2012 Cottrell Scholars Conference
Problem Solving in the Real World :
Estimation & Approximation Methods
Kyle Shen
Department of Physics &
Kavli Institute for Nanoscale Science
Cornell University
Friday, July 13, 12
issue to address :
most “real world” problems either in research or industry do not have
simple analytical solutions
•
traditional curricula typically lack classes which provide exposure and
the proper tools for attacking such kinds of problems
•
proposal :
develop a class which will provide students with approximation and
numerical problem solving tools
•
•
apply these various tools to “live” problems
Friday, July 13, 12
outline of class
fits into existing curriculum as Physics 3317 : Applications of Quantum
Mechanics (currently an assortment of atomic & molecular physics,
solid state physics, nuclear physics, quantum statistics; no formalism
taught!)
Proposed structure
1. Back-of-the-Envelope Calculations (1-2 weeks)
2. Introduction to Numerical Tools (2-3 weeks)
3. Application to problems (primarily in atomic, molecular, solid-state
physics) (7-9 weeks)
Friday, July 13, 12
back-of-the-envelope calculations
Simple estimations (i.e. Fermi problems)
how many piano tuners are there in Chicago?
about 200 (350 according to Wolfram Alpha)
Dimensional Analysis
what is the approximate radius of a hydrogen atom?
~
a0 ⇡
me2
Friday, July 13, 12
(in CGS; about 0.5 angstroms)
numerical & approximation techniques
Want students to learn to write simple code & simulations without needing
to learn a complex programming language
Possible Platforms
•
MATLAB (commercial, existing familiarity, built-in GUI and plotting)
•
Python (free, simple syntax, installation and library issues)
suggestions are welcome!
Friday, July 13, 12
a simple example
You are in Vegas and down to your last $100, but
need to win another $100 to get home. You decide
to play roulette (betting on red or black) to win the
$100 needed.
How should you apportion your bets to maximize
your chance of winning the needed $100?
18 reds,
18 blacks,
1 green
simulation : 100,000 trials
betting...
Friday, July 13, 12
$100
per spin
$20
per spin
$5
per spin
$2
per spin
49%
43%
25%
6%
a more advanced problem
Using electron beam lithography, your roommate
fabricates a “nanocross” which can be used to
confine an electron.
What is the ground state energy of an electron
confined within the cross?
a first attempt : back-of-the-envelope calculation
Heisenberg Uncertainty Principle
x p⇡~
Solution to Infinite Square Well
~ ⇡
=
mL2
2 2
E2D
Friday, July 13, 12
p2
E⇡
2m
better estimates using numerical techniques
Variational Wavefunctions
(x, y) = f (x, y){a, b, c, d...}
construct an analytic function with the correct
boundary conditions with variational parameters
{a,b,c,d....} and minimize the ground state energy
Combining Monte Carlo & Variational Methods
•
discretize a “guess” wavefunction into a 2D mesh
•
numerically calculate the ground state energy
change each ‘pixel’ value randomly; use Monte Carlo algorithm to keep
or reject change based on change in energy
•
•
repeat and iterate
Friday, July 13, 12
the “dream outcome” of this project
solving of “real world” problems are implemented throughout the
curriculum
•
dedicated and required class for development of numerical tools &
methods
•
•
“trickle down” effect to later classes (i.e. advanced lab)
•
adoption by other institutions
Friday, July 13, 12