Book contents
- Frontmatter
- Contents
- Foreword
- Preface
- 1 Historical perspective
- 2 Present situation, remaining conceptual difficulties
- 3 The theorem of Einstein, Podolsky, and Rosen
- 4 Bell theorem
- 5 More theorems
- 6 Quantum entanglement
- 7 Applications of quantum entanglement
- 8 Quantum measurement
- 9 Experiments: quantum reduction seen in real time
- 10 Various interpretations
- 11 Annex: Basic mathematical tools of quantum mechanics
- Appendix A Mental content of the state vector
- Appendix B Bell inequalities in non-deterministic local theories
- Appendix C An attempt for constructing a “separable” quantum theory (non-deterministic but local)
- Appendix D Maximal probability for a state
- Appendix E The influence of pair selection
- Appendix F Impossibility of superluminal communication
- Appendix G Quantum measurements at different times
- Appendix H Manipulating and preparing additional variables
- Appendix I Correlations in Bohmian theory
- Appendix J Models for spontaneous reduction of the state vector
- Appendix K Consistent families of histories
- References
- Index
Appendix J - Models for spontaneous reduction of the state vector
Published online by Cambridge University Press: 05 September 2012
- Frontmatter
- Contents
- Foreword
- Preface
- 1 Historical perspective
- 2 Present situation, remaining conceptual difficulties
- 3 The theorem of Einstein, Podolsky, and Rosen
- 4 Bell theorem
- 5 More theorems
- 6 Quantum entanglement
- 7 Applications of quantum entanglement
- 8 Quantum measurement
- 9 Experiments: quantum reduction seen in real time
- 10 Various interpretations
- 11 Annex: Basic mathematical tools of quantum mechanics
- Appendix A Mental content of the state vector
- Appendix B Bell inequalities in non-deterministic local theories
- Appendix C An attempt for constructing a “separable” quantum theory (non-deterministic but local)
- Appendix D Maximal probability for a state
- Appendix E The influence of pair selection
- Appendix F Impossibility of superluminal communication
- Appendix G Quantum measurements at different times
- Appendix H Manipulating and preparing additional variables
- Appendix I Correlations in Bohmian theory
- Appendix J Models for spontaneous reduction of the state vector
- Appendix K Consistent families of histories
- References
- Index
Summary
In this appendix, we introduce a few simple models involving modified Schrödinger dynamics with stochasticity, in order to illustrate how such models may lead to an evolution that reproduces the reduction of the state vector during a measurement (emergence of a single eigenvalue during a single realization, with a random value). For the sake of simplicity, weignore the usual Hamiltonian evolution during the time of measurement, assuming for instance that this time is too short for this evolution to be significant; otherwise, it would be necessary to use the interaction representation with respect to the Hamiltonian, which does not change the calculations much, except that this introduces a time dependence of the operators.
Single operator
We consider the measurement of some quantum observable associated with an Hermitian operator A; we look for an equation of evolution containing a state vector reduction process associated with this particular measurement. Since the final eigenvector must vary randomly from one realization to the next, the evolution equation necessarily contains a random component. In our case, it will take the form of a random function of time (as opposed to the GRWtheory where the stochasticity is introduced by the discontinuous “hitting processes”, see §10.8.1.b).
Equation of evolution
We assume that the state vector ∣ψ(t)〈 evolves according to:
where w(t) is a real random function of time. In order to simplify the model as much as possible, we may discretize time into small finite intervals Δt, during which we assume that w(t) remains constant; moreover, we may also assume that the possible values of w(t) belong to a finite discrete ensemble w1, w2, …, wN.
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- Information
- Do We Really Understand Quantum Mechanics? , pp. 357 - 361Publisher: Cambridge University PressPrint publication year: 2012