Quantum Mechanics
Instructor: Dr. B. Mitrovic
Textbook:

There is no required text for this course. The library has many texts on this subject and I leave it to students to pick their favorite one. I don't follow closely any particular text (perhaps Sakurai's Modern Quantum Mechanics is closest in spirit to my lecture notes) and I view the textbook as a complement to the lecture material. A copy of my lecture notes will be available to students in the main office (MC B210).

Syllabus:

Postulates about (pure) states, observables, probabilities, change of state in a filtration measurement, quantization of a classical system and the time evolution of a quantum mechanical system.

Dirac's bra and ket notation; representation and transformation theory (coordinate and momentum representations as the important special cases); the eigenvalue problem and the spectral form of a Hermitian operator (observable); general Heisenberg's uncertainty relations; commuting observables and a complete set of commuting observables-classification of states in terms of the compatible eigenvalues of the observables from a complete set; solution of the eigenvalue problem for general angular momentum; analytic functions of operators (definition in terms of the power series expansions) and functions of Hermitian operators (definition in terms of a spectral form of a Hermitian operator); Baker-Hausdorff theorem and some other important operator identities; commutator algebra; symmetries; angular momentum as a generator of spatial rotations, momentum as a generator of spatial translations; position operator as a generator of boosts; Hamiltonian as a generator of translations in time; using symmetries in solving the eigenvalue problem of a Hamiltonian; stationary states and the solution of a time-dependent Schrödinger equation for a conservative system; solution of a time-dependent Schrödinger equation for a two-state system in a harmonic field; operator method for simple harmonic oscillator; nondegenerate time-independent perturbation theory.

Most of the material is illustrated in the case of a two-state system (spin-1/2, ammonia molecule, benzene molecule, H2+-ion). Stern-Gerlach experiment is used as a prototype for filtration measurement. How symmetries are used to simplify the solution of the eigenvalue problem of a Hamiltonian is illustrated by solving the eigenvalue problem of an electron moving in a one-dimensional lattice via the nearest neighbor hopping ($\hat{H}=\sum_{n}(\epsilon_{0}\vert n\rangle\langle n\vert-t\vert n+1\rangle\langle n\vert-t\vert n-1\rangle\langle n\vert)$; the case which also includes the second nearest neighbor hopping or the two-dimensional case are given as homework. The time dependence of a spin-1/2 in a uniform magnetic field is examined in detail (the time dependence of other two-state systems is assigned as homework). The basic principles of the Electron Spin Resonance (ESR) and Nuclear Magnetic Resonance (NMR).

Course Policies:
  • Penalty for Late Assignments: 10% deduction per day.
  • A student must achieve 50% on the final exam to pass the course.
  • Note that the last day to withdraw without academic penalty is Monday, Nov. 6, 2017.
Marking scheme:

10 assignments (45%), 1 midterm test (10%) on Monday, October 23, final exam (45%)