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Graduate Quantum Mechanics

Text Syllabus

Meeting Time

CHM 6938-005 (Quantum Mechanics I) will meet Mondays and Wednesdays from 10:45am-12:00pm in NES 104.

The instructor may be reached anytime by phone 4-4298 or email (username: davidrogers on

Course Overview & Objectives

Nanoscience has already reached its natural quantum limit, bringing the 'spooky' behavior of quantum systems into the center of a revolution. This course will rigorously develop the fundamentals of quantum mechanics required for understanding the interplay of electricity and magnetism with electronic structure.

By the end of the course, you will be able to determine which problems in chemistry and physics have an essentially quantum nature. For those that do, you will know the rules for formulating the solution mathematically and the tools and special tricks for completing the solution. You will also learn aspects of numerical solutions and partial differential equations that apply to many aspect of modern technology outside of QM. As a research course, you will be expected to finish with a high-level understanding of where to find worked projects similar to new and open questions, and to mix, match, and evaluate solution techniques before diving into differentiation or trial and error.

Grading & Due Dates

Your work will be graded based on homework assignments (80%) and one student project (20%).

  • Homework 1: Problems from Bes, 2007 Chapter 2 - due Monday, September 9
  • Homework 2: Problems based on Bes, Ch. 4, 5 and 9 - due Monday, October 7
  • Homework 3: Worked Examples of the Schrodinger Equation - due Monday, October 28
  • Homework 4: Introduction to Group Theory - due Monday, November 18

Some Useful Preliminaries

Partial differential equations, linear algebra, Fourier series

These will be reviewed in the course, but undergraduate calculus is required.


  • Quantum Mechanics: A Modern and Concise Introductory Course, Daniel Bes (Springer, 2007 (2nd ed) or 2012 (3rd ed))

Bes gives an excellent survey of the myriad applications of QM, and is detailed enough to provide all the tools and intuition needed to do the math. It does as well as anyone should expect in getting at the whys and hows of the confusion that is quantum, including various manifestations of the measurement problem.


  • McQuarrie (Quantum Chemistry, 2nd ed., Univ. Sci. 2007)

McQuarrie focuses in on molecular electronic structure, providing a chemistry compliment to Bes' physics-oriented book. It covers what we need to understand molecular computations, including spectroscopy, thermochemistry, Hartree-Fock and Slater determinants, molecular orbitals, density-functional theory.

Leung and Marshall have written a solutions manual, Problems and Solutions for McQuarrie's QM, Univ. Sci., 2007.

Both these texts are available at the library (possibly in course reserves), and purchasing them is not required for the course.

Other QM Texts

  • Pauling (free book on
    • Early, but presciently maps out the next ~30 y. of developments
  • Atkins
  • Griffiths
  • Levine
  • Shankar (Principles of QM, 2nd ed., Plenum 1994)
    • Introduces Quantization using commutation...
  • Landau & Lifschitz 3 & 4
  • Advanced Quantum Mechanics, Dick Rainer (Springer, 2012) - chem/phys.
  • Bes, QM 2012 edition

Course Outline

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(01) Aug 26

Class intro., failures of classical mechanics, Planck's hypothesis. Notes

Weeks 1-4 correspond roughly to Bes' Chapters 1-4 and Chapter 13 on the history of QM. Pages 11-28 of the DOE Grand Challenges report also make for great related reading.

(02) Sep 2 (Mon. = Labor Day)

Review of the Schrodinger equation, Heisenberg states and all that. Basic linear algebra. Typed Class Notes

(03) Sep 9

Eigenvalue decompositions and measurements in QM.

(04) Sep 16

Worked solutions to HW1.

(05) Sep 23

One-dimensional problems - review of the potential well. New applications for the Helmholtz equation - bound and scattering states. Variational solutions. Interspersed with background on group projects.

(06) Sep 30

Applications of the Helmholtz equation, continued - tunneling, radioactive decay and diffraction. Time-dependent perturbation theory and the interpretation of off-diagonal matrix elements. This covers material in Bes, Ch. 9.

(07) Oct 7

Review of HW2.

(08) Oct 14

Mathematics of rotation and solution of the 2-body Kepler problem. Application to rotational (far infrared) spectroscopy. Week 8 corresponds roughly to Bes' Chapter 5 and rotational spectroscopy in McQuarrie.

(09) Oct 21

Review of the harmonic oscillator and hydrogen atom, introduction to multi-particle statistics. Week 9 corresponds roughly to Bes' Chapters 6-7.

(10) Oct 28

Introduction to molecular symmetry operations. Electronic (UV) and vibrational (IR) polarizability. Molecular point groups and symmetry properties. Your reference for the group theory material is McQuarrie's book.

(11) Nov 4

Numerical solution of Schrodinger's equation and QM software intro. NWChem

Review of HW3 solutions.

(12) Nov 11 (Mon. = Veteran's Day)

Group theory and spectroscopy review.

(13) Nov 18

Example group project presentation - EPR paradox, the Bell inequalities, and Bohm's suggestion.

(14) Nov 25 (Fri. Nov 29 = Day after Thanksgiving)

Nov. 25 - Understanding HF output and visualizing orbitals with NWChem.

Nov. 27 - Variational calculation of H2 energy.

(15) Dec 2

Dec. 2 - Transition state theory and numerical solution of Schrodinger.

Dec. 4 - Working problems with the class.

(16) Dec 9 - final exam week (no class)

Coming up next semester:

Running large parallel electronic structure calculations and plotting molecular orbitals. The role of basis functions and convergence.

Electronic polarization and its role in UV absorption and dispersion forces.

QM/MM methods applicable to the condensed phase - the many representations of solvent.

Basic statistics of Boson and Fermion energy distributions - statistics on top of statistics.

Statistics of an electron gas. The Kohn-Sham decomposition and the resulting alphabet soup of density functionals.

Thermochemistry, chemical reactions and kinetics.

Feynman's path integral representation of QM - derivation of classical mechanics, Heisenberg and Schrodinger. Elementary path integrals, and the concepts of coupled-cluster perturbation.

Emerging research areas in molecular physics simulation and statistics.

Optional material: Quaternion representation of rotations and the Dirac equation.

List of Group Study Topics

  1. Einstein, Podolsky and Rosen's paradox and Bell's inequality.
  2. Bohm's quantum interpretation of quantum mechanics (see also Bell's later response)
  3. Gaussian-type orbitals, exponents, and integrals[1] for the H<math>_2</math> atom.
  4. Calculating electron hopping rates using the Non-equilibrium Green's function method
    • see Keldysh, "Diagram Technique for Nonequilibrium Processes," Keldysh, Sov. Phys. 1965. and Ch. 10 of Physical Kinetics, Lifshitz
  5. Eyring's transition-state-theory[2] [3] and reaction diagrams using Marcus Theory lit ref: Warshel and Parson [4].
  6. Formulation of QM expectations using path integrals
  7. Numerical solution methods for 1D problems.

Choose an interesting topic in groups of no more than 3 students by Friday, Oct. 4.

Group Assignment Instructions