This course will focus on the explanation of fundamental concepts, mathematical structure, and calculation methods of quantum mechanics. The course covers the following topics: fundamental concepts of quantum mechanics and its mathematical structure, Exactly solvable quantum systems, symmetries in quantum mechanics, approximation methods, atomic and Molecular structure, scattering theory, quantum manyparticle systems, relativistic wave equations.
Course Requirements
: There will be regularly assigned problem sets. Your letter grade will be based on the two exams (Midterm 20%; Final: 40%), homework (30%), and class participation (10%).
Prerequisites: The basic mathematical prerequisites are linear algebra and calculus. It would be useful to have a previous course on introductory quantum mechanics at the undergraduate level, but that is not an essential requirement. Some results from group theory will be used for discussion on symmetries, but I will explain the results before I use them.
Books and References
Textbook



Other
recommended
books


 P.A.M. Dirac, The Principles of Quantum Mechanics
 E. Merzbacher, Quantum Mechanics (1998)
 J.J. Sakurai, Modern Quantum Mechanics (1982)

Further Readings


 C. CohenTannoudji, B. Diu, and F. Laloe, Quantum Mechanics (1997)
 R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals
 L. Landau and E. Lifshitz, Quantum Mechanics: Nonrelativistic Theory
 J. Preskill, Lecture Notes on quantum inforamtion and computation (the
first four chapters, which are general quantum mechanics), see
http://www.theory.caltech.edu/people/preskill/p29/#lecture

Course Outline: Second term (Winter)
 Brief introduction and review
Review of the course structure and the firstsemester contents  Approximation methods in quantum mechanics
 Overview of approximation methods in QM
 Timeindependent perturbation methods
 The general method and classification of perturbation theory
 Boundstate nondegenerate perturbation
Basic recursion relations for arbitrary order perturbation, explicit
formula for 1st and 2nd order perturbation  The degenerate
perturbation method
 Convergence, asymptotic series, and limit of perturbation
theory
 Example applications of the timeindependent perturbation
 d.c. Stark shift (from a static electric field) of the
ground state of
the hydrogenlike atoms (nondegenerate perturbation)
 d.c. Stark shift of excited states of the hydrogenlike
atoms
(degenerate
perturbation)
 Atom's polarizability and atom's trap
 a.c. Stark shift (from a laser) of the ground state of the
hydrogenlike
atoms
 Design of optical lattice from a.c. Stark shift
 Timedependent perturbation, transition, and Fermi's golden
rule
 Timedependent perturbation (formalism)
 Timeenergy uncertainty relation
 Fermi's golden rule for transition (discrete spectrum to
continuous
spectrum)
 Atomic transition through broadband incoherent (thermal)
light
Stimulated emission and absorption, explain of spontaneous emission  Quantum
Zeno effect (coherent vs incoherent transitions)
 The variational method
 The general idea
 The variational principle for the stationary Schrodinger
equation
 The variational principle for the dynamical Schrodinger
equation
 A simple illustrative example: Harmonic potential
 Example applications of the variational method
 The ground state of the Heliumlike atoms
 The bandgap structure for atoms in the optical lattice
 Quantum phase transitions in quantum magnetism: meanfield
theory for the anisotropic Heisenberg model
 The semiclassical (WKB) method
 The WKB approximation
 Turning points and the connection formula
 Example applications of the WKB method
 Quantization condition for a singleminimum potential
 Quantization condition and the eigenenergies of the
doublewell
potential
 Tunneling through any potential barrier
 The adiabatic approximation and the Berry's phase
 The fast and slow evolution: sudden versus adiabatic
approximation
 The adiabatic theorem and the Berry's phase
Proof the adiabatic theorem, dynamical and Berry's (geometric) phase,
Estimation of the transition probability  The adiabatic passage
with counterintuitive pulses
 The adiabatic quantum algorithm
 The Berry's phase for a twolevel system
 Structure of atoms and molecules
 Overview
 Overview of the atomic and molecular structure
 About the units
 Fine structure of the hydrogenlike atoms
 Spinorbital coupling
 Splitting of the energy levels due to the spinorbital coupling
 The relativistic correction to eigenenergies and the whole fine structure
 The hyperfine structure of the hydorgenlike atoms
 The basic picture
 The hyperfine coupling Hamiltonian
N, L coupling, N,S coupling (contact vs. dipole terms)  The
hyperfine splitting of the ground state
Level splitting, applications in radio astronomy
 Influence of magnetic fields on atomic structure: Zeeman
effects
 Atomic magnetic moment
 Zeeman effects within Fine structure
 Zeeman effects within hyperfine structure
 The molecular structure
 The BornOppenheimer approximati