Physics 460-461 Quantum Mechanics I-II :  Course Outline

 

 

 

I.  Schrödinger Wave Mechanics

 

            A)  Postulates and Discussion

            B)  Consequences and Physical Interpretation

                        1)  Conservation of Probability

                        2)  Free Particle Wave Functions

                        3)  Expectation Values

                        4)  CCR

                        5)  Ehrenfest's Thm. and Classical Mechanics

                        6)  Heisenberg Uncertainty Relations

                        7)  Stationary States, Hermitian Operators, Eigenvalues and Eigenfunctions

                        8)  Boundary Conditions on the Wavefunction

 

II.  Applications in One-Dimension

 

            A)  Bound States

                        1)  Infinite Square Well Potential

                        2)  SHO

                        3)  Dirac δ-function Potential

                        4)  Finite Square Well Potential

                        5)  Krönig-Penney Model

            B)  Scattering Off a Potential Step-Barrier Penetration

 

III.  Applications in Three-Dimensions

 

            A)  Central Potential, Center of Momentum and Spherical Polar Coordinates

            B)  Orbital Angular Momentum

            C)  Hydrogen Atom

 

IV.  Abstract Formulation of Quantum Mechanics

 

            A)  Hilbert Space and Dirac Bra-Ket Notation

            B)  Operators, Eigenvalues and Observables

            C)  The Postulates

            D)  Position, Momentum and Energy Representations (Bases)

            E)  Consequences and Physical Interpretation

            F)  SHO

            G)  Symmetries in Quantum Mechanics

                        1)  Space Translations

                        2)  Time Translations

                        3)  Rotation Group and Angular Momentum

                        4)  Supersymmetry

            H)  Angular Momentum and Spin

             I)   Addition of Angular Momentum

 

 

V.  Identical Particles

 

            A)  Two-Particle Systems

            B)  Atoms and Solids

 

VI.  Time Independent Perturbation Theory

 

            A)  Rayleigh-Schrödinger (Non-) Degenerate Perturbation Theory

            B)  Fine Structure and Hyperfine Splitting of Hydrogen

            C)  Zeeman Effect

            D)  Stark Effect

 

VII.  Variational Principle

 

            A)  Ground State of Helium

 

VIII.  WKB Approximation

 

            A)  Tunneling and Lifetime

 

IX.  Time-Dependent Perturbation Theory

 

            A)  Dirac Interaction Picture and Time Evolution Operator

            B)  Energy Loss of Charged Particle Travelling Through Matter

            C)  Scattering Theory

                        1)  Fermi's Golden Rule

                        2)  Potential Scattering-Born Approximation and Rutherford Cross Section

            D)  Semi-Classical Treatment of Electromagnetic Radiation

                        1)  Fermi's Golden Rule

                        2)  Stimulated Emission and Absorption

                        3)  Spontaneous Emission and Lifetime of Excited State

 

X.  Potential Scattering

 

            A)  Born Approximation

            B)  Partial Wave Analysis

            C)  Examples

 

XI.  Feynman Path Integral Formulation of Quantum Mechanics

 

            A)  Feynman Path Integral

            B)  Free Particle

            C)  SHO

            D)  Instantons and Tunneling

            E)  Supersymmetry and Grassmann (Spinning) Coordinates

            F)  Entanglement and Quantum Computing