This is a two-semester graduate course that covers most of the essential topics of modern non-relativistic quantum mechanics. The course is primarily intended for graduate students with background in Physics and aims to prepare such students for taking further advanced courses in Physics and Material Science offered in OIST, such as Solid State and Condensed Matter Physics, Advanced Quantum Mechanics, Advances in Atomic Physics, Quantum Field Theory, etc.
Students who take this course are expected to be familiar with general topics in Classical Mechanics, Electrodynamics and Calculus.
Quantum Mechanics I
1. Early crisis of classical physics: black body radiation and the “ultraviolet catastrophe”. Plank’s hypothesis. Einstein’s explanation of photoelectric effect. Bohr’s model of hydrogen atom.
2. Brief review of analytical mechanics: Newtonian mechanics and conservation laws, constrains and Lagrange reformulation of classical mechanics. Hamiltonian formalism. Poisson brackets and canonical transformations. The Hamilton-Jacoby equation.
3. Brief review of classical electrodynamics: Maxwell equations and boundary conditions, effect of continuous medium, propagation of electromagnetic waves. Ray optics and eikonal approximation. Charged particle in electric and magnetic fields.
4. Motivations for postulates of quantum mechanics: Young’s double-slit experiment. de Broglie’s hypothesis of matter waves.
5. Bra-ket formalism, Hilbert space, operators, and their matrix representation. Postulates of quantum mechanics. General uncertainty relation.
6. Canonical transformation in quantum mechanics as a main approach to describe motion of a physical system. Translation in space and operator of momentum. Coordinate and momentum representations. Coordinate-momentum uncertainty relation and the Standard Quantum Limit.
7. Time-evolution operator. Energy-time uncertainty relation. The Schrodinger equation of motion and continuity equation. The Heisenberg picture and equation of motion for operators. The Ehrenfest theorem.
8. Some exactly solvable problems in wave mechanics: particle in free space and motion of the Gaussian packet, particle in the box, linear potential, potential barriers and tunneling.
9. Quantum harmonic oscillator: two approaches in solving the problem, coherent and squeezed states of the quantum harmonic oscillator.
10. The WKB approximation. Feynman’s path integral and classical limit of the quantum mechanics.
11. Quantum particle in static electric and magnetic fields. Gauge transformation and the Aharonov-Bohm effect. Macroscopic quantum coherence and the Josephson effect. Charged particle in the uniform magnetic field: Landau states and their degeneracy. The Quantum Hall effect.
12. Rotations in space and operator of angular momentum. Orbital and spin angular momentums. Coordinate representation of orbital angular momentum. Spherical harmonics.
13. The Schrodinger equation of motion in 2D and 3D. Particle in central potential: 2D and 3D rigid rotators, particle in a spherical box, 3D quantum harmonic oscillator, the hydrogen atom and emission spectrum.
14. Scattering of quantum particles from 3D potentials. Green function method and the Born approximation. Expansion into partial waves and the Optical theorem.
15. Spin-1/2 particle and the Stern-Gerlach experiment. Matrix representation of spin-1/2 states and Pauli matrices. Bloch sphere representation. Motion of spin-1/2 particle in a uniform magnetic field.
Students who take the course are expected to be familiar with general topics in Classical Mechanics, Electrodynamics and Calculus.