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 many-particle 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

Course Outline: Second term (Winter)

• Brief introduction and review
        Review of the course structure and the first-semester contents
• Approximation methods in quantum mechanics
1. Overview of approximation methods in QM
2. Time-independent perturbation methods
• The general method and classification of perturbation theory
• Bound-state non-degenerate 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
3. Example applications of the time-independent perturbation
• d.c. Stark shift (from a static electric field) of the ground state of the hydrogen-like atoms (non-degenerate perturbation)
• d.c. Stark shift of excited states of the hydrogen-like atoms (degenerate perturbation)
• Atom's polarizability and atom's trap
• a.c. Stark shift (from a laser) of the ground state of the hydrogen-like atoms
• Design of optical lattice from a.c. Stark shift
4. Time-dependent perturbation, transition, and Fermi's golden rule
• Time-dependent perturbation (formalism)
• Time-energy 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)
5. 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
6. Example applications of the variational method
• The ground state of the Helium-like atoms
• The band-gap structure for atoms in the optical lattice
• Quantum phase transitions in quantum magnetism: mean-field theory for the anisotropic Heisenberg model
7. The semi-classical (WKB) method
• The WKB approximation
• Turning points and the connection formula
8. Example applications of the WKB method
• Quantization condition for a single-minimum potential
• Quantization condition and the eigen-energies of the double-well potential
• Tunneling through any potential barrier
9. 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 counter-intuitive pulses
• The adiabatic quantum algorithm
• The Berry's phase for a two-level system
• Structure of atoms and molecules
1. Overview
• Overview of the atomic and molecular structure
• About the units
2. Fine structure of the hydrogen-like atoms
• Spin-orbital coupling
• Splitting of the energy levels due to the spin-orbital coupling
• The relativistic correction to eigen-energies and the whole fine structure
3. The hyperfine structure of the hydorgen-like 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
4. Influence of magnetic fields on atomic structure: Zeeman effects
• Atomic magnetic moment
• Zeeman effects within Fine structure
• Zeeman effects within hyperfine structure
5. The molecular structure
• The Born-Oppenheimer approximati