Phy5645

Welcome to the Quantum Mechanics A PHY5645 Fall2008/2009

Schrödinger Equation
The most fundamental equation of quantum mechanics; given a Hamiltonian ${\mathcal {H}}$, it describes how a state $|\Psi \rangle$ evolves in time.

This is the first semester of a two-semester graduate level sequence, the second being PHY5646 Quantum B. Its goal is to explain the concepts and mathematical methods of Quantum Mechanics, and to prepare a student to solve quantum mechanics problems arising in different physical applications. The emphasis of the courses is equally on conceptual grasp of the subject as well as on problem solving. This sequence of courses builds the foundation for more advanced courses and graduate research in experimental or theoretical physics.

The key component of the course is the collaborative student contribution to the course Wiki-textbook. Each team of students is responsible for BOTH writing the assigned chapter AND editing chapters of others.

Team assignments: Fall 2009 student teams

Fall 2009 Midterm is October 15

Outline of the Course

Chapter 1: Physical Basis of Quantum Mechanics

• Basic Concepts and Theory of Motion
• UV Catastrophe (Black-Body Radiation)
• Photoelectric Effect
• Stability of Matter
• Double Slit Experiment
• Stern-Gerlach Experiment
• The Principle of Complementarity
• The Correspondence Principle
• The Philosophy of Quantum Theory

Chapter 2: Schrödinger Equation

• Brief Derivation of Schrödinger Equation
• Relation Between the Wave Function and Probability Density
• Stationary States
• Heisenberg Uncertainty Principle
• Some Consequences of the Uncertainty Principle

Chapter 3: Operators, Eigenfunctions, and Symmetry

• Linear Vector Space and Operators
• Commutation Relations and Simultaneous Eigenvalues
• The Schrödinger Equation in Dirac Notation
• Transformations of Operators and Symmetry

Chapter 4: Motion in One Dimension

• One-Dimensional Bound States
• Oscillation Theorem
• The Dirac Delta Function Potential
• Scattering States, Transmission and Reflection
• Motion in a Periodic Potential
• Summary of 1D Systems

Chapter 5: Discrete Eigenvalues and Bound States; Harmonic Oscillator and WKB Approximation

• Harmonic Oscillator Spectrum and Eigenstates
• Analytical Method for Solving the Simple Harmonic Oscillator
• Coherent States
• Charged Particles in an Electromagnetic Field
• WKB Approximation

Chapter 6: Time Evolution and the Pictures of Quantum Mechanics

• The Heisenberg Picture: Equations of Motion for Operators
• The Interaction Picture
• Time Evolution of Expectation Values and Ehrenfest's Theorem
• The Virial Theorem

Chapter 7: Angular Momentum

• Commutation Relations
• Angular Momentum as a Generator of Rotations in 3D
• Spherical Coordinates
• Eigenvalue Quantization
• Orbital Angular Momentum Eigenfunctions

Chapter 8: Central Forces

• Generalized Derivation
• Free Particle in Spherical Coordinates
• Spherical Well
• Isotropic Harmonic Oscillator
• Hydrogen Atom
• WKB in Spherical Coordinates

Chapter 9: Path Integral Evaluation of the Free-Particle Propagator

• Feynman Path Integrals
• Saddle-Point Action
• Harmonic Fluctuations

Chapter 10: Continuous Eigenvalues and Collision Theory

• Differential Cross-Section and the Green's Function Formulation of Scattering
• Central Potential Scattering and Phase Shifts
• Born Approximation and Examples of Cross-Section Calculations
• Coulomb Potential Scattering
• Two-Particle Scattering