An Introduction to Quantum Mechanics: From Facts to Formalism

Description

The core content of even the most intricate intellectual edifices is often a simple fact or idea. So is it with quantum mechanics; the entire mathematical fabric of the formal description of quantum mechanics stems essentially from the fact that quantum probabilities interfere (i.e., from the superposition principle). This book is dedicated to substantiating this claim. In the process, the book tries to demonstrate how the factual content of quantum mechanics can be transcribed in the formal language of vector spaces and linear transformations by disentangling the empirical content from the usual formal description. More importantly, it tries to bring out what this transcription achieves. The book uses a pedagogic strategy which reverse engineers the postulates of quantum mechanics to device a schematic outline of the empirical content of quantum mechanics from which the postulates are then reconstructed step by step. This strategy is adopted to avoid the disconcerting details of actual experiments (however simplified) to spare the beginner of issues that lurk in the fragile foundations of the subject. In the Copenhagen interpretation of quantum mechanics, the key idea is measurement. But measurement carries an entirely different meaning from the connotation that the term carries elsewhere in physics. This book strives to underline this as strongly as possible. The book is intended as an undergraduate text for a first course in quantum mechanics. Since the book is self contained, it may also be used by enthusiastic outsiders interested to get a glimpse of the core content of the subject. Features: Demonstrates why linear algebra is the appropriate mathematical language for quantum mechanics. Uses a reconstructive approach to motivate the postulates of quantum mechanics. Builds the vocabulary of quantum mechanics by showing how the entire body of its conceptual ingredients can be constructed from the single notion of quantum measurement. Cover Half Title Title Page Copyright Page Dedication Contents Preface Prologue 1. Theoretical Framework - A Working Definition What Is a Theory? Formal Description of a Theory Abstraction and Generalization Philosophy and Fundamental Theorem Theory vs Theoretical Framework 2. The Empirical Basis of Quantum Mechanics Professor Funnyman’s Ghosts An Experiment with Bullets Basic Concepts in Quantum Mechanics Notation Laws of Quantum Mechanics The Quantum State Redefined 3. States as Vectors Vectors in Cn Towards a Formal Description Vector Spaces and Inner Product Spaces Postulates of Quantum Mechanics - Version 1 4. Observables as Operators Linear Operators Describing Observables by Linear Operators Postulates of Quantum Mechanics - Version 2 Quantization Expectation Values 5. Imprecise Measurements and Degeneracy Precise and Imprecise Measurements Postulates of Quantum Mechanics - Version 3 Complete and Incomplete Measurements Commuting Operators Compatibility and Commutativity 6. Time Evolution Unitary Operators Time Evolution of Quantum States Postulates of Quantum Mechanics - Version 4 The Programme in Quantum Mechanics Formal Solution of Time Evolution 7. Continuous Spectra Description using Function Spaces Modified Postulates Quantization of Systems with Classical Analogues 8. Three Archetypal Eigenvalue Problems Harmonic Oscillator Angular Momentum Hydrogen Atom 9. Composite Systems The Meaning of a Composite System Tensor Product Space Description of Composite Systems Interaction between Subsystems Measurement on a Subsystem Examples of Composite Systems Identical Particles A. Probability Preliminary Background Frequentist Interpretation Alternative Interpretations Formal Description B. Linear Algebra Vector Space Inner Product Space Linear Operators Eigen Systems Direct Sum of Subspaces Tensor Product Space Bibliography Index