Quantum Mechanics: A Paradigms Approach

✍ Scribed by David H. McIntyre

• Publisher: Cambridge University Press
• Year: 2022
• Tongue: English
• Leaves: 598
• Edition: 1
• Category: Library

✦ Synopsis

This popular undergraduate quantum mechanics textbook is now available in a more affordable printing from Cambridge University Press. Unlike many other books on quantum mechanics, this text begins by examining experimental quantum phenomena such as the Stern-Gerlach experiment and spin measurements, using them as the basis for developing the theoretical principles of quantum mechanics. Dirac notation is developed from the outset, offering an intuitive and powerful mathematical toolset for calculation, and familiarizing students with this important notational system. This non-traditional approach is designed to deepen students' conceptual understanding of the subject, and has been extensively class tested. Suitable for undergraduate physics students, worked examples are included throughout and end of chapter problems act to reinforce and extend important concepts. Additional activities for students are provided online, including interactive simulations of Stern-Gerlach experiments, and a fully worked solutions manual is available for instructors.

✦ Table of Contents

Cover
Front Endmatter
Half-title
Title page
Copyright information
Brief Contents
Contents
Preface
Prologue
1 Stern-Gerlach Experiments
1.1 Stern-Gerlach Experiment
1.1.1 Experiment
1
1.1.2 Experiment 2
1.1.3 Experiment 3
1.1.4 Experiment 4
1.2 Quantum State Vectors
1.2.1 Analysis of Experiment 1
1.2.2 Analysis of Experiment 2
1.2.3 Superposition States
1.3 Matrix Notation
1.4 General Quantum Systems
1.5 Postulates
Summary
Problems
Resources
Activities
Further Reading
2 Operators and Measurement
2.1 Operators, Eigenvalues, and Eigenvectors
2.1.1 Matrix Representation of Operators
2.1.2 Diagonalization of Operators
2.2 New Operators
2.2.1 Spin Component in a General Direction
2.2.2 Hermitian Operators
2.2.3 Projection Operators
2.2.4 Analysis of Experiments 3 and 4
2.3 Measurement
2.4 Commuting Observables
2.5 Uncertainty Principle
2.6 S[sup(2)]
Operator
2.7 Spin-1 System
2.8 General Quantum Systems
Summary
Problems
Resources
Activities
3 Schrödinger Time Evolution
3.1 Schrödinger Equation
3.2 Spin Precession
3.2.1 Magnetic Field in the z-Direction
3.2.2 Magnetic Field in a General Direction
3.3 Neutrino Oscillations
3.4 Time-Dependent Hamiltonians
3.4.1 Magnetic Resonance
Summary
Problems
Resources
Activities
Further Reading
4 Quantum Spookiness
4.1 Einstein-Podolsky-Rosen Paradox
4.2 Schrödinger Cat Paradox
Problems
Resources
Further Reading
5 Quantized Energies: Particle in a Box
5.1 Spectroscopy
5.2 Energy Eigenvalue Equation
5.3 The Wave Function
5.4 Infinite Square Well
5.5 Finite Square Well
5.6 Compare and Contrast
5.6.1 Wave Function Curvature
5.6.2 Nodes
5.6.3 Barrier Penetration
5.6.4 Inversion Symmetry and Parity
5.6.5 Orthonormality
5.6.6 Completeness
5.7 Superposition States and Time Dependence
5.8 Modern Application: Quantum Wells and Dots
5.9 Asymmetric Square Well: Sneak Peek at Perturbations
5.10 Fitting Energy Eigenstates by Eye or by Computer
5.10.1 Qualitative (Eyeball) Solutions
5.10.2 Numerical Solutions
5.10.3 General Potential Wells
Summary
Problems
Resources
Activities
Further Reading
6 Unbound States
6.1 Free Particle Eigenstates
6.1.1 Energy Eigenstates
6.1.2 Momentum Eigenstates
6.2 Wave Packets
6.2.1 Discrete Superposition
6.2.2 Continuous Superposition
6.3 Uncertainty Principle
6.3.1 Energy Estimation
6.4 Unbound States and Scattering
6.5 Tunneling Through Barriers
6.6 Atom Interferometry
Summary
Problems
Resources
Activities
Further Reading
7 Angular Momentum
7.1 Separating Center-of-Mass and Relative Motion
7.2 Energy Eigenvalue Equation in Spherical Coordinates
7.3 Angular Momentum
7.3.1 Classical Angular Momentum
7.3.2 Quantum Mechanical Angular Momentum
7.4 Separation of Variables: Spherical Coordinates
7.5 Motion of a Particle on a Ring
7.5.1 Azimuthal Solution
7.5.2 Quantum Measurements on a Particle Confined to a Ring
7.5.3 Superposition States
7.6 Motion on a Sphere
7.6.1 Series Solution of Legendre’s Equation
7.6.2 Associated Legendre Functions
7.6.3 Energy Eigenvalues of a Rigid Rotor
7.6.4 Spherical Harmonics
7.6.5 Visualization of Spherical Harmonics
Summary
Problems
Resources
Activities
8 Hydrogen Atom
8.1 The Radial Eigenvalue Equation
8.2 Solving the Radial Equation
8.2.1 Asymptotic Solutions to the Radial Equation
8.2.2 Series Solution to the Radial Equation
8.3 Hydrogen Energies and Spectrum
8.4 The Radial Wave Functions
8.5 The Full Hydrogen Wave Functions
8.6 Superposition States
Summary
Problems
Resources
Activities
Further Reading
9 Harmonic Oscillator
9.1 Classical Harmonic Oscillator
9.2 Quantum Mechanical Harmonic Oscillator
9.3 Wave Functions
9.4 Dirac Notation
9.5 Matrix Representations
9.6 Momentum Space Wave Function
9.7 The Uncertainty Principle
9.8 Time Dependence
9.9 Molecular Vibrations
Summary
Problems
Resources
Activities
Further Reading
10 Perturbation Theory
10.1 Spin-1/2 Example
10.2 General Two-Level Example
10.3 Nondegenerate Perturbation Theory
10.3.1 First-Order Energy Correction
10.3.2 First-Order State Vector Correction
10.4 Second-Order Nondegenerate Perturbation Theory
10.5 Degenerate Perturbation Theory
10.6 More Examples
10.6.1 Harmonic Oscillator
10.6.2 Stark Effect in Hydrogen
Summary
Problems
11 Hyperfine Structure and the Addition of Angular Momenta
11.1 Hyperfine Interaction
11.2 Angular Momentum Review
11.3 Angular Momentum Ladder Operators
11.4 Diagonalization of the Hyperfine Perturbation
11.5 The Coupled Basis
11.6 Addition of Generalized Angular Momenta
11.7 Angular Momentum in Atoms and Spectroscopic Notation
Summary
Problems
Resources
Activities
Further Reading
12 Perturbation of Hydrogen
12.1 Hydrogen Energy Levels
12.2 Fine Structure of Hydrogen
12.2.1 Relativistic Correction
12.2.2 Spin-Orbit Coupling
12.3 Zeeman Effect
12.3.1 Zeeman Effect without Spin
12.3.2 Zeeman Effect with Spin
12.3.2.1 Weak magnetic field
12.3.2.2 Strong magnetic field
12.3.2.3 Intermediate magnetic field
12.3.3 Zeeman Perturbation of the 1s Hyperfine Structure
Summary
Problems
Resources
Activities
Further Reading
13 Identical Particles
13.1 Two Spin-1/2 Particles
13.2 Two Identical Particles in One Dimension
13.2.1 Two-Particle Ground State
13.2.2 Two-Particle Excited State
13.2.3 Visualization of States
13.2.4 Exchange Interaction
13.2.5 Consequences of the Symmetrization Postulate
13.3 Interacting Particles
13.4 Example: The Helium Atom
13.4.1 Helium Ground State
13.4.2 Helium Excited States
13.5 The Periodic Table
13.6 Example: The Hydrogen Molecule
13.6.1 The Hydrogen Molecular Ion H[sub(2)sup(+)]
13.6.2 The Hydrogen Molecule H[sub(2)]
Summary
Problems
Resources
Further Reading
14 Time-Dependent Perturbation Theory
14.1 Transition Probability
14.2 Harmonic Perturbation
14.3 Electric Dipole Interaction
14.3.1 Einstein Model: Broadband Excitation
14.3.2 Laser Excitation
14.4 Selection Rules
Summary
Problems
Resources
Further Reading
15 Periodic Systems
15.1 The Energy Eigenvalues and Eigenstates of a Periodic Chain of Wells
15.1.1 A Two-Well Chain
15.1.2 N-Well Chain
15.2 Boundary Conditions and the Allowed Values of k
15.3 The Brillouin Zones
15.4 Multiple Bands from Multiple Atomic Levels
15.5 Bloch’s Theorem and the Molecular States
15.6 Molecular Wave Functions—a Gallery
15.7 The Density of States
15.8 Calculation of the Model Parameters
15.8.1 LCAO Summary
15.9 The Kronig-Penney Model
15.10 Practical Applications: Metals, Insulators, and Semiconductors
15.11 Effective Mass
15.12 Direct and Indirect Band Gaps
15.13 New Directions—Low-Dimensional Carbon
Summary
Problems
Resources
Activities
Further Reading
16 Modern Applications of Quantum Mechanics
16.1 Manipulating Atoms with Quantum Mechanical Forces
16.1.1 Magnetic Trapping
16.1.2 Laser Cooling
16.2 Quantum Information Processing
16.2.1 Quantum Bits—Qubits
16.2.2 Quantum Gates
16.2.3 Quantum Teleportation
Summary
Problems
Resources
Further Reading
Appendix A: Probability
Appendix B: Complex Numbers
Appendix C: Matrices
Appendix D: Waves and Fourier Analysis
Appendix E: Separation of Variables
Appendix F: Integrals
Appendix G: Physical Constants
Index
Back Endmatter