An Introductory Path to Quantum Theory: Using Mathematics to Understand the Ideas of Physics

Preface
Prerequisites
Contents
Notations and Abbreviations
1 Introduction to this Path
	1.1 A New Physics
	1.2 Notes
2 Viewpoint
	2.1 A Bit of Motivation
	2.2 Notes
3 Neither Particle nor Wave
	3.1 Basics of Quantum Theory
	3.2 Black Bodies
	3.3 Dimensions and Units
	3.4 Notes
4 Schrödinger\'s Equation
	4.1  Some Classical Physics
	4.2  Introducing Schrödinger\'s Equation
	4.3  The Eigenvalue Problem
	4.4  Notes
5 Operators and Canonical Quantization
	5.1 First Quantization
	5.2 The Quantum Plane
	5.3 Notes
6 The Harmonic Oscillator
	6.1 The Classical Case
	6.2 The Quantum Case
	6.3 Notes
7 Interpreting ψ: Mathematics
	7.1 A Cauchy Problem
	7.2 Notes
8 Interpreting ψ: Physics
	8.1 The Entrance of Probability
	8.2 Expected Value
	8.3 Copenhagen and Philosophy
	8.4 Trajectories (Optional)
	8.5 Notes
9 The Language of Hilbert Space
	9.1 Facts and Definitions, No Proofs
	9.2 Unitary Operators
	9.3 More Facts without Proofs
	9.4 States Revisited (Optional)
	9.5 The Spectrum (Optional)
	9.6 Densely Defined Operators (Optional)
	9.7 Dirac Notation (Optional)
	9.8 Notes
10 Interpreting ψ: Measurement
	10.1 Some Statements
	10.2 Some Controversies
	10.3 Simultaneous Eigenvectors,  Commuting Operators
	10.4 The Inner Product—Physics Viewpoint
	10.5 Notes
11 The Hydrogen Atom
	11.1 A Long and Winding Road
	11.2 The Role of Symmetry
	11.3 A Geometrical Problem
	11.4 The Radial Problem
	11.5 Spherical Coordinates (Optional)
	11.6 Two-body Problems (Optional)
	11.7 A Moral or Two
	11.8 Notes
12 Angular Momentum
	12.1 Basics
	12.2 Spherical Symmetry
	12.3 Ladder Operators
	12.4 Relation to Laplacian on mathbbR3
	12.5 Notes
13 The Rotation Group SO(3)
	13.1 Basic Definitions
	13.2 Euler\'s Theorem (Optional)
	13.3 One-parameter Subgroups
	13.4 Commutation Relations, so(3)  and All That
	13.5 Notes
14 Spin and SU(2)
	14.1 Basics of SU(2)
	14.2 A Crash Course on Spin
	14.3 The Map p
	14.4 The Representations ρs
	14.5 ρs and Angular Momentum
	14.6 Magnetism
	14.7 Notes
15 Bosons and Fermions
	15.1 Multi-particle Statistics
	15.2 Notes
16 Classical and Quantum Probability
	16.1 Classical Kolmogorov Probability
	16.2 Quantum Probability
	16.3 States as Quantum Events
	16.4 The Case of Spin 1/2
	16.5 Expected Value (Revisited)
	16.6 Dispersion
	16.7 Probability  1
	16.8 A Feature, Not a Bug
	16.9 Spectral Theorem as Diagonalization
	16.10 Two Observables
	16.11 Notes
17 The Heisenberg Picture
	17.1 Kinetics and Dynamics chez Heisenberg
	17.2 Notes
18 Uncertainty (Optional)
	18.1 Moments of a Probability Distribution
	18.2 Incompatible Measurements
	18.3 Example: Harmonic Oscillator
	18.4 Proof of the Uncertainty Inequality
	18.5 Notes
19 Speaking of Quantum Theory (Optional)
	19.1 True vs. False
	19.2 FALSE: Two places at same time
	19.3 FALSE: Any state can be determined
	19.4 Expected Values—One Last Time
	19.5 Tunneling
	19.6 Superposition
	19.7 Quantum Fluctuations
	19.8 Notes
20 Complementarity (Optional)
	20.1  Some Ideas and Comments
	20.2 Notes
21 Axioms (Optional)
	21.1 A List of Axioms
	21.2 A Few Comments
	21.3 Notes
22 And Gravity?
A Measure Theory: A Crash Course
	A.1  Measures
	A.2  Integrals
	A.3  Absolute Integrability
	A.4  The Big Three
	A.5  Counting Measure and Infinite Series
	A.6  Fubini\'s Theorem
	A.7   L1(Ω)
	A.8  Concluding Remarks
	A.9  Notes
Index