Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Applying the CSL Stratonovich Equation to the Free Particle Undergoing Collapse in Position

Philip Pearle

Published Jan 30, 2024• 0 citations
Publisher

Abstract

Abstract This chapter addresses the problem of applying the continuous spontaneous localization (CSL) Stratonovich equation to the free particle undergoing collapse in position, by finding the solution in Gaussian form. This solution was previously obtained in Chapter 10 by using the linear Schrödinger equation/probability rule formulation of CSL (which was used in the first seven chapters of this book). This chapter shows how the same result is obtained from the stochastic differential equation formulation of CSL.

DOI
10.1093/oso/9780198901372.003.0017

Related papers

Applying the CSL Stratonovich Equation to the Harmonic Oscillator Undergoing Collapse in Position

Philip Pearle

Abstract This chapter addresses the problem of applying the continuous spontaneous localization (CSL) Stratonovich equation to a harmonic oscillator particle undergoing collapse in position, by finding the solution in Gaussian form. This solution was not previously obtained. It is quite complicated, and provides a demonstration of the power of the SDE approach to CSL.

•Jan 30, 2024•0 citations

CSL Expressed as a Schrödinger Stochastic DE

Philip Pearle

Abstract Chapter 16 uses the results of Chapter 15 to show how to construct the CSL Stratonovich and Ito Schrödinger equations for the normalized state vector. It is emphasized that there is a non-linear dependence on the state vector. It then goes on to show how the (linear) density matrix evolution equation follows is constructed from that. The chapter ends with a discussion regarding specialization to non-relativistic CSL.

•Jan 30, 2024•0 citations

Continuous Spontaneous Localization (CSL) Theory

Philip Pearle

Abstract This chapter introduces the two equations that completely define CSL, for the simplest case, that is, a state vector in a two-dimensional Hilbert space. One equation is the modified evolution equation for the state vector which depends upon a Brownian motion function of time, B(t), a Hermitian operator A called the “collapse-generating operator,” and a collapse rate parameter λ. The state vector evolution is linear, but not unitary, so its norm changes with time. The second equation is called the probability rule. It says the probability a particular B(t) occurs in Nature is proportional to the squared norm of the state vector, that is, high norm state vectors are most likely to occur. The chapter shows how an initial state vector that is a superposition of the eigenstates of A evolves to one eigenstate, with the Born probability, as predicted by standard quantum theory. Finally, it discusses the density matrix description of the ensemble of evolutions under all possible B(t).

•Jan 30, 2024•0 citations

Introduction

Philip Pearle

Abstract This chapter introduces the problem (that the wave function of standard quantum theory (SQT) cannot correspond to reality in nature) and its resolution (change Schrodinger’s equation). The chapter begins by citing Schrodinger’s argument as to how the collapse postulate of SQT leads to the cited problem. It then presents a resolution embodied in the continuous spontaneous localization (CSL) theory‘s altered Schrodinger equation. The chapter shows how the “Gambler’s Ruin” game, where two gamblers bet on a coin toss until one wins all the money, is a precise and intuitively appealing analogy to the needed collapse behavior for a superposition of two states evolving under random influence to one or another of those states. Following this, the chapter ends by presenting the structure of the book. The first seven chapters show how CSL works, discusses experimentally testable and already tested predictions, and covers interpretive features of CSL. The purpose of the remaining eleven chapters, called “Supplementary,” is summarized as well.

•Jan 30, 2024•1 citation

BOUNDS ON SPONTANEOUS COLLAPSE MODEL OF QUANTUM MECHANICS FROM FORMATION OF CMBR AND STANDARD COSMOLOGY

Suratna Das, Kinjalk Lochan, Angelo Bassi

Continuous Spontaneous Localization (CSL) model of Quantum Mechanics modifies Schrödinger equation by adding non-linear stochastic terms due to which the total energy of a system increases with a constant rate which is proportional to the collapse rate $λ$. Thus applying CSL model to cosmological scenarios can change the thermal behaviour of the particles during evolution and we will put constraints on $λ$ by considering several cosmological scenarios.

arXiv:1302.0171•The Ninth Marcel Grossmann Meeting•Jan 29, 2015•1 citation

Dissipative Continuous Spontaneous Localization (CSL) model

Andrea Smirne, Angelo Bassi

Collapse models explain the absence of quantum superpositions at the macroscopic scale, while giving practically the same predictions as quantum mechanics for microscopic systems. The Continuous Spontaneous Localization (CSL) model is the most refined and studied among collapse models. A well-known problem of this model, and of similar ones, is the steady and unlimited increase of the energy induced by the collapse noise. Here we present the dissipative version of the CSL model, which guarantees a finite energy during the entire system's evolution, thus making a crucial step toward a realistic energy-conserving collapse model. This is achieved by introducing a non-linear stochastic modification of the Schrödinger equation, which represents the action of a dissipative finite-temperature collapse noise. The possibility to introduce dissipation within collapse models in a consistent way will have relevant impact on the experimental investigations of the CSL model, and therefore also on the testability of the quantum superposition principle.

arXiv:1408.6446•Scientific Reports•Aug 5, 2015•61 citations
Abstract This chapter addresses the problem of applying the continuous spontaneous localization (CSL) Stratonovich equation to the free particle undergoing collapse in position, by finding the solution in Gaussian form. This solution was previously obtained in Chapter 10 by using the linear Schrödinger equation/probability rule formulation of CSL (which was used in the first seven chapters of this book). This chapter shows how the same result is obtained from the stochastic differential equation formulation of CSL.
Abstract This chapter introduces the two equations that completely define CSL, for the simplest case, that is, a state vector in a two-dimensional Hilbert space. One equation is the modified evolution equation for the state vector which depends upon a Brownian motion function of time, B(t), a Hermitian operator A called the “collapse-generating operator,” and a collapse rate parameter λ. The state vector evolution is linear, but not unitary, so its norm changes with time. The second equation is called the probability rule. It says the probability a particular B(t) occurs in Nature is proportional to the squared norm of the state vector, that is, high norm state vectors are most likely to occur. The chapter shows how an initial state vector that is a superposition of the eigenstates of A evolves to one eigenstate, with the Born probability, as predicted by standard quantum theory. Finally, it discusses the density matrix description of the ensemble of evolutions under all possible B(t).
Abstract This chapter introduces the problem (that the wave function of standard quantum theory (SQT) cannot correspond to reality in nature) and its resolution (change Schrodinger’s equation). The chapter begins by citing Schrodinger’s argument as to how the collapse postulate of SQT leads to the cited problem. It then presents a resolution embodied in the continuous spontaneous localization (CSL) theory‘s altered Schrodinger equation. The chapter shows how the “Gambler’s Ruin” game, where two gamblers bet on a coin toss until one wins all the money, is a precise and intuitively appealing analogy to the needed collapse behavior for a superposition of two states evolving under random influence to one or another of those states. Following this, the chapter ends by presenting the structure of the book. The first seven chapters show how CSL works, discusses experimentally testable and already tested predictions, and covers interpretive features of CSL. The purpose of the remaining eleven chapters, called “Supplementary,” is summarized as well.
Continuous Spontaneous Localization (CSL) model of Quantum Mechanics modifies Schrödinger equation by adding non-linear stochastic terms due to which the total energy of a system increases with a constant rate which is proportional to the collapse rate λλλ. Thus applying CSL model to cosmological scenarios can change the thermal behaviour of the particles during evolution and we will put constraints on λλλ by considering several cosmological scenarios.
Collapse models explain the absence of quantum superpositions at the macroscopic scale, while giving practically the same predictions as quantum mechanics for microscopic systems. The Continuous Spontaneous Localization (CSL) model is the most refined and studied among collapse models. A well-known problem of this model, and of similar ones, is the steady and unlimited increase of the energy induced by the collapse noise. Here we present the dissipative version of the CSL model, which guarantees a finite energy during the entire system's evolution, thus making a crucial step toward a realistic energy-conserving collapse model. This is achieved by introducing a non-linear stochastic modification of the Schrödinger equation, which represents the action of a dissipative finite-temperature collapse noise. The possibility to introduce dissipation within collapse models in a consistent way will have relevant impact on the experimental investigations of the CSL model, and therefore also on the testability of the quantum superposition principle.