Paradigms in Physics: Quantum Mechanics

CONTENTS Chapter 1 Stern-Gerlach Experiments Chapter 2 Operators And Measurement Chapter 3 Schrödinger Time Evolution Chapter 4 Quantum Spookiness Chapter 5 Quantized Energies: Particle in a Box Chapter 6 Unbound States Chapter 7 Angular Momentum Chapter 8 Hydrogen Atom Chapter 9 Harmonic Oscillator Chapter 10 Perturbation Theory Chapter 11 Hyperfine Structure and the Addition of Angular Momentum Chapter 12 Perturbation of Hydrogen Chapter 13 Identical Particles Chapter 14 Time dependent perturbation theory Chapter 15 Periodic Systems Chapter 16 Modern Applications Appendices Chapter 1 STERN-GERLACH EXPERIMENTS 1-1 1.1 Stern-Gerlach experiment 1-1 1.1.1 Experiment 1 1-6 1.1.2 Experiment 2 1-7 1.1.3 Experiment 3 1-8 1.1.4 Experiment 4 1-9 1.2 Quantum State Vectors 1-11 1.2.1 Analysis of Experiment 1 1-18 1.2.2 Analysis of Experiment 2 1-18 1.2.3 Superposition states 1-21 1.3 Matrix notation 1-24 1.4 General Quantum Systems 1-28 1.5 Postulates 1-30 1.6 Summary 1-31 1.7 Problems 1-32 1.8 Resources 1-36 1.8.1 Activities 1-36 1.8.2 Further reading 1-37

Chapter 2 OPERATORS AND MEASUREMENT 2-1 2.1 Operators, Eigenvalues, and Eigenvectors 2-1 2.1.1 Matrix Representation of Operators 2-4 2.1.2 Diagonalization of Operators 2-6 2.2 New Operators 2-9 2.2.1 Spin Component in General Direction 2-9 2.2.2 Hermitian Operators 2-12 2.2.3 Projection Operators 2-13 2.2.4 Analysis of Experiments 3 and 4 2-17 2.3 Measurement 2-20 2.4 Commuting Observables 2-25 2.5 Uncertainty Principle 2-27 2.6 S 2 Operator 2-29 2.7 Spin 1 System 2-30 2.8 General Quantum Systems 2-34 2.9 Summary 2-35 2.10 Problems 2-36 2.11 Resources 2-40 2.11.1 Activities 2-40 2.11.2 Further reading 2-41 Chapter 3 SCHRÖDINGER TIME EVOLUTION 3-1 3.1 Schrödinger Equation 3-1 3.2 Spin Precession 3-5 3.2.1 Magnetic Field in z-direction 3-6 3.2.2 Magnetic field in a general direction 3-13 3.3 Neutrino Oscillations 3-20 3.4 Time-Dependent Hamiltonians 3-23 3.4.1 Magnetic Resonance 3-24 3.4.2 Light-Matter Interactions 3-30 3.5 Summary 3-31 3.6 Problems 3-32 3.7 Resources 3-36 3.7.1 Activities 3-36 3.7.2 Further reading 3-36 Chapter 4 Quantum Spookiness 4-1 4.1 Einstein-Podolsky-Rosen Paradox 4-1 4.2 Schrödinger Cat Paradox 4-8 4.3 Problems 4-11 4.4 Resources 4-12 4.4.1 Further reading 4-12

Chapter 5 Quantized Energies: Particle in a Box 5-1 5.1 Spectroscopy 5-1 5.2 Energy Eigenvalue Equation 5-4 5.3 The Wave Function 5-6 5.4 Infinite Square Well 5-14 5.5 Finite Square Well 5-24 5.6 Compare and Contrast 5-30 5.6.1 Wave function curvature 5-30 5.6.2 Nodes 5-32 5.6.3 Barrier penetration 5-32 5.6.4 Inversion Symmetry and Parity 5-33 5.6.5 Orthonormality 5-33 5.6.6 Completeness 5-34 5.7 Superposition states and time dependence 5-34 5.8 Modern Application: Quantum Wells and Dots 5-44 5.9 Asymmetric square well: sneak peek at perturbations 5-45 5.10 Fitting energy eigenstates by eye or by computer 5-48 5.10.1 Qualitative (eyeball) solutions 5-48 5.10.2 Numerical solutions 5-49 5.10.3 General potential wells 5-52 5.11 Summary 5-53 5.12 Problems 5-55 5.13 Resources 5-58 5.13.1 Activities 5-58 5.13.2 Further reading 5-59 Chapter 6 Unbound States 6-1 6.1 Free particle eigenstates 6-1 6.1.1 Energy eigenstates 6-1 6.1.2 Momentum eigenstates 6-3 6.2 Wave packets 6-9 6.2.1 Discrete superposition 6-9 6.2.2 Continuous superposition 6-12 6.3 Uncertainty Principle 6-18 6.3.1 Energy estimation 6-23 6.4 Unbound States and Scattering 6-24 6.5 Tunneling through Barriers 6-32 6.6 Atom Interferometry 6-37 6.7 Summary 6-41 6.8 Problems 6-42 6.9 Resources 6-47 6.9.1 Activities 6-47 6.9.2 Further reading 6-47

Chapter 7 Angular Momentum 7-1 7.1 Separating Center-of-Mass and Relative Motion 7-3 7.2 Energy Eigenvalue Equation In Spherical Coordinates 7-7 7.3 Angular Momentum 7-9 7.3.1 Classical Angular Momentum 7-9 7.3.2 Quantum Mechanical Angular Momentum 7-10 7.4 Separation Of Variables: Spherical Coordinates 7-15 7.5 Motion of a Particle on a Ring 7-19 7.5.1 Azimuthal Solution 7-21 7.5.2 Quantum Meas. on a Particle Confined to a Ring 7-24 7.5.3 Superposition states 7-25 7.6 Motion on a Sphere 7-28 7.6.1 Series Solution of Legendre's Equation 7-30 7.6.2 Associated Legendre Functions 7-35 7.6.3 Energy Eigenvalues of a Rigid Rotor 7-38 7.6.4 Spherical Harmonics 7-40 7.6.5 Visualization of Spherical Harmonics 7-44 7.7 Summary 7-49 7.8 Problems 7-50 7.9 Resources 7-56 7.9.1 Activities 7-56 Chapter 8 Hydrogen Atom 8-1 8.1 The Radial Eigenvalue Equation 8-1 8.2 Solving The Radial Equation 8-3 8.2.1 Asymptotic Solutions of the Radial Equation 8-3 8.2.2 Series Solution of the Radial Equation 8-4 8.3 Hydrogen Energies and Spectrum 8-8 8.4 The Radial Wave Functions 8-13 8.5 The Full Hydrogen Wave Functions 8-16 8.6 Superposition States 8-23 8.7 Summary 8-26 8.8 Problems 8-27 8.9 Resources 8-29 8.9.1 Activities 8-29 8.9.2 Further reading 8-29 Chapter 9 Harmonic Oscillator 9-1 9.1 Classical Harmonic Oscillator 9-1 9.2 Quantum Mechanical Harmonic Oscillator 9-3 9.3 Wave functions 9-11 9.4 Dirac Notation 9-17 9.5 Matrix representations 9-21 9.6 Momentum space wave function 9-24

9.7 The Uncertainty Principle 9-27 9.8 Time dependence 9-29 9.9 Molecular Vibrations 9-35 9.10 Summary 9-38 9.11 Problems 9-39 9.12 Resources 9-43 9.12.1 Activities 9-43 9.12.2 Further reading 9-43 Chapter 10 Perturbation Theory 10-1 10.1 Spin ½ example 10-1 10.2 General two-level example 10-6 10.3 Nondegenerate Perturbation Theory 10-9 10.3.1 First-order energy correction 10-10 10.3.2 First-order state vector correction 10-15 10.4 Second-order nondegenerate perturbation theory 10-20 10.5 Degenerate perturbation theory 10-27 10.6 More Examples 10-35 10.6.1 Harmonic Oscillator 10-35 10.6.2 Stark effect in hydrogen 10-39 10.7 Summary 10-44 10.8 Problems 10-45 Chapter 11 Hyperfine Structure and the Addition of Angular Momentum11-1 11.1 Hyperfine Interaction 11-1 11.2 Angular Momentum Review 11-3 11.3 Angular Momentum Ladder Operators 11-5 11.4 Diagonalization of the hyperfine perturbation 11-7 11.5 The Coupled Basis 11-12 11.6 Addition of Generalized Angular Momenta 11-18 11.7 Angular momentum in atoms and spectroscopic notation11-24 11.8 Summary 11-25 11.9 Problems 11-27 11.10 Resources 11-30 11.10.1 Activities 11-30 11.10.2 Further reading 11-30 Chapter 12 Perturbation of Hydrogen 12-1 12.1 Hydrogen Energy Levels 12-1 12.2 Fine Structure of Hydrogen 12-5 12.2.1 Relativistic Correction 12-5 12.2.2 Spin-orbit coupling 12-8 12.3 Zeeman effect 12-13

12.3.1 Zeeman effect without spin 12-14 12.3.2 Zeeman effect with spin 12-16 12.3.3 Zeeman perturbation of the 1s hyperfine structure12-26 12.4 Summary 12-28 12.5 Problems 12-29 12.6 Resources 12-31 12.6.1 Activities 12-31 12.6.2 Further reading 12-31 Chapter 13 Identical Particles 13-1 13.1 Two spin-½ particles 13-1 13.2 Two identical particles in one dimension 13-5 13.2.1 Two-particle ground state 13-6 13.2.2 Two-particle excited state 13-7 13.2.3 Visualization of states 13-8 13.2.4 Exchange Interaction 13-12 13.2.5 Consequences of the symmetrization postulate 13-14 13.3 Interacting particles 13-16 13.4 Example: The Helium Atom 13-20 13.4.1 Helium ground state 13-23 13.4.2 Helium excited states 13-25 13.5 The Periodic Table 13-29 13.6 Example: The Hydrogen Molecule 13-33 + 13.6.1 The hydrogen molecular ion H 2 13-34 13.6.2 The hydrogen molecule H 2 13-36 13.7 Problems 13-38 13.8 Resources 13-41 13.8.1 Further reading 13-41 Chapter 14 Time-dependent perturbation theory 14-1 14.1 Transition probability 14-1 14.2 Harmonic perturbation 14-7 14.3 Electric dipole interaction 14-11 14.3.1 Einstein model: Broadband excitation 14-13 14.3.2 Laser excitation 14-18 14.4 Selection rules 14-21 14.5 Summary 14-24 14.6 Problems 14-25 14.7 Resources 14-28 14.7.1 Further reading 14-28

Chapter 15 Periodic Systems 15-1 15.1 Introduction 15-1 15.2 The energy eigenvalues and eigenstates of a periodic chain of wells 15-3 15.2.1 A 2-well chain 15-3 15.2.2 N-well chain 15-6 15.3 Boundary Conditions and the Allowed Values of k 15-9 15.4 The Brillouin Zones 15-11 15.5 Multiple Bands from Multiple Atomic Levels 15-12 15.6 Bloch s Theorem and the Molecular States 15-14 15.7 Molecular Wave Functions a Gallery 15-16 15.8 The Density of States 15-18 15.9 Calculation of the Model Parameters 15-20 15.9.1 LCAO summary 15-22 15.10 The Kronig-Penney Model 15-23 15.11 Practical Applications: Metals, Insulators and Semiconductors 15-26 15.12 Effective Mass 15-30 15.13 Direct and Indirect Band Gaps 15-32 15.14 New directions low-dimensional carbon 15-34 15.15 Summary 15-34 15.16 Problems 15-35 15.17 Resources 15-37 15.17.1 Activities 15-37 15.17.2 Further reading 15-37 Chapter 16 Modern Applications of Quantum Mechanics 16-1 16.1 Manipulating atoms with quantum mechanical forces 16-1 16.1.1 Magnetic trapping 16-1 16.1.2 Laser cooling 16-6 16.2 Quantum Information Processing 16-15 16.2.1 Quantum bits Qubits 16-16 16.2.2 Quantum gates 16-19 16.2.3 Quantum teleportation 16-27 16.3 Summary 16-30 16.4 Problems 16-30 16.5 Resources 16-32 16.5.1 Further reading 16-32 Appendices Appendix A: Probability A-1 A.1 Discrete Probability Distribution D-1 A.2 Continuous Probability Distribution D-3 Appendix B: Complex Numbers B-1

Appendix C: Matrices C-1 Appendix D: Waves and Fourier Analysis D-1 D.1 Classical Waves D-1 D.2 Fourier Analysis D-3 D.3 Quantum Mechanics D-6 Appendix E: Separation of Variables E-1 Appendix F: Integrals F-1 Appendix G: Physical constants G-1

PROLOGUE It was a dark and stormy night. Erwin huddled under his covers as he had done numerous times that summer. As the wind and rain lashed at the window, he feared having to retreat to the storm cellar once again. The residents of Erwin's apartment building sought shelter whenever there were threats of tornadoes in the area. While it was safe down there, Erwin feared the ridicule he would face once again from the other school boys. In the rush to the cellar, Erwin seemed to always end up with a random pair of socks, and the other boys teased him about it mercilessly. Not that Erwin hadn't tried hard to solve this problem. He had a very simple collection of socks black or white, for either school or play; short or long, for either trousers or lederhosen. After the first few teasing episodes from the other boys, Erwin had sorted his socks into two separate drawers. He placed all the black socks in one drawer and all the white socks in another drawer. Erwin figured he could determine an individual sock's length in the dark of night simply by feeling it, but he had to have them presorted into white and black because the apartment generally lost power before the call to the shelter. Unfortunately, Erwin found that this presorting of the socks by color was ineffective. Whenever he reached into the white sock drawer and chose two long socks, or two short socks, there was a 50% probability of any one sock being black or white. The results from the black sock drawer were the same. The socks seemed to have "forgotten" the color that Erwin had determined previously. Erwin also tried sorting the socks into two drawers based upon their length, without regard to color. When he chose black or white socks from these long and short drawers, the socks had also "forgotten" whether they were long or short. After these fruitless attempts to solve his problem through experiments, Erwin decided to save himself the fashion embarrassment, and he replaced his sock collection with a set of medium length brown socks. However, he continued to ponder the mysteries of the socks throughout his childhood. After many years of daydreaming about the mystery socks, Erwin Schrödinger proposed his theory of "Quantum Socks" and become famous. And that is the beginning of the story of the quantum socks. The End. Farfetched?? You bet. But Erwin's adventure with his socks is the way quantum mechanics works. Read on.