Quantum Mechanics: A Fundamental Approach

Description

The mathematical formalism of quantum theory in terms of vectors and operators in infinite-dimensional complex vector spaces is very abstract. The definitions of many mathematical quantities used do not seem to have an intuitive meaning, which makes it difficult to appreciate the mathematical formalism and understand quantum mechanics. This book provides intuition and motivation to the mathematics of quantum theory, introducing the mathematics in its simplest and familiar form, for instance, with three-dimensional vectors and operators, which can be readily understood. Feeling confident about and comfortable with the mathematics used helps readers appreciate and understand the concepts and formalism of quantum mechanics. This book is divided into four parts. Part I is a brief review of the general properties of classical and quantum systems. A general discussion of probability theory is also included which aims to help in understanding the probability theories relevant to quantum mechanics. Part II is a detailed study of the mathematics for quantum mechanics. Part III presents quantum mechanics in a series of postulates. Six groups of postulates are presented to describe orthodox quantum systems. Each statement of a postulate is supplemented with a detailed discussion. To make them easier to understand, the postulates for discrete observables are presented before those for continuous observables. Part IV presents several illustrative applications, which include harmonic and isotropic oscillators, charged particle in external magnetic fields and the Aharonov–Bohm effect. For easy reference, definitions, theorems, examples, comments, properties and results are labelled with section numbers. Various symbols and notations are adopted to distinguish different quantities explicitly and to avoid misrepresentation. Self-contained both mathematically and physically, the book is accessible to a wide readership, including astrophysicists, mathematicians and philosophers of science who are interested in the foundations of quantum mechanics. Cover Half Title Title Page Copyright Page Dedication Table of Contents Preface SECTION I: CLASSICAL AND QUANTUM SYSTEMS 1: Structure of Physical Theories 2: Classical Systems 2.1 Discrete Systems 2.1.1 Intuition and Description 2.1.2 Determinism 2.1.3 Unambiguous Objective Reality 2.1.4 Structure of Classical Mechanics 2.2 Continuous Systems 3: Probability Theory for Discrete Variables 3.1 Physical Concept of Probability 3.2 Sets, Mappings and Functions 3.2.1 Set Operations 3.2.2 Mappings and Functions 3.3 Discrete Sample Spaces and Events 3.4 Probability Mass Functions and Measures 3.5 Expectation Values, Variances and Uncertainties 3.6 Probability Distribution Functions Exercises and Problems 4: Probability Theory for Continuous Variables 4.1 Borel Sets, Borel Functions and Measures 4.1.1 Borel Sets of the Reals and Borel Functions 4.1.2 Lebesgue and Lebesgue–Stieltjes Measures 4.2 Riemann and Lebesgue Integrals 4.2.1 Riemann Integrals 4.2.2 Lebesgue Integrals 4.2.3 Riemann–Stieltjes Integrals 4.2.4 Lebesgue–Stieltjes Integrals 4.3 Probability Distributions and Measures 4.3.1 Probability Distribution Functions 4.3.2 Distribution Functions and Measures 4.3.3 Expectation Values and Uncertainties Exercises and Problems 5: Quantum Mechanical Systems 5.1 Experimental Measurability 5.2 Observable Values 5.3 Observables, States and Probabilistic Behaviour 5.4 Structure of Quantum Mechanics Exercises and Problems SECTION II: MATHEMATICAL FRAMEWORK 6: Three-Dimensional Real Vectors 6.1 Properties 1: Algebraic Properties 6.1.1 Addition 6.1.2 Scalar Multiplication 6.2 Properties 2: Dimensions and Bases 6.2.1 Linear Dependence and Independence 6.2.2 Dimensions, Bases and Complete Sets 6.3 Properties 3: Scalar Product 6.3.1 Scalar Product 6.3.2 Orthonormality 6.3.3 Orthonormal Bases and Complete Sets 6.3.4 Pythagoras Theorem 6.3.5 Gram-Schmidt Orthogonalisation 6.3.6 Inequalities on Scalar Product and Norm 6.4 Properties 4: Scalar Product and Projections 6.4.1 Projection onto i 6.4.2 Projection onto Arbitrary Unit Vector e 6.4.3 Planes and Projection onto Planes 6.4.4 Subspaces and Projection onto Subspaces Exercises and Problems 7: Matrices and their Relations with Vectors 7.1 Basic Definitions 7.2 Square Matrices 7.3 Transpose and Adjoint of a Matrix 7.3.1 The Transpose of a Matrix 7.3.2 The Adjoint of a Square Matrix 7.4 The Inverse of a Matrix 7.5 Matrix Representation of Vectors in E3 7.5.1 Scalar Product of Column Matrices 7.5.2 Column Matrices and Column Vectors 7.6 Eigenvalue Problem for Matrices 7.7 Special Matrices 7.7.1 Introduction 7.7.2 Orthogonal Matrices 7.7.3 Unitary Matrices 7.7.4 Selfadjoint Matrices 7.7.5 Projection Matrices 7.7.6 Spectral Decomposition of Selfadjoint Matrices Exercises and Problems 8: Operations on Vectors in E3 8.1 Functionals on 𝔼 3 and the Riesz Theorem 8.2 Linear Operators 8.2.1 The Concept 8.2.2 General Definitions 8.2.2.1 Domain and range 8.2.2.2 Norm of operators 8.2.2.3 Algebraic operations 8.2.2.4 Commutators and anticommutators 8.2.2.5 Inverse operators 8.2.2.6 Adjoint operators 8.2.2.7 Quadratic form 8.2.3 Matrix Representation of Operators 8.2.4 Eigenvalue Problem for Operators Exercises and Problems 9: Special Operators on E3 9.1 Scalar Multiplication Operators 9.2 Rotations and Orthogonal Operators 9.2.1 Rotations of Vectors 9.2.2 Orthogonal Operators 9.3 Projections and Projection Operators 9.3.1 Projectors onto Unit Vectors 9.3.2 Projectors onto Subspaces 9.3.3 Eigenvalue Problem 9.4 Selfadjoint Operators 9.4.1 Concept and Definition 9.4.2 Properties and Relations with Projectors 9.4.3 Eigenvalue Problem and Quadratic Form 9.4.4 Eigensubspaces and Eigenprojectors 9.4.5 Spectral Theorem of Selfadjoint Operators on E3 9.4.6 Functions of Selfadjoint Operators Exercises and Problems 10: Probability, Selfadjoint Operators, Unit Vectors and the Need for Complexness 10.1 Generating Probability Distributions on E3 10.2 A Model Probability Theory Based on E3 10.2.1 The Model 10.2.2 A Need for Generalisation 10.3 Need for Complexness Exercises and Problems 11: Complex Vectors 11.1 Complex Numbers 11.2 Complexification of the Vector Space E3 11.2.1 Complex Vectors 11.2.2 Scalar Product and Orthonormal Bases Exercises and Problems 12: N-Dimensional Complex Vector Spaces 12.1 Introductory Remarks 12.2 Definitions 12.3 Examples 12.3.1 Complex Column Matrices as Vectors 12.3.2 Complex N × N Matrices as Vectors 12.3.3 Complex-Valued Functions as Vectors 12.4 Isomorphism between Spaces 12.5 Concluding Remarks Exercises and Problems 13: Operators on N-Dimensional Complex Vector Spaces 13.1 Introduction 13.2 Subspaces, Projections and Projectors 13.2.1 Subspaces 13.2.2 Projections and Projectors 13.3 Selfadjoint Operators 13.3.1 Properties 13.3.2 Spectral Theorem 13.3.3 Functions of a Selfadjoint Operator 13.3.4 Commuting Selfadjoint Operators 13.4 Unitary Operators 13.4.1 Definition and Spectral Decomposition 13.4.2 Unitary Transformations 13.4.3 Stone’s Theorem 13.5 Matrix Representation of Operators Exercises and Problems 14: Model Theories Based on Complex Vector Spaces VN 14.1 Model Theories of Spin 14.1.1 Electron Spin 14.1.1.1 On states 14.1.1.2 On observable Sz 14.1.1.3 On probability distributions 14.1.1.4 On observable Sx 14.1.1.5 On observable Sy 14.1.2 Spin-1 Particles 14.2 Generating Probability Distribution Functions on VN Exercises and Problems 15: Spectral Theory in VN in Terms of Stieltjes Integrals 15.1 Spectral Functions and Spectral Measures 15.2 Spectral Measures in Terms of Riemann-Stieltjes Integrals 15.3 Spectral Theorem and Spectrum 15.4 Functions of Commuting Selfadjoint Operators 15.5 Probability Distribution and Expectation Values Exercises and Problems 16: Infinite-Dimensional Complex Vectors and Hilbert Spaces 16.1 Infinite-Dimensional Vector Spaces 16.1.1 The Space l2 16.1.2 Spaces of Complex-Valued Functions 16.1.2.1 Continuous functions 16.1.2.2 Absolutely continuous functions 16.1.2.3 Smooth functions 16.1.2.4 Schwartz functions 16.1.2.5 Smooth functions of compact support 16.1.2.6 Riemann square-integrable functions 16.1.2.7 Lebesgue square-integrable functions 16.1.2.8 Functions on a circle and L2(Ca) 16.1.2.9 Functions on a unit sphere 16.2 Hilbert Spaces 16.2.1 Cauchy Sequences and Separability 16.2.2 Hilbert spaces 16.2.3 Subspaces, Projections and Projectors Exercises and Problems 17: Operators in a Hilbert space H 17.1 Boundedness, Continuity and Closedness 17.2 Multiplication Operators 17.3 Differential Operators and Boundary Conditions 17.3.1 Introduction 17.3.2 Specific operators 17.3.2.1 Differential operators in L2( ⋀ ), ⋀ = [0, L] 17.3.2.2 Differential operators in L2(Ca) 17.3.2.3 Differential operators in L2(Su) 17.3.2.4 Differential operators in L2(IR+) 17.3.2.5 Differential operators in L2(IR) 17.3.2.6 Differential operators in L2(IR2) and L2(IR3) 17.4 Algebraic Operations of Operators 17.5 Invertible Operators 17.6 Extensions and Restrictions of Operators 17.7 Commutation Relations 17.8 Adjoints of Operators 17.9 Reduction of Operators 17.10 Annihilation and Creation Operators Exercises and Problems 18: Bounded Operators on H 18.1 Selfadjoint Operators and Projectors 18.2 Density Operators 18.3 Unitary Operators 18.4 Fourier Transformations in L2(IR) 18.4.1 Notation and Preliminaries 18.4.2 Fourier transform as unitary transform 18.4.2.1 Coordinate space and momentum space 18.4.2.2 Fourier transforms of operators Exercises and Problems 19: Symmetric and Selfadjoint Operators in H 19.1 Definitions and Examples 19.2 Symmetric Differential Operators 19.2.1 In L2( ⋀ ), ⋀ = [0, L] 19.2.2 In L2(IR+) and L2(IR) 19.3 First Order Selfadjoint Differential Operators 19.3.1 In L2( ⋀ ) 19.3.2 In L2(Ca) and L2(Su) 19.3.3 In L2(IR+) 19.3.4 In L2(IR) 19.3.5 In L2(IR2) and L2(IR3) 19.4 Second Order Selfadjoint Differential Operators 19.5 Essentially Selfadjoint Operators Exercises and Problems 20: Spectral Theory of Selfadjoint Operators in H 20.1 Spectral Functions and Spectral Measures 20.2 Spectral Theorem and Spectrum 20.3 Operators with a Discrete Spectrum 20.4 Operators with a Continuous Spectrum 20.4.1 The Position Operator x in L2(IR) 20.4.2 The Momentum Operator p in L2(IR) 20.5 Functions of Selfadjoint Operators 20.5.1 Characteristic Functions 20.5.2 Complex-Valued Functions 20.5.3 Spectral Functions and Spectral Measures 20.6 Complete Set of Commuting Selfadjoint Operators 20.7 Irreducible Sets of Selfadjoint Operators Exercises and Problems 21: Spectral Theory of Unitary Operators on H 21.1 Spectral Decomposition of Unitary Operators 21.2 Stone’s Theorem Exercises and Problems 22: Probability, Selfadjoint Operators and Unit Vectors 22.1 Generating Probability Distributions on H 22.2 Operators with a Discrete Spectrum 22.3 Operators with a Continuous Spectrum 22.3.1 Position Operator in L2(IR) 22.3.2 Momentum Operator in L2(IR) Exercises and Problems 23: Physics of Unitary Transformations Exercises and Problems 24: Direct Sums and Tensor Products of Hilbert Spaces and Operators 24.1 Direct Sums of Hilbert Spaces and Operators 24.1.1 Direct Sums of Hilbert Spaces 24.1.2 Direct Sums of Operators 24.2 Tensor Products of Hilbert Spaces and Operators 24.2.1 Definitions 24.2.2 Examples Exercises and Problems SECTION III: QUANTUM FORMALISM 25: Pure States 25.1 Postulate (PS) Exercises and Problems 26: Observables and Their Values 26.1 Postulate (OV) 26.2 On Propositions 26.2.1 Definition 26.2.2 Observables and Their Propositions 26.2.2.1 Discrete observables 26.2.2.2 Continuous observables 26.2.3 Propositions and States Exercises and Problems 27: Canonical Quantisation 27.1 Hamiltonian Formulation of Classical Mechanics 27.1.1 Conservative Mechanical Systems 27.1.2 Charged Particle in Magnetic Field 27.1.3 Poisson Bracket and Structure of Classical Observables 27.2 Postulate (CQ) 27.3 Canonical Quantisation Scheme 27.4 Quantisation of Orbital Angular Momentum 27.5 Quantisation of Hamiltonians 27.6 Charged Particles in Magnetic Field 27.7 Manipulations of Commutation Relations 27.8 A Particle in Circular Motion 27.9 Characterisation of Annihilation and Creation Operators 27.10 Limitations of Canonical Quantisation Scheme 27.10.1 Problems due to Non-Commuting Factors 27.10.2 Problems due to Constraints on Position 27.10.3 Problems with Commutation Relations Exercises and Problems 28: States, Observables and Probability Distributions 28.1 Postulate (PDDO) on Discrete Observables 28.2 Postulate (PDCO) on Continuous Obsevables and Postulate (PD) 28.3 Position and Momentum 28.3.1 The Position Observable 28.3.2 The Momentum Observable 28.3.3 Uncertainty Relations 28.4 Compatible and Incompatible Observables 28.4.1 Discrete Observables 28.4.2 Continuous Observables Exercises and Problems 29: Time Evolution 29.1 The Schrödinger Picture 29.1.1 Schrödinger Equation 29.1.2 Unitary Evolution 29.2 The Heisenberg Picture 29.2.1 Unitary Evolution 29.2.2 Heisenberg Equation of Motion 29.3 Equivalence of the Schrödinger and the Heisenberg Pictures 29.4 Interacting Systems and the Interaction Picture 29.4.1 Derivation 29.4.2 Equations of Motion Exercises and Problems 30: State after Measurement 30.1 Discrete Observables 30.1.1 Postulate (PPDO) 30.1.2 Nondegenerate Eigenvalues 30.1.3 Degenerate Eigenvalues and Propositions 30.2 Continuous Observables 30.2.1 Postulate (PPCO) and Postulate (PP) 30.2.2 Position Measurement 30.3 Complete Sets of Compatible Observables 30.3.1 Discrete Observables 30.3.2 Continuous Observables Exercises and Problems 31: Pure and Mixed States 31.1 Classical Mixtures of States 31.2 Quantum Mixtures of States 31.3 Changes of States 31.4 Postulates Incorporating Mixed States 31.5 Correlations and the Superposition Principle Exercises and Problems 32: Superselection Rules 32.1 Superselection Rules 32.2 Supersectors and Direct Sum Decomposition 32.3 An Example Exercises and Problems 33: Many-Particle Systems 33.1 Identity and Distinguishability 33.2 Distinguishable Particles 33.3 Identical Particles 33.3.1 Bosons and Fermions 33.3.2 The Pauli Exclusion Principle 33.4 Indefinite Number of Particles Exercises and Problems 34: Conceptual Issues 34.1 Understanding Quantum Behaviour 34.2 Particle-Wave Duality 34.3 Classical and Quantum Divide 34.4 Schrödinger’s Cat Paradox 34.5 De Broglie Paradox and Non-Locality 34.6 Entanglement and EPR Paradox 34.7 Quantum Measurement 34.7.1 The Measurement Problem 34.7.2 Measuring Devices and Processes 34.7.2.1 Introduction 34.7.2.2 Mathematical formulation 34.7.2.3 Concluding remarks 34.8 Quantum Theory, Relativity and Photons Exercise and Problem SECTION IV: ILLUSTRATIVE APPLICATIONS 35: Harmonic and Isotropic Oscillators 35.1 Harmonic Oscillators 35.2 Energy Eigenvalues and Eigenvectors 35.2.1 Derivation 35.2.2 Mathematical Discussion 35.3 Time Evolution 35.3.1 In the Schrödinger Picture 35.3.2 In the Heisenberg Picture 35.3.3 In the Interaction Picture 35.4 Isotropic Oscillators Exercises and Problems 36: Angular Momenta 36.1 Orbital Angular Momentum 36.1.1 Orbital Angular Momentum Operators 36.1.2 Eigenvalues and Eigenfunctions 36.2 Annihilation and Creation Operators 36.2.1 Introduction 36.2.2 Notation 36.2.3 Summary of Results 36.3 Spin Angular Momentum 36.3.1 Introduction 36.3.2 Two-Component Wave Functions 36.3.3 Spin Functions and Spin Vectors 36.3.4 Spin Operator Sz 36.3.5 Spin Operators Sx and Sy 36.3.6 Matrix Representation 36.3.6.1 Two-component column vectors 36.3.6.2 Spin operators as 2 × 2 matrices Exercises and Problems 37: Particles in Static Magnetic Field 37.1 Static Magnetic Fields 37.1.1 Vector Potentials 37.1.2 Uniform Field in Cylindrical Coordinates 37.1.3 Field Confined in a Cylindrical Region 37.2 Charged Quantum Particles in Uniform Field 37.3 Magnetic Moment and Magnetic Energy 37.3.1 For Circular and Orbital Motion 37.3.2 For Spin Motion 37.4 Pauli–Schrödinger Equation 37.5 The Simple Zeeman Effect 37.5.1 The Hydrogen Atom 37.5.2 Hydrogen Atom in Magnetic Field 37.6 Aharonov–Bohm Effect 37.6.1 Circular Motion 37.6.2 The Aharonov–Bohm Effect Exercises and Problems References Notation Index