QUANTUM MECHANICS I - IIIPHYS 516 - 518Jan 1 - Dec. 31, 2011Prof. R. Gilmore12-918 [email protected], [email protected] Schedule: MWF 11:00 - 11:50, Disque 919Objective: To provide the foundations for modern physics.Course Topics• Schr¨odinger’s Papers1. Quantization as an Eigenvalue Problem: Part I2. Quantization as an Eigenvalue Problem: Part II3. Quantization as an Eigenvalue Problem: Part III4. Quantization as an Eigenvalue Problem: Part IV• Forms of Quantum Theory: Matrix Mechanics, Wave Mechanics, PathIntegrals• Separation of Variables:1. Klein-Gordan Equation2. Schr¨odinger Equation• Frobenius’s Method• Eigenvalues and Eigenvectors• Brief Remarks: Spherical Harmonics1• Time-Independent Perturbation Theory• Applictions:1. Finite nuclear size2. Zeeman Effect3. Stark Effect4. Crossed Fields• Harmonic Oscillator1. Analytic solution: Frobenius’ Method2. Operator solution3. Discretization and Matrix Diagonalization4. Ginzburg-Landau Quartic Potential• Coupled Oscillators1. Linear Molecules and Normal Modes2. One-Dimensional Solids(a) One atom/unit cell(b) Two atoms/unit cell(c) Three atoms/unit cell3. Two-dimensional solids4. Three-dimensional solids• Electromagnetic Field1. Maxwell’s Equations2. Vector and Scalar Potentials3. Normal Modes4. Independent Oscillators5. Quantization• Time Dependence• Time-dependent perturbation theory• Representations:1. Schr¨odinger2. Interaction3. Heisenberg2• Applictions:1. Perturbed harmonic oscillator2. Fermi golden Rule3. Lorentzians• Angular Momentum1. Analytic representation, angular variables: L2. Algebraic representation, |l, mli3. J ' a†a4. Spin angular momentum: S5. Total angular momentum: J6. Spherical harmonics7. Clebsch-Gordan coefficients• Angular Momentum Applications1. Shielded Coulomb Potential → Mendelyeev2. Harmonic + Square Well + Spin Orbit = Nuclear Shell Model3. Hydrogen → Positronium → Charmonium → BottomoniumQuantization as an Eigenvalue Problem. I1. Variational formulation.2. Standard formulation.3. Hydrogen atom: Bound states.4. Hydrogen atom: Scattering states.Quantization as an Eigenvalue Problem. II1. Harmonic oscillator.2. Rotator with fixed axis (2D).3. Rigid rotator with free axis (3D).4. Diatomic molecule.5. Two-dimensional oscillators.6. Three-dimensional oscillators.7. Coupled oscillations.8. Coherent states. (After the first of his 2 intermediate papers.)Quantization as an Eigenvalue Problem. III1. Perturbation theory.2. Stark effect.3. Line strengths.Quantization as an Eigenvalue Problem. IV31. Time-dependent wave equation.2. Perturbation theory (time-dependent).3. Resonance phenomena.4. Minimal electromagnetic coupling.Ehrenfest Theorems:1. Expectation values and density matrices/operators.2. Newton’s Equations.3. Harmonic motion.4. Orbital angular momentum and torque.5. Angular momentum and precession.6. Lorentz force.7. Hamilton’s Equations.8. The Virial.9. Quadrupoles.10. Euler’s Equations.11. Runge Lenz vector and precession (S.R. & G.R.)Matrix Mechanics1. Born, Heisenberg, and Jordan.2. Schr¨odinger’s demonstration of equivalence.3. Then and Now: the Swing of the Pendulum.4. Matrix computations.5. FEMFeynman’s Path Integrals1. A particle goes along all possible paths.2. The Action Integral.3. Equivalence with Schr¨odinger’s Equation (time-dependent).4. 2-Slit interference pattern (Young diffration pattern).5. Single-Slit interference pattern (Fraunhofer diffration pattern).6. Diffraction gratings.7. Interferometers: Matrix methods.8. Resonators: Matrix methods.9. Networks: S-matrices.10. Networks: eigenstates.Broad Historical Sweep1. The light dialogue: From Newton to Einstein (?) and Beyond?2. The gravity dialogue: From Newton to Einstein (?) and Beyond?3. Problems with ∞: Planck ¯h; Bohr atom; Renormalization; Casimir.4. The Phases of Quantum Theory: 1913, 1926, 1964.5. 1913: Correspondence Principle.6. 1926: Ehrenfest Theorems.47. 1935: EPR and Schr¨odinger’s Cat.8. 1964: Bell’s Theorem unlocks the flood.9. 2000 → “At last, we’re free from our classical manacles.” (“The Quantumworld is weirder that we could possibly have imagined.”)Uncertainties1. Position and momentum: ∆x∆p ≥ ¯h/2.2. Time and energy: ∆t∆E ≥ ¯h/2.3. Angle and angular momentum: ∆θ∆Lθ≥ ¯h/2.4. Number and phase: ∆N ∆φ ≥ 1.5. Amplitude and phase: ∆A∆φ ≥ π.6. Light Blitz Box: 2 ships passing in the night.7. Squeezed states: trading uncertainties.8. COBE and an absolute rest frame.9. Nyquist Theorem.10. Cramer-Rao Uncertainty Relations.11. Uncertainty Relations of Statistical Mechanics: ∆U∆1T≥ k.Symmetry1. Solving equations.2. Symmetry ⇒ degeneracy.3. Dynamical symmetry.4. Classification of states.5. Point Groups, Space Groups.6. SU(2) and rotations.7. SU(3) and particles.8. SU(5)9. Symmetry-breaking.Gauge Theories1. Measuring the gravitational field.2. Measuring the phase of an electric field.3. Global gauge transformations: U (1).4. Local gauge transformations: U (1).5. Yang-Mills, Nuclear Forces and Mesons: SU (2).6. Utiyama.7. Groups and gauge theories: gauge bosons.8. Renormalizable gauge theories.Troublesome Infinities1. The Ultraviolet Catastrophe: Planck and ¯h.2. The Hydrogen Catastrophe: Bohr and the Old Quantum Theory.3. Electron Self-Energy Catastrophe: Renormalization Group Theory.4. Zero-Point Fluctuation Catastrophes: Casimir Effect.5Quantum Theory: Phase III1. Phase I: The Old Quantum Theory.2. Phase II: Quantum theory: 1925 → present.3. Phase III: The Great Smokey Dragon.4. EPR & Schr¨odinger: Entanglement and Decoherence.5. von Neumann’s “proof”6. Bohm’s Hidden Variables: A Counterexample.7. Bell’s theorem (1964).8. The first three measurements.9. Later Measurements (Aspect).10. The Floodgates are Opened: GHZ and others.11. Entanglement at a Distance: The Danube.12. Measuring Decoherence.13. Looking at Pilot Waves (Yves Couder).14. Delayed choice Experiment.15. Quantum Eraser.16. Bounding the speed of Quantum Information: VQI/c.C3: Quantum Cryptography, Computing, Communication(to be