Syllabus - PHY-401 (Spring
'23)
Chapter
1: Introduction to Progamming and Numerical Solutions (Jan 18,20,23)
- 1.Order differential equations
(DEQs) and Euler method; nuclear decay
- Code
design and construction
- Code
debugging+checking; numerical parameters and accuracy
- Runge-Kutta Method
Chapter 2: Drag Forces in 1- and 2-D Motion (Jan 25,27,30)
- 1-D:
car/bicycle with velocity-dependent force
- 2-D: smooth cannon shell
- Effects of height, turbulence, wind,
spinning
Chapter 3: Oscillatory Motion and Chaos (Feb 01,03,06,08)
- 2.Order
DEQ: Simple Harmonic Motion, Euler-Cromer method
- Driven Harmonic Motion,
transition to Chaos, Lyapunov exponent
- Bifurcation, Logistic Map
- Billard
problem
Chapter 4: The Solar System (Feb 10,13,15,17)
- Kepler's laws in cartesian
coordinates
- Elliptic orbits and stability of the
inverse-square force law
- Correction to 1/r2-orbits: general relativity, 3-body;
least-square fit
- 3-Body problem, Kirkwood Gaps
Chapter 5: Potentials and Fields (Feb 20,22,24,27)
- Partial differential equations
(PDEs): Laplace equation and relaxation algorithm
- Inclusion of charges: Poisson
equation
- Magnetic fields (wire, solenoid),
numerical integration
Chapter 6: Waves and Spectral Analysis (Mar 01,03,06,08)
- Ideal wave equation (in
space-time, DEQ), stability
- Frequency spectra, (fast) Fourier analysis, power
spectrum
- Realistic string: stiffness and friction
- Spectral methods
---- Midterm Exam: Mar 10 in class
----
Chapter 7: Random Systems (Mar 20,22,24,27,29,31 Apr 03,05)
- Random
number generators
- Random Walk, Self-Avoiding Walk, Flory Exponent
- Diffusion equation, entropy
- Cluster growth
- Fractal dimensions
- Percolation and 2. order phase transition
Chapter 8: Statistical Mechanics, Ising Model and Phase
Transitions (Apr 10,12,14,17)
- Ising Model, Mean-Field Theory
- Monte-Carlo method
- Ising model and 2.order phase transition, critical exponents,
correlation function
- External magnetic field, 1.order phase tranition
Chapter
9: Molecular Dynamics (Apr 19,21,24,26)
- Dilute gas, equations of motion,
Verlet method
- Boundary conditions and initialization
- Equilibrium and equipartition theorem
- Solidification and melting transition
Chapter 10: Quantum Mechanics (Apr 28, May 01,02)
- Time-independent Schroedinger
equation (SEQ)
- Numerical solutions (shooting+matching method)
- Matrix methods
- Energy minimization and variational approach