OPTI 570A
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Quantum Mechanics (3). This is a one-semester course designed to provide students with a solid understanding of quantum mechanics formalism, techniques, and important example problems. With this background, students will be prepared for subsequent in- depth studies in optical physics, quantum optics, relativistic quantum mechanics and other advanced quantum mechanics topics, condensed matter physics, laser physics, and semiconductor physics and optics. The course emphasizes a formal mathematical treatment of quantum mechanics, and is therefore intended for students who have already completed at least a one-semester course in quantum mechanics where the basic concepts, symbols, and mathematical approaches have been introduced.  

Course Objectives:
OPTI 570A is intended for students who have the following goals:

  1. learning formal techniques for solving problems in quantum mechanics (and related areas) for experimental and theoretical research;

  2. learning advanced quantum mechanics and quantum optics techniques and concepts, as would be encountered in PHYS 570B and OPTI 544 (for example) or other advanced courses.

Prerequisites
It is expected that students enrolling in this course have already studied the following topics in an introductory quantum mechanics course:
  • deBroglie wavelength of a particle
  • Schrodinger’s equation
  • energy eigenstates of example potential wells (particle in a box, harmonic oscillator, hydrogen atom)
  • Dirac notation (not essential, since this is not always covered in a one-semester undergraduate quantum mechanics course)
  • Commutators and operator algebra
  • Angular momentum in quantum mechanics (spin, electron orbital angular momentum)
However, it is not necessary for students to be experts already in these preceding topics, as they will be covered in detail in OPTI 570A.

It is also expected that students are familiar with the following mathematical concepts:
  • matrix and vector multiplication
  • finding the eigenvalues and eigenvectors of simple matrices
  • working with complex numbers
  • basic formalism of Fourier transform integrals
  • differential equations (although not necessarily the various means to solve a wide range of equations)
Students will be expected to review on their own (as needed) these or similar background topics that will be used in OPTI  570A.

Course Expectations
OPTI  570A involves required reading assignments (at least weekly) in which students will be challenged to learn many of the intricate details of quantum mechanics from the required textbook (which will be closely followed throughout the semester). Challenging homework problems, both required and optional, will be assigned almost weekly (but not all problems will be graded).

Class time will be focused on
  • summarizing and explaining the topics covered in the textbook,
  • discussion of concepts that are unclear or difficult to understand,
  • working example problems, and
  • discussion of material and applications not covered in the textbook.
Students will be expected to fully participate in classroom discussions.

Class schedule: Three 50-minute lectures per week (3 credit-hour course).

Grading
Final course grades will be based on a 100-point scale, with the following percentage breakdown:
  In-class participation: 15%
  Homework sets and special assignments: 15%
  First Mid-term exam: 20%
  Second Mid-term exam or project: 20%
  Final exam: 30%

Each student’s final course grade will be based on the total points accumulated over the semester. A grade of “A” will be given for 90 – 100 total points, “B” for 80 – 89 points, “C” for 70 – 79 points, etc. Extra credit points may be given for the completion of certain assignments, but should not be expected. Grading will not be done on a curve: you should not feel like you are in competition for grades with other students in this course.

Required Textbook:
Cohen-Tannoudji C., Diu B. & Laloë F. (1992). Quantum Mechanics (vol. 1 & 2). Wiley & Sons.
ISBN 9780471569527 | A used copy of either the first or second edition is OK

Topics
OPTI  570A aims to cover the following topics (numbers in parentheses indicate approximate number of 50-minute lectures for each topic):
  1. Mathematical formalism I. State space and state vectors, scalar product, Dirac notation. Linear operators, Hermitian operators. Representations and bases. Eigenvalue equations, observables, commuting observables. Position and momentum operators. Unitary operators and unitary transformations, transformation between representations. (5)
  2. Basic postulates of quantum mechanics. Physical implications, Copenhagen interpretation of quantum mechanics, wavefunctions and wavepackets, evolution operator, Schrödinger and Heisenberg pictures. (4)
  3. The harmonic oscillator. Creation and annihilation operators, operator algebra. Solution of the eigenvalue problem. Stationary states in position and momentum representations. Quasi-classical states, time evolution of expectation values, comparison to classical harmonic oscillator. (5)
  4. Angular momentum. Commutation relations. Raising and lowering operators, operator algebra. Solution of the eigenvalue problem. Orbital angular momentum, spherical harmonics. Addition of angular momentum, Clebsch-Gordan coefficients (4)
  5. Spin.-1/2. Stern-Gerlach experiment, Pauli matrices, Bloch picture. Rotations and the spin-1/2 problem. Rotations and time evolution for a two-level system. (8)
  6. Mathematical formalism II. Tensor product of state spaces, separable and entangled states, the density operator, pure and mixed states. (2)
  7. Hydrogen atom and atomic structure. Eigenvalue problem for the central potential, separation in angular and radial equations. The hydrogen atom, solution to the eigenvalue problem. Spectroscopic notation. (3)
  8. Stationary perturbation theory. Perturbation equations. Non-degenerate perturbation theory. Degenerate perturbation theory. (3)
  9. Fine structure and other perturbations to the Hydrogen problem. Fine structure of the n=2 shell in hydrogen. Hyperfine structure. Introduction to Zeeman effect and Stark effect. (3)
  10. Time-dependent perturbation theory. Perturbation equations, solution to first order, transitions between discrete states, limits of validity, relationship to the exact 2-level problem, example problems. (4)