Cornell University

Department of Physics

Phys 480/680, Astro 690 February 16, 2007

Computational Physics, Spring 2007

Homework Assignment # 3

(Note change due to snow: Due Mon, February 26 at 5:00pm.)

Agenda and readings:

Readings marked NR are from Numerical Recipes: The Art of Scientific Computing, 2$^{\mathrm nd}$ edition (in C). Readings marked LN are from the course lecture notes to be found at http://www.muchomas.lassp.cornell.edu/P480.
Goal: This problem set develops a fully automated solver which finds all of the energy states for any specified external potential ``V[]'' to high precision.

Readings marked NR are from Numerical Recipes: The Art of Scientific Computing, 2$^{\mathrm nd}$ edition (in C). Readings marked LN are from the course lecture notes to be found at http://www.muchomas.lassp.cornell.edu/P480.

Recap of last week:

• Lec 5 02/06 (Tue): Solution of ODE's in standard form; 1st order, 2nd order, and 4th order methods; correspondences to numeric integration; DFT minimization problem, constraints, real-valued functions of a complex variable
  Reading: NR 16.1.

• Lec 6 02/08 (Thu): Calculus of variations; derivation of Kohn-Sham equations; overall plan for solution; Schrödinger's equation for atoms, spherical solutions; Mathematical properties of effective Scrödinger equation.
  Reading: LN: ``Overview of quantum mechanics ...'': 4,5

• Lec 7 02/13 (Tue): Schrödinger's equation for atoms (spherically symmetric solutions); boundary value problems; standard form for root finding; bisection; convergence rates; brackets for Schrödinger's equation; overview of coming methods; begin secant/false position methods
  Reading: NR 17.0-17.1, 9.0-9.1

• Lec 8 02/15 (Thu): Cancelled due to snow.

Preview of coming lectures:

• Lec 8 02/20 (Tue) Analysis of convergence of secant method; Ridders method
  Reading: NR 9.2

• Lab 2 02/20 (Tue) IEEE arithmetic; convergence of solution of ODE's; numerical solutions of Schrödinger's equation.

• Lec 9, 02/22 (Thu) Numerical interpolation; Numerical stability of solution of ODE's, shooting to a fitting point; Non-spherical states (higher angular momenta).
  Readings: NR 17.2, 3.0-3.2.

• Lec 10, 02/27 (Tue): Normalization conventions for atoms; Calculation of total energy in density functional theory; Functional form for exchange-correlation energy.
  Readings: Sections ``Introduction,'' ``Procedure'' (you can skip the relativistic part), ``Results,'' and ``The LDA approximation,'' from
  http://physics.nist.gov/PhysRefData/DFTdata/contents.html

• Lab 3, 02/27 (Tue): Stability of solutions, effectiveness of root finders; convergence of eigenvalues with N.

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