Foundations of Quantum Mechanics: (Theoretical and Mathematical Physics)

In this second volume on the Foundations of Quantum Mechanics we shall show how it is possible, using the methodology presented in Volume I, to deduce some of the most important applications of quantum mechanics. These deductions are concerned with the structures of the micro systems rather than the technical details of the construction of preparation and registration devices. Accordingly, the only new axioms (relative to Volume I) which are introduced are concerned with the relationship between ensemble operators W, effect operators F, and certain construction principles of the preparation and registration devices. The applications described here are concerned with the measurement of atomic and molecular structure and of collision experiments. An additional and essential step towards a theoretical description of the preparation and registration procedures is carried out in Chapter XVII. Here we demonstrate how microscopic collision processes (that is, processes which can be described by quantum mechanics) can be used to obtain novel preparation and registration procedures if we take for granted the knowledge of only a few macroscopic preparation and registration procedures. By clever use of collision processes we are often able to obtain very precise results for the operators Wand F which describe the total procedures from a very imprecise knowledge of the macroscopic parts of the preparation and regis- tration processes. In this regard experimental physicists have done brilliant work. In this sense Chapter XVII represents a general theoretical foundation for the procedures used by experimental physicists.

Table of Contents:
(Volume II).- IX Representation of Hilbert Spaces by Function Spaces.- 1 Maximal Decision Observables.- 2 Representation of ? as ?2(Sp(A), ?) where Sp(A) is the Spectrum of a Scale Observable A.- 3 Improper Scalar and Vector Functions Defined on Sp(A).- 4 Transformation of One Representation into Another.- 5 Position and Momentum Representation.- 6 Degenerate Spectra.- X Equations of Motion.- 1 The Heisenberg Picture.- 2 The Schrodinger Picture.- 3 The Interaction Picture.- 4 Time Reversal Transformations.- XI The Spectrum of One-Electron Systems.- 1 The Effect of the Emission of a Photon.- 2 Ensembles Consisting of Bound States.- 3 The Spectrum of Hydrogen-like Atoms.- 4 The Eigenfunctions for the Discrete Spectrum.- 5 The Continuous Spectrum.- 6 Perturbation Theory.- 7 Perturbation Computations and Symmetry.- 8 The Spectrum of Alkali Atoms.- 9 Electron Spin.- 10 Addition of Angular Momentum.- 11 Fine Structure of Hydrogen and Alkali Metals.- XII Spectrum of Two-Electron Systems.- 1 The Hilbert Space and the Hamiltonian Operator for the Internal Motion of Atoms with n Electrons.- 2 The Spectrum of Two-Electron Atoms.- 3 Ritz Variational Principle.- 4 The Fine Structure of the Helium Spectrum.- XIII Selection Rules and the Intensity of Spectral Lines.- 1 Intensity of Spectral Lines.- 2 Representation Theory and Matrix Elements.- 3 Selection Rules for One-Electron Spectra.- 4 Selection Rules for the Helium Spectrum.- XIV Spectra of Many-Electron Systems.- 1 Energy Terms in the Absence of Spin.- 2 Fine Structure Splitting of Spectral Lines.- 3 Structure Principles.- 4 The Periodic System of the Elements.- 5 Selection and Intensity Rules.- 6 Zeeman Effect.- 7 f Electron Problems and the Symmetric Group.- 8 The Characters for the Representations of Sf and Un.- 9 Perturbation Computations.- XV Molecular Spectra and the Chemical Bond.- 1 The Hamiltonian Operator for a Molecule.- 2 The Form of the Eigenfunctions.- 3 The Ionized Hydrogen Molecule.- 4 Structure Principles for Molecular Energy Levels.- 5 Formation of a Molecule from Two Atoms.- 6 The Hydrogen Molecule.- 7 The Chemical Bond.- 8 Spectra of Diatomic Molecules.- 9 The Effect of Electron Spin on Molecular Energy Levels.- XVI Scattering Theory.- 1 General Properties of Ensembles Used in Scattering Experiments.- 2 General Properties of Effects Used in Scattering Experiments.- 3 Separation of Center of Mass Motion.- 4 Wave Operators and the Scattering Operator.- 4.1 Definition of the Wave Operators.- 4.2 Some General Properties of Wave Operators.- 4.3 Wave Operators and the Spectral Representation of the Hamiltonian Operators.- 4.4 The S Operator.- 4.5 A Sufficient Condition for the Existence of Normal Wave Operators.- 4.6 The Existence of Complete Wave Operators.- 4.7 Stationary Scattering Theory.- 4.8 Scattering of a Pair of Identical Elementary Systems.- 4.9 Multiple-Channel Scattering Theory.- 5 Examples of Wave Operators and Scattering Operators.- 5.1 Scattering of an Elementary System of Spin $$ \frac{1}{2} $$ by an Elementary System of Spin 0.- 5.2 The Born Approximation.- 5.3 Scattering of an Electron by a Hydrogen Atom.- 6 Examples of Registrations in Scattering Experiments.- 6.1 The Effect of the "Impact"(Volume II).- IX Representation of Hilbert Spaces by Function Spaces.- 1 Maximal Decision Observables.- 2 Representation of ? as ?2(Sp(A), ?) where Sp(A) is the Spectrum of a Scale Observable A.- 3 Improper Scalar and Vector Functions Defined on Sp(A).- 4 Transformation of One Representation into Another.- 5 Position and Momentum Representation.- 6 Degenerate Spectra.- X Equations of Motion.- 1 The Heisenberg Picture.- 2 The Schrodinger Picture.- 3 The Interaction Picture.- 4 Time Reversal Transformations.- XI The Spectrum of One-Electron Systems.- 1 The Effect of the Emission of a Photon.- 2 Ensembles Consisting of Bound States.- 3 The Spectrum of Hydrogen-like Atoms.- 4 The Eigenfunctions for the Discrete Spectrum.- 5 The Continuous Spectrum.- 6 Perturbation Theory.- 7 Perturbation Computations and Symmetry.- 8 The Spectrum of Alkali Atoms.- 9 Electron Spin.- 10 Addition of Angular Momentum.- 11 Fine Structure of Hydrogen and Alkali Metals.- XII Spectrum of Two-Electron Systems.- 1 The Hilbert Space and the Hamiltonian Operator for the Internal Motion of Atoms with n Electrons.- 2 The Spectrum of Two-Electron Atoms.- 3 Ritz Variational Principle.- 4 The Fine Structure of the Helium Spectrum.- XIII Selection Rules and the Intensity of Spectral Lines.- 1 Intensity of Spectral Lines.- 2 Representation Theory and Matrix Elements.- 3 Selection Rules for One-Electron Spectra.- 4 Selection Rules for the Helium Spectrum.- XIV Spectra of Many-Electron Systems.- 1 Energy Terms in the Absence of Spin.- 2 Fine Structure Splitting of Spectral Lines.- 3 Structure Principles.- 4 The Periodic System of the Elements.- 5 Selection and Intensity Rules.- 6 Zeeman Effect.- 7 f Electron Problems and the Symmetric Group.- 8 The Characters for the Representations of Sf and Un.- 9 Perturbation Computations.- XV Molecular Spectra and the Chemical Bond.- 1 The Hamiltonian Operator for a Molecule.- 2 The Form of the Eigenfunctions.- 3 The Ionized Hydrogen Molecule.- 4 Structure Principles for Molecular Energy Levels.- 5 Formation of a Molecule from Two Atoms.- 6 The Hydrogen Molecule.- 7 The Chemical Bond.- 8 Spectra of Diatomic Molecules.- 9 The Effect of Electron Spin on Molecular Energy Levels.- XVI Scattering Theory.- 1 General Properties of Ensembles Used in Scattering Experiments.- 2 General Properties of Effects Used in Scattering Experiments.- 3 Separation of Center of Mass Motion.- 4 Wave Operators and the Scattering Operator.- 4.1 Definition of the Wave Operators.- 4.2 Some General Properties of Wave Operators.- 4.3 Wave Operators and the Spectral Representation of the Hamiltonian Operators.- 4.4 The S Operator.- 4.5 A Sufficient Condition for the Existence of Normal Wave Operators.- 4.6 The Existence of Complete Wave Operators.- 4.7 Stationary Scattering Theory.- 4.8 Scattering of a Pair of Identical Elementary Systems.- 4.9 Multiple-Channel Scattering Theory.- 5 Examples of Wave Operators and Scattering Operators.- 5.1 Scattering of an Elementary System of Spin $$ \frac{1}{2} $$ by an Elementary System of Spin 0.- 5.2 The Born Approximation.- 5.3 Scattering of an Electron by a Hydrogen Atom.- 6 Examples of Registrations in Scattering Experiments.- 6.1 The Effect of the "Impact" of a Microsystem on a Surface.- 6.2 Counting Microsystems Scattered into a Solid Angle.- 6.3 The Scattering Cross Section.- 7 Survey of Other Problems in Scattering Theory.- XVII The Measurement Process and the Preparation Process.- 1 The Problem of Consistency.- 2 Measurement Scattering Processes.- 2.1 Measurement with a Microscope.- 2.2 Measurement Scattering Morphisms.- 2.3 Properties of Measurement Scattering Morphisms.- 3 Measurement Transformations.- 3.1 Measurement Transformation Morphisms.- 3.2 Properties of Measurement Transformation Morphisms.- 4 Transpreparations.- 4.1 Reduction of a Preparation Procedure by Means of a Registration Procedure.- 4.2 Transpreparation by Means of Scattering.- 4.3 Collapse of Wave Packets?.- 4.4 The Einstein-Podolski-Rosen Paradox.- 5 Measurements of the First Kind.- 6 The Physical Importance of Scattering Processes Used for Registration and Preparation.- 6.1 Sequences of Measurement Scatterings and Measurement Transformations.- 6.2 Physical Importance of Measurement Scattering and Measurement Transformations.- 6.3 Chains of Transpreparations.- 6.4 The Importance of Transpreparators for the Preparation Process.- 6.5 Unstable States.- 7 Complex Preparation and Registration Processes.- XVIII Quantum Mechanics, Macrophysics and Physical World Views.- 1 Universality of Quantum Mechanics?.- 2 Macroscopic Systems.- 3 Compatibility of the Measurement Process with Preparation and Registration Procedures.- 4 "Point in Time" of Measurement in Quantum Mechanics?.- 5 Relationships Between Different Theories and Quantum Mechanics.- 6 Quantum Mechanics and Cosmology.- 7 Quantum Mechanics and Physical World Views.- Appendix V Groups and Their Representations.- 1 Groups.- 2 Cosets and Invariant Subgroups.- 3 Isomorphisms and Homomorphisms.- 4 Isomorphism Theorem.- 5 Direct Products.- 6 Representations of Groups.- 7 The Irreducible Representations of a Finite Group.- 8 Orthogonality Relations for the Elements of Irreducible Representation Matrices.- 9 Representations of the Symmetric Group.- 10 Topological Groups.- 10.1 The Species of Structure: Topological Group.- 10.2 Uniform Structures of Groups.- 10.3 Lie Groups.- 10.4 Representations of Topological Groups.- 10.5 Group Rings of Compact Lie Groups.- 10.6 Representations in Hilbert Space.- 10.7 Representations up to a Factor.- References.