate a modified Schrödinger equation or an explicit modified Hamiltonian, or more complicated mechanisms for reduction and decoherence to introduce transition to classical stochasticity, we propose the novel use of couplings to the environment, and random walks in the product Hilbert space of the combined system, with first passage stopping rules, which seem intuitively simple, as quantum weights and related stochasticity is a commonality that must be preserved under the widest range of applications, independent of the measured quantity and the specific properties of the measuring device.

Motivation & Objective

• To provide a physically intuitive and dynamic explanation for quantum measurement and wave function collapse, avoiding ad hoc modifications to the Schrödinger equation or Hamiltonian.
• To resolve the puzzle of how superposed quantum states reduce to a single definite outcome in a classical measurement, preserving quantum probabilities.
• To model the transition from mixed quantum states to pure eigenstates via stochastic processes rooted in Hilbert space dynamics.
• To explain decoherence and state reduction simultaneously through local couplings and information dissipation in a hierarchical system of quantum and mesoscopic components.
• To offer a framework where the choice of measurement basis emerges from detector-environment structure rather than being externally imposed.

Proposed method

• The model introduces a preparation stage where the quantum system and detector form a transient, entangled superposed state before the random walk begins.
• It employs first passage random walks in the product Hilbert space of the combined system, with absorbing boundaries corresponding to eigenstates.
• Couplings between the quantum system and mesoscopic detector components drive a stochastic walk toward one eigenstate, with amplification via internal couplings.
• Non-selected states are dissipated into the environment through random phase cancellation and energy leakage, losing coherence and macroscopic expression.
• The diffusion coefficient D in the random walk equations governs the average time to collapse, depending on system-device coupling strength.
• The model avoids explicit decoherence mechanisms by letting decoherence emerge passively from the loss of coherence in non-amplified states.

Experimental results

Research questions

• RQ1 How can quantum indeterminism and the selection of a single eigenstate be explained without modifying the Schrödinger equation or introducing a collapse postulate?
• RQ2 What dynamical mechanism enables the transition from a superposition to a definite measurement outcome while preserving Born rule probabilities?
• RQ3 How do environmental degrees of freedom and local couplings lead to the irreversible loss of coherence for non-selected states?
• RQ4 In what way does the choice of measurement basis emerge from the detector-environment system rather than being externally defined?
• RQ5 Can the time scale of collapse be derived from first passage statistics in a Hilbert space random walk framework?

Key findings

• The model successfully reproduces the Born rule probabilities by ensuring that the probability of first passage to a given eigenstate matches |a|² and |b|² for superposed amplitudes a and b.
• The average time to collapse is calculable via first passage equations and depends on the diffusion coefficient D, which reflects the strength of system-device coupling.
• Decoherence and state reduction occur simultaneously and dynamically: the non-selected states are dissipated into the environment and lose coherence through phase cancellation and energy leakage.
• The choice of basis is determined by the detector’s internal structure, which preferentially couples to certain quantum states, making the axis of measurement intrinsic to the apparatus.
• The model avoids the need for pre-existing orthogonal macroscopic states in the environment, resolving concerns about the sudden creation of such states during measurement.
• The framework naturally accommodates entangled systems by treating them collectively in Hilbert space, consistent with nonlocal correlations in quantum mechanics.