Instructor:
E-mail: klikharev@notes.cc.sunysb.edu
Office: B-135
Phone: 2-8159
Office hours: Thu 3:00 to 5:00 pm
Preliminary Syllabus:
1. Introduction
Summary of experimental
motivations for quantum mechanics. Basic concepts of wave
mechanics; statistical ensembles; wavefunctions and probability. The 1D Schrödinger equation,
its general solution by linear superposition. Eigenvalues and
eigenstates; discrete and continuous spectra; confined and unconfined 1D
motion. Differential linear operators, expectation
values. Operator of momentum; Heisenberg's uncertainty
relation. Continuity equation, probability current.
2. 1D quantum particle
Plane waves, wave packets. Reflection from a potential step. Tunneling through delta-functional and rectangular barriers.
1D scattering and transfer matrix. Resonant tunneling.
Motion in periodic potentials; Bloch theorem, energy bands
and gaps. Rectangular quantum well. WKB approximation, classical turning points, the Bohr-Sommerfeld
quantization rule. Double quantum well, the Kimble formula.
Propagator, Feynman path integral. Metastable
states and their decay. Quantum oscillations in double
well potentials. Harmonic oscillator, the Fock (stationary) states.
3.
Bra-ket formalism
Bra and ket vectors. Scalar (inner) product. Linear operators, commutators and anticommutators.
Identity, adjoint and self-adjoint (Hermitian) operators.
Compatible and incompatible observables. Orthonormal sets and matrix formalism. Outer
products and projection operators. Change of basis. Matrix
diagonalization. Coordinate operator, reduction to wave mechanics.
Hamiltonian operator, The Schrödinger and Heisenberg pictures of quantum
dynamics. The Ehrenfest theorem. Back to the harmonic
oscillator: creation and annihilation operators, the Fock states, the Glauber
(coherent) states, squeezed states.
4. 2D and 3D problems
Generalization to higher dimensions.
2D and 3D harmonic oscillators, rotators and spherical
quantum wells. Symmetry at rotation, angular momentum;
Bohr’s atom. Partial quantum confinement, two-slit
interference description. Motion in EM field; the Aharonov-Bohm
effect; the Landau levels. 2D and 3D scattering
characterization. The Born approximation, optical
theorem, eikonal approximation. Partial phase method,
hard sphere scattering, resonant scattering.
5. Perturbation theories
Constant perturbation in
non-degenerate and degenerate systems; anharmonic oscillator, Stark effect. Back to the coupled quantum wells. Time-dependent
perturbation theory; Rabi oscillations. Transitions in
continuous spectrum, the "Fermi" Golden Rule.
6.
Spin
Insufficiency of wave mechanics. Spin operator; the Stern-Gerlach
experiment and its description. Spin dynamics. The coupled quantum
wells again; qubits and quantum computing. Spin addition to orbital
momentum; Clebsh-Gordan coefficients; Zeeman effect.
7. Open systems, quantum statistics, and
quantum measurements
Coupling to environment. Pure
and mixed quantum states. Density matrix. Classical mixture in thermal equilibrium. The Wigner
function. Density matrix dynamics without and with interaction with
environment, dephasing. Quantum measurements and ensemble
redefinition. QND. The Bayes
theorem.
8. Identical particles
Permutation symmetry,
indistinguishability principle, bosons and fermions. Two-electron
systems, singlet and triplet states, helium atom, covalent (chemical) bond.
Atoms, periodic table of elements. Second
quantization for bosons and fermions, Fermi gas of interacting electrons.
Hartri and Hartri-Fock approximations.
9. Quantum theory of EM field
Electromagnetic field modes and
their quantization. The Casimir effect. The notion of photon; its energy, momentum, and angular momentum.
EM field statistics, coherence, 2nd order correlation functions, photon
bunching and antibunching. Quantum EM field interaction with charged particles.
Spontaneous and induced transitions, rate of electric dipole transitions, the Einstein
coefficients.
10. Quantum theory of relativistic particles
The relativistic Schrödinger equation,
particles and antiparticles. Dirac equation, introduction of
spin. Relativistic Fermi particles in EM field, spin-orbit interaction,
application to atomic spectra. Relativistic theory of the hydrogen
atom.
11. Quantum field theory preview
Recommended textbooks:
E. Merzbacher, Quantum
Mechanics, Wiley, 1998
L. Landau and E.
Lifshitz, Quantum Mechanics, Non-Relativistic Theory, 3rd ed.
Pergamon, 1977
J. Sakurai, Modern
Quantum Mechanics, Addison-Wesley, 1994
Lectures:
Twice a week (1 hour 20 minutes each)
Homeworks:
Weekly (with just a few exceptions); completed work due in a week
after the assignment
Exams:
Two midterm exams (1
hour 20 minutes each) and a final (2 hours 30 minutes)
All
exams: books open
Final grade components:
Homeworks: 15%
Midterms:
25+25%
Final
exam: 35%
Logistics for Fall 2008:
Lectures: Tue-Thu
11:20-12:40, Rm. P-112 (first
lecture: Tue Sept. 2)
Grader: TBA
Exams
(all in the lecture room):
midterms: preliminary, Oct. 14 and Nov. 11 (lecture time)
final: Dec. 11 (11:00 am – 1:30 pm)
Americans with Disabilities Act: If you have a physical, psychological, medical or learning
disability that may impact your course work, please contact Disability Support
Services, ECC (
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his or her academic goals honestly and be personally accountable for all
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Faculty are required to report and suspected instances of academic dishonesty
to the Academic Judiciary. For more comprehensive information on academic
integrity, including categories of academic dishonesty, please refer to the
academic judiciary website at http://www.stonybrook.edu/uaa/academicjudiciary/ .
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Management: Stony Brook University expects students to respect the rights,
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Office of Judicial Affairs any disruptive behavior that interrupts their
ability to teach, compromises the safety of the learning environment, or
inhibits students' ability to learn.