This book investigates two possibilities for describing classical-mechanical physical systems along with their Hamiltonian dynamics in the framework of quantum mechanics.The first possibility consists in exploiting the geometrical properties of the set of quantum pure states of "microsystems" and of the Lie groups characterizing the specific classical system. The second approach is to consider quantal systems of a large number of interacting subsystems – i.e. macrosystems, so as to study the quantum mechanics of an infinite number of degrees of freedom and to look for the behaviour of their collective variables. The final chapter contains some solvable models of “quantum measurement" describing dynamical transitions from "microsystems" to "macrosystems".
Author(s): Pavel Bóna
Publisher: Springer
Year: 2020
Language: English
Pages: 244
Preface
Contents
1 Introduction
1.1 Motivation and Summary
1.2 Quantum Mechanics
1.3 Classical Hamiltonian Mechanics
1.4 Quantum Theory of Large Systems
2 Geometry of the State Space of Quantum Mechanics
2.1 Manifold Structure of P(mathcalH)
2.2 Symplectic Structure
2.3 Quantum Mechanics as a Classical Hamiltonian Field Theory
3 Classical Mechanical Projections of QM
3.1 Orbits of Lie Group Actions on P(calH)
3.2 Classical Phase Spaces from the Quantal State Space
3.3 Classical Mechanical Projections of Quantal Dynamics
4 Examples of Classical Mechanical Projections
4.1 The Heisenberg Group (CCR)
4.2 Extension of CCR by a Quadratic Generator
4.3 Notes on Other Examples
5 Macroscopic Limits
5.1 Multiple Systems
5.2 Generalized Macroscopic Limits
6 Mathematical Structure of QM Mean-Field Theories
6.1 General Considerations
6.2 Spin Systems with Polynomial Local Hamiltonians QN
6.3 Time Evolution in Generalized Mean-Field Theories
6.4 Equilibrium States
6.5 An Example: The B.C.S. Model of Superconductivity
7 Some Models of ``Quantum Measurement''
7.1 Introductory Notes
7.2 On `Philosophy' of ``Models''
7.3 Quantum Domino
7.4 Particle Detection—A ``Nonideal'' Measurement
7.5 The X-Y Chain as a Measuring Device
7.6 Radiating Finite Spin Chain
7.7 On the ``Measurement Problem'' in QM
Appendix References
Index
