Paradigms in Physics: Quantum Mechanics

Transcription

1 Paradigms in Physics: Quantum Mechanics David H. McIntyre Corinne A. Manogue Janet Tate Oregon State University 23 November 2010 Copyright 2010 by David H. McIntyre, Corinne A. Manogue, Janet Tate

2 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 Stern-Gerlach experiment Experiment Experiment Experiment Experiment Quantum State Vectors Analysis of Experiment Analysis of Experiment Superposition states Matrix notation General Quantum Systems Postulates Summary Problems Resources Activities Further reading 1-37

3 Chapter 2 OPERATORS AND MEASUREMENT Operators, Eigenvalues, and Eigenvectors Matrix Representation of Operators Diagonalization of Operators New Operators Spin Component in General Direction Hermitian Operators Projection Operators Analysis of Experiments 3 and Measurement Commuting Observables Uncertainty Principle S 2 Operator Spin 1 System General Quantum Systems Summary Problems Resources Activities Further reading 2-41 Chapter 3 SCHRÖDINGER TIME EVOLUTION Schrödinger Equation Spin Precession Magnetic Field in z-direction Magnetic field in a general direction Neutrino Oscillations Time-Dependent Hamiltonians Magnetic Resonance Light-Matter Interactions Summary Problems Resources Activities Further reading 3-36 Chapter 4 Quantum Spookiness Einstein-Podolsky-Rosen Paradox Schrödinger Cat Paradox Problems Resources Further reading 4-12

4 Chapter 5 Quantized Energies: Particle in a Box Spectroscopy Energy Eigenvalue Equation The Wave Function Infinite Square Well Finite Square Well Compare and Contrast Wave function curvature Nodes Barrier penetration Inversion Symmetry and Parity Orthonormality Completeness Superposition states and time dependence Modern Application: Quantum Wells and Dots Asymmetric square well: sneak peek at perturbations Fitting energy eigenstates by eye or by computer Qualitative (eyeball) solutions Numerical solutions General potential wells Summary Problems Resources Activities Further reading 5-59 Chapter 6 Unbound States Free particle eigenstates Energy eigenstates Momentum eigenstates Wave packets Discrete superposition Continuous superposition Uncertainty Principle Energy estimation Unbound States and Scattering Tunneling through Barriers Atom Interferometry Summary Problems Resources Activities Further reading 6-47

5 Chapter 7 Angular Momentum Separating Center-of-Mass and Relative Motion Energy Eigenvalue Equation In Spherical Coordinates Angular Momentum Classical Angular Momentum Quantum Mechanical Angular Momentum Separation Of Variables: Spherical Coordinates Motion of a Particle on a Ring Azimuthal Solution Quantum Meas. on a Particle Confined to a Ring Superposition states Motion on a Sphere Series Solution of Legendre's Equation Associated Legendre Functions Energy Eigenvalues of a Rigid Rotor Spherical Harmonics Visualization of Spherical Harmonics Summary Problems Resources Activities 7-56 Chapter 8 Hydrogen Atom The Radial Eigenvalue Equation Solving The Radial Equation Asymptotic Solutions of the Radial Equation Series Solution of the Radial Equation Hydrogen Energies and Spectrum The Radial Wave Functions The Full Hydrogen Wave Functions Superposition States Summary Problems Resources Activities Further reading 8-29 Chapter 9 Harmonic Oscillator Classical Harmonic Oscillator Quantum Mechanical Harmonic Oscillator Wave functions Dirac Notation Matrix representations Momentum space wave function 9-24

6 9.7 The Uncertainty Principle Time dependence Molecular Vibrations Summary Problems Resources Activities Further reading 9-43 Chapter 10 Perturbation Theory Spin ½ example General two-level example Nondegenerate Perturbation Theory First-order energy correction First-order state vector correction Second-order nondegenerate perturbation theory Degenerate perturbation theory More Examples Harmonic Oscillator Stark effect in hydrogen Summary Problems Chapter 11 Hyperfine Structure and the Addition of Angular Momentum Hyperfine Interaction Angular Momentum Review Angular Momentum Ladder Operators Diagonalization of the hyperfine perturbation The Coupled Basis Addition of Generalized Angular Momenta Angular momentum in atoms and spectroscopic notation Summary Problems Resources Activities Further reading Chapter 12 Perturbation of Hydrogen Hydrogen Energy Levels Fine Structure of Hydrogen Relativistic Correction Spin-orbit coupling Zeeman effect 12-13

7 Zeeman effect without spin Zeeman effect with spin Zeeman perturbation of the 1s hyperfine structure Summary Problems Resources Activities Further reading Chapter 13 Identical Particles Two spin-½ particles Two identical particles in one dimension Two-particle ground state Two-particle excited state Visualization of states Exchange Interaction Consequences of the symmetrization postulate Interacting particles Example: The Helium Atom Helium ground state Helium excited states The Periodic Table Example: The Hydrogen Molecule The hydrogen molecular ion H The hydrogen molecule H Problems Resources Further reading Chapter 14 Time-dependent perturbation theory Transition probability Harmonic perturbation Electric dipole interaction Einstein model: Broadband excitation Laser excitation Selection rules Summary Problems Resources Further reading 14-28

8 Chapter 15 Periodic Systems Introduction The energy eigenvalues and eigenstates of a periodic chain of wells A 2-well chain N-well chain Boundary Conditions and the Allowed Values of k The Brillouin Zones Multiple Bands from Multiple Atomic Levels Bloch s Theorem and the Molecular States Molecular Wave Functions a Gallery The Density of States Calculation of the Model Parameters LCAO summary The Kronig-Penney Model Practical Applications: Metals, Insulators and Semiconductors Effective Mass Direct and Indirect Band Gaps New directions low-dimensional carbon Summary Problems Resources Activities Further reading Chapter 16 Modern Applications of Quantum Mechanics Manipulating atoms with quantum mechanical forces Magnetic trapping Laser cooling Quantum Information Processing Quantum bits Qubits Quantum gates Quantum teleportation Summary Problems Resources Further reading 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

9 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

10 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.