Chapter 10: Quantum Mechanics

Lecture 1: Wednesday February 20

• Boundary-value and eigenvalue problems
• Shooting methods and relaxation methods
• Shooting example
• Relaxation example

Lecture 2: Friday February 22

• Linear equations and the Sturm-Liouville problem
• The regular Sturm Liouville problem
• The Numerov algorithm
• Time-independent Schr\"odinger Equation
• Equation and normalizable solutions
• Boundary conditions
• Integrating the equation
• Instabilities and need for a matching point
• Matching at a turning point
• Program to find harmonic oscillator eigenvalues and eigenfunctions

Lecture 3: Monday February 25

• Time-dependent Schr\"odinger Equation
• Discretizing the Schr\"odinger Equation
• Forward Time Centered Space (FTCS) or Explicit Differencing
• Backward Time Centered Space (BTCS) or Implicit Differencing
• Symmetric Time Centered Space (STCS) Crank-Nicolson Differencing
• Implementing the Crank-Nicholson Method
• Dirichlet Boundary Conditions: Tridiagonal Matrix
• Periodic Boundary Conditions: Cyclic-Tridiagonal Matrix

Lecture 4: Wednesday February 27

• Quantum Wave Packet Scattering from a Potential
• Simplified Crank-Nicholson Formula
• Solving a Tridiagonal System of Equations
• Periodic Boundary Conditions: Sherman-Morrison Formula
• OpenGL Wave Packet Propagation Program

Lecture 5: Friday February 29

• Variational Methods
• The basic problem
• The variational theorem
• Variational Monte Carlo
• VMC Program for the Harmonic Oscillator
• The program uses $N$ Metropolis random walkers
• Variables to measure observables
• Initialization
• Probability function and local energy
• One Metropolis step
• Steering the computation with the {\tt main} function

Lecture 6: Wednesday March 5

• Variational Monte Carlo for the Hydrogen Atom
• Reduction to a one-dimensional problem
• Exact solution for the ground state
• Variational Monte Carlo for the Helium Atom
• A simple choice of variational trial wave function
• Pad\'e-Jastrow wave function
• VMC program for the Helium Atom
• Results from running the program
• Appendix: Derivation of Local Energy for Helium

Lecture 7: Friday March 7

• Spectral Method for Time-dependent Schroedinger Equation
• Discrete time approximation
• Algorithm for time evolution
• C++ FFT Wave Packet Program
• Leap-frog Method

Lecture 8: Monday March 17

• Diffusion Monte Carlo
• Connection with quantum mechanics
• Diffusion leads the system into its ground state
• Diffusion with a potential energy term
• Diffusion Monte Carlo algorithm
• Diffusion Monte Carlo program for the 3-D harmonic oscillator
• Potential energy function
• Dynamical adjustment of array storage
• One Diffusion Monte Carlo step
• One time step $\Delta t$
• The {\tt main} function to steer the calculation
• Output of the program

Lecture 9: Wednesday March 19

• Solving Linear Systems
• The problem
• Examples
• Singular versus nonsingular
• Properties of matrices
• Tasks of computational linear algebra

Lecture 10: Friday March 21

• Gaussian Elimination
• Cramer's permutation sum formula
• Reduce to upper triangular form
• Backsubstitution
• Partial pivoting
• Other Useful Linear Algebra Algorithms

Lecture 11: Monday March 24

• Basic Decompositions for Linear Algebra
• Decomposition of Non-singular Matrices
• LU Decomposition
• LU algorithms
• Cholesky Decomposition
• QR Decomposition
• Singular Value Decomposition
• Eigenvalue Decomposition
• Algorithms for eigensystems
• Matrix Eigenvalue Problems
• Left and right eigenvectors
• Normal, Hermitian, and symmetric matrices
• Similarity transformations and diagonalization
• Numerical eigensolver routines