Since the 17th century, physical theories have been expressed in the language of mathematical equations. This introduction to quantum theory uses that language to enable the reader to comprehend the notoriously non-intuitive ideas of quantum physics.
The mathematical knowledge needed for using this book comes from standard undergraduate mathematics courses and is described in detail in the section Prerequisites. This text is especially aimed at advanced undergraduate and graduate students of mathematics, computer science, engineering and chemistry among other disciplines, provided they have the math background even though lacking preparation in physics. In fact, no previous formal study of physics is assumed.
Author(s): Stephen Bruce Sontz
Edition: 1
Publisher: Springer
Year: 2020
Language: English
Tags: Introduction Quantum Theory
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
