Course Outline
Computational Quantum Mechanics,
Physics 498CQM, Spring 2001
Version of January 2001 --- may be modified as semester progresses
PART I: Introduction and Basic Methods
Overview - See notes and Thijssen, Ch. 1
Computational Aspects.
Basic Numerical Methods.
(Thijssen, Appendix; see also Koonin Ch. 1 and 2.)
PART II: Single-Particle and Many-Independent-Particle Problems
Solving the time-independent Schr. Eq. in 1 dimension
(radial or linear)
(ordinary diff. eq.)
- Scattering from a spherical potential
(Thijssen, Ch. 2; see also Koonin, ch. 4)
Born approximation
Application: scattering of H from Kr
- Bound State solutions
Solution using "shooting" method
- Semiclassical approximation (may be omitted)
- Applications: vibrations of bound H-Kr, electron in a quantum well
Solving the single-particle time-dependent Schr. Eq.
(partial diff. eq.)
- Unitary discrete step methods that conserve energy
- 1-dimensional examples of evolution
Hartree-Fock theory
- General variational theory for many particles
- Symmetry of the wavefunction; Exchange for Fermions
- Koopman's theorem
- Restricted/unrestricted theory applied to atoms
"Hartree-Fock" theory for Bose condensates
- Recent interest in Bose condensates
- Reduces to self-consistent equation in the Hartree approximation
Hartree-Fock solution for atoms
- Working programs for atoms
- Examples of solutions for ground and excited states
- Comparison of energies from Koopman's theorem and self-consistent
solutions ("Delta H-F")
Density Functional Theory
- Functionals in quantum mechanics
- The Hohenberg-Kohn Theorem for many-body interacting systems
- Kohn-Sham approach for solving for ground state of
a many-body interacting particle system as a self-consistent problem
of many-non-interacting particles
- Methods for self-consistency
- Application to the atomic problem
- Modifying Hartree-Fock program for Hartree-density-functional calculations
- Matching logrithmic derivative x=d (log psi)/dr for wavefunction
- Theorem relating dx/dE to integrated charge
- Algorithm for solving radial Schr. Eq.
Scattering theory, phase shifts, and ab initio pseudopotentials
- Calculation of cross sections, transport
- Eliminating core electrons to define pseudopotentials
- Modifying atomic program for pseudopotentials (may be omitted)
Solving Schroedinger Eq. in a basis: Matrix operations
- Comparison of Finite difference discrete algorithms with matrix representations in a basis
- Variational properties
- Basis functions for atomic calculations. (Gaussian, Slater)
- Calculations for molecules (and atoms) -- Beyond this course to construct general programs
- Calculations with Gaussian program "Quantum Chemistry" package
Independent electrons in crystals
- The reciprocal lattice and Brillouin Zone
- The Bloch theorem for excitations in a periodic system
- Solution for electrons in solids in a plane wave basis
- Construction of a working program for
crystals, using pseudopotentials and matrix diagonalization
- Approximate solutions using the Harris functional
- Discussion of full self-consistent solutions
Iterative methods
- General iterative methods for eigenvectors, related to time dependence
Thermal simulations of matter
- Car-Parrinello molecular dynamics equation of motion.
PART III: Many-Interacting-Particle Problems
The problem: basis sizes which grow as N!
Prototype many-body problems
- Hubbard and Anderson model for electrons
- Heisenberg model for spins
The Lanczos method (exact diagonalization) for the lowest states
- Applications to finite temperature statistical quantum mechanics
(may be omitted)
Monte Carlo methods
- High dimensional integration by statistical sampling
- Variational Monte Carlo with a trial correlated wavefunction
- Slater-Jastrow two-body correlations
- Gutzwiller wavefunctions for the Hubbard model
- Random walks, the Metropolis algorithm.
Estimation of correlation and convergence
- Diffusion Monte Carlo methods
- Exact solution of hydrogen molecule
- Comparison with Hartree-Fock, DFT (using Gaussian program)
- Discussion of the "sign problem" for Fermions
- Bose Condensates
Perturbation Methods
- The idea of quasiparticles and Fermi liquid theory
- Calculation of quasiparticle energies using low-order perturbation
expressions with approximate dynamically screened interactions
- The GW method for quasiparticle energies
- Simplifications in the limit of large degeneracy and large dimension
Other
Student project reports.
Special Topics which may be included at intervals during course
- Perturbation theory: 2n+1 theorem
- Optimization Techniques;
Global minimization methods such as conjugate gradient and simulated annealing.
- Variational Quantum Monte Carlo calculation on a lattice (e.g.
Hubbard Model) or continuum problem (e.g. 4He)
- Lattice Gauge Calculation and Theory
- Relativistic models, Klein Gordon Equation for atoms
- Berry's phases and polarization in extended systems
- Charged particles in magnetic fields
- Quantum Hall Effect
- Particles in random potentials - disorder
- Simple quantum system coupled to heat bath
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