### CHEMISTRY 680 Syllabus Spring, 1999

```Review of classical mechanics

Newton's second law
Generalized coordinates
Lagrange's equations (from N's laws)
Conjugate momentum
Canonically conjugate variables
Hamiltonian (as Legendre transformation)
Poisson brackets
{ , } of canonically conjugate variables
Equations of motion in Poisson bracket form
Time dependence of an observable in Poisson bracket form

Quantization

Poisson brackets & commutators
Commutators of canonical pairs
Operators
Representations of operators
Construction of operators for observables
Hamiltonian operator
State spaces
Function spaces
Vector spaces
Dirac notation
closure, etc
Hermitian operators
Unitary operators
x and p representations, delta functions
Relation between kets and wave functions
Fourier transforms in x and p
Heisenberg equations of motions
Time development operators (propagators)
Schroedinger equation for U
Schroedinger equation for functions and vectors
Stationary states
Eigenstates, eigenvalues
Completeness
Expectation values
Pictures ("Representations")
Schroedinger
Heisenberg
Interaction
Others

Solution of the time-dependent Schroedinger equation
Power series expansion
Time-dependent perturbation theory
Magnus (and etc.) expansions
polarizability

Harmonic Oscillator

Operator (matrix) approach
Creation and annihilation operators
Eigenvalue spectrum
Matrices and matrix elements
Harmonically driven harmonic oscillator
Relationship to classical harmonic oscillator

Angular momentum

Operator (matrix) approach
Raising and lowering operators
Eigenvalue spectra
Matrices and matrix elements
Pauli matrices
Symmetric top
Configuration space eigenfunctions (from matrices)

How to quantize a new system

Lagrangian formulation
Examples
Particle on a helix (optical activity)
Particle in a (classical) E-M field
Coulomb and Lorentz gauges
Goeppert-Mayer transformation
Quantization of the E-M field
field modes
photons

Jaynes theory (wrong, but cute and instructive)
Einstein "A" coefficient

Rotating wave approximation

Dynamics of a two-level system
Rabi theory

Formal Hartree-Fock theory

Fock space
Fermi creation and annihilation operators
One- and two-electron operators
Hamiltonian in Fock space representation
core electrons
Fock matrix
definition of the Hartree-Fock orbitals
Brillouin's theorem
Koopman's theorem

Applied ab initio theory

Practical Hartree-Fock
Configuration interaction
Coupled-Cluster theory
Density Functional theory

Exams
Midterm, 11 Mar (tentative)
Final, 11 May, (alternate, 7 May)

Homework
12 - 14 problem sets

20% midterm
30% final
50% problem sets

References

The following books are on reserve in the Main Library

Our text book is:
C. Cohen-Tannoudji, et al. Quantum Mechanics, vols 1 and 2, QC 174.12 C6313

Other useful references:
H. Goldstein, Classical Mechanics, QA 805 G6

A, Messiah, Quantum Mechanics vols 1 and 2, QC 174.1 M413 v1 & v2

P. A. M. Dirac, Principles of Quantum Mechanics, QC 174.3 D5 1958

H. Eyring, et al, Quantum Chemistry, QD 453 E9

L. Schiff, Quantum Mechanics, QC 174.1 S34

S. Schweber, An Introduction to Relativistic Quantum Field Theory, 530.12 S412
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