[Paper Review] Collapse models: analysis of the free particle dynamics

Angelo Bassi|arXiv (Cornell University)|Oct 27, 2004

Quantum Mechanics and Applications4 citations

TL;DR

This paper analyzes a stochastic collapse model for a free quantum particle, showing that the dynamics induces spatial localization of the wavefunction via a stochastic Schrödinger equation. With a mass-dependent collapse rate, the model reproduces standard quantum mechanics for microscopic systems and classical Newtonian mechanics for macroscopic systems, demonstrating a consistent quantum-to-classical transition.

ABSTRACT

We study a model of spontaneous wavefunction collapse for a free quantum particle. We analyze in detail the time evolution of the single-Gaussian solution and the double-Gaussian solution, showing how the reduction mechanism induces the localization of the wavefunction in space; we also study the asymptotic behavior of the general solution. With an appropriate choice for the parameter $λ$ which sets the strength of the collapse mechanism, we prove that: i) the effects of the reducing terms on the dynamics of microscopic systems are negligible, the physical predictions of the model being very close to those of standard quantum mechanics; ii) at the macroscopic scale, the model reproduces classical mechanics: the wavefunction of the center of mass of a macro-object behaves, with high accuracy, like a point moving in space according to Newton's laws.

Motivation & Objective

• To analyze the time evolution of wavefunction solutions in a stochastic collapse model for a free particle.
• To investigate how the collapse mechanism induces spatial localization of the wavefunction.
• To demonstrate that the model reproduces standard quantum mechanics at the microscopic scale and classical mechanics at the macroscopic scale.
• To establish the asymptotic behavior of general solutions and their convergence to stationary Gaussian states.
• To validate the model's consistency with experimental observations by showing negligible deviations from standard quantum mechanics for microscopic systems.

Proposed method

• Uses a stochastic Schrödinger equation with non-linear, non-Markovian terms to model spontaneous wavefunction collapse.
• Applies a mass-proportional collapse rate parameter λ = (m/m₀)λ₀, with λ₀ ≈ 10⁻² m⁻²s⁻¹, based on the GRW model.
• Analyzes single-Gaussian and double-Gaussian wavefunction solutions to study localization dynamics.
• Employs linearization techniques via a change of measure to derive stochastic differential equations for the wavefunction parameters.
• Derives and solves Fokker-Planck-type equations for the moments of the wavefunction to study asymptotic behavior.
• Uses Itô calculus and stochastic processes to analyze the time evolution of expectation values and variances.

Experimental results

Research questions

• RQ1 How does the collapse model induce spatial localization of a free particle’s wavefunction?
• RQ2 What is the time evolution of single-Gaussian and double-Gaussian wavefunction solutions under the stochastic dynamics?
• RQ3 How do the model’s predictions compare with standard quantum mechanics for microscopic systems?
• RQ4 To what extent does the wavefunction of a macroscopic system behave like a classical point particle?
• RQ5 What is the asymptotic behavior of general wavefunction solutions in the model?

Key findings

• The single-Gaussian solution evolves to a stationary state with finite, non-zero spread, indicating effective localization.
• The double-Gaussian solution rapidly collapses into a single peak, demonstrating suppression of macroscopic superpositions.
• For microscopic systems, the collapse effects are negligible: predictions are indistinguishable from standard quantum mechanics.
• For macroscopic systems, the center-of-mass wavefunction behaves like a classical point particle following Newton’s laws with high accuracy.
• The stochastic average of the position and momentum follows classical equations of motion, confirming classical behavior in the macroscopic limit.
• The model’s predictions are robust under parameter variations, with the collapse rate λ₀ ≈ 10⁻² m⁻²s⁻¹ ensuring consistency with experimental bounds.