Classes which prepare students for research, and visa-versa

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Classes which prepare students for research, and
visa-versa

• John Boccio and Amy Bug
• Dept. of Physics and Astronomy
• Swarthmore College

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Undergraduate Student Research is a good thing
(proof forthcoming )

• Excitement of being engaged in pushing the limits
  of our current knowledge S. Brubaker-Cole,
  Stanford Univ. Newsletter, Spring 2002
• A good way to connect faculty and students
• J. Mervis, Science Aug. 31, 2001
• The goal of integrating research and education is
  to teach students to understand and apply the
  scientific method Reed College NSF-AIRE website,
  download June, 2002

Undergraduate student research in physics
(specifically theoretical/computational quantum
physics) Special benefits? Special
difficulties?
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Benefits

• Recruitment and Retention of physics majors
• R. and R. of nontraditional students
• Intellectual ones lab and classroom are
  compli-mentary settings in which to learn the
  same physics
• (We) teach students how to learn from a variety
  of sources, including books, journals,
  experiments, colleagues, modeling, and
  simulations. Encourage independent
  investigation. - Harvey Mudd College Physics
  website, June 2002
• Proposal The earlier research is integrated into
  a students career, the better.

Difficulties
A relatively slow maturation rate under the
standard physics curriculum. When should we teach
quantum mechanics and at what level?
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There is no typical first-two-years-curriculum.
Here is one
Classical physics I, II and III (Serway and
Beichner, Wolfson and Pasachoff, etc.) followed
by ...
2XX. MODERN PHYSICS (1 credit ) An introduction
to the elementary theoretical aspects of special
relativity, quantum mechanics, atomic and nuclear
structure, and a few selected topics from solid
state physics and particle physics. The course
presents the structure of these theories and how
they differ from corresponding classical
theories, and some historical and philosophical
aspects of the theories. Class discussion and
demonstrations. Prerequisites PHY-2XX,
concurrent enrollment in PHY-2XX, or permission
of instructor. Spring semester. 2XX. APPLIED
MATHEMATICS FOR PHYSICS AND ENGINEERING (.5
credit)  An introduction to Fourier series,
series solutions to differential equations,
special functions, partial differential
equations, linear equations, vectors, matrices
and determinants, and coordinate transformations.
Emphasis will be placed on the mathematics
needed to describe physical systems. Three
lectures weekly. Prerequisites XXX. Spring
semester.
Research-friendly teaching Ideas Permute the
order teach quantum physics to freshmen Formal
techniques come first not history, or even
phenomenology Be state-of-the-art dont avoid
foundational questions Programming is a part of
every course (Computational lab within a
sophomore math methods course) Require
nontraditional activities programs, lab
projects, essays
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The Character of Physical Law teaches
freshman-level relativity and quantum mechanics
Permute the order teach quantum physics to
freshmen
texts Mermin, Moore, ...
texts various, none optimal lecture notes!
Formal techniques come first
Excerpts from The Stanford Encyclopedia of
Philosophy
The heart and soul of quantum mechanics is
contained in the Hilbert spaces that represent
the state-spaces of quantum mechanical systems.
The internal relations among states and
quantities, and everything this entails about the
ways quantum mechanical systems behave, are all
woven into the structure of these spaces,
embodied in the relations among the mathematical
objects which represent them.4 This means that
understanding what a system is like according to
quantum mechanics is inseparable from familiarity
with the internal structure of those spaces..
Graduate students in physics spend long years
gaining familiarity with the nooks and crannies
of Hilbert space, locating familiar landmarks,
treading its beaten paths, learning where secret
passages and dead ends lie, and developing a
sense of the overall lay of the land. They learn
how to navigate Hilbert space in the way a cab
driver learns to navigate his city ... Vectors
and vector spaces A vector A, written Agt, is a
mathematical object characterized by a length,
A, and a direction. A normalized vector is a
vector of length 1 i.e., A 1. Vectors can
be added together, multiplied by constants
(including complex numbers), and multiplied
together. Vector addition maps any pair of
vectors onto another vector, specifically, the
one you get by moving the second vector so that
its tail coincides with the tip of the first,
without altering the length or direction of
either ... So, for example, adding vectors Agt
and Bgt yields vector Cgt ( Agt Bgt) as in
Figure 1
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Teach algebra of states and operators using Dirac
notation
Operators act on states to produce new states
(sometimes just an eigenvalue times the old state
) O Agt Bgt
Quon states are labeled by quantum numbers like
color
E.g. PG Agt PG ( aG Ggt aM Mgt ) aG
Ggt fgt
CGgt 1 Ggt CMgt -1 Mgt
One can show that non-commuting operators exist
CTRgt C (1)Rgt CRgt (G gt/v2 - M gt/v2)
S gt and TCRgt T(G gt/v2 - M gt/v2) TSgt
(-1) S gt -S gt
Various important ideas emerge quantization,
coherent superposition, Heisenberg uncertainty
principle, measurements give one member of an
eigenspectrum, ... Students can do novel
calculations learn of, but are not dependant on
particular instantiations krazy kaons,
electrons, double slits, polarized
photons... are not asked to unlearn classical
physics
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Be state of the art dont avoid foundational
questions
QND, teleportation, macroscopic superposition ...
experiments
Describing entangled states and EPR paradox is
a culminating exercise
Bell inequality implies a statistical
prediction n(C-, P-) n(C, T) gt n(P-,
T) Identical pairs of socks are tested
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Programming is a part of the course
Socks are in entangled, Bell state
do the same thing state
Probability we observe G1, R2 1lt G 2lt RBgt2
(1/2) cos2 q
Programming, during sophomore year, is taught
using a structured environment like Matlab or
IDL, ...
Basic control flow graphics numerical
derivatives numerical integration random
number generation Monte Carlo root-finding
Solving ODEs 1D eigenvalues Fourier
transforms finite-element solution of PDEs
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Require nontraditional activities journal
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Case study involving computational physics
research Goal Simulation of Positron
Annihilation in Materials
An undergraduate student, with the aforementioned
experience in quantum mechanics and some
programming ability (or willingness to learn) is
able to participate very fully in this research.

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Motivation Positron Annihilation Spectroscopy
(PAS) provides information on size and chemical
content of pores and defects in solids

• e enters solid e selects
  electron e in Ps is picked
  two gamma rays
• and thermalizes of solid and forms
  off prematurely by with
  characteristic
• within tens of ps positronium (Ps)
  with another electron energy of 511
  keV
• e preferentially natural lifetime
  of of solid are emitted
• locates in 125 ps (para) or
• channels and cavities 140 ns (ortho)

Lifetime, t , of Ps is properly determined by
electronic density r-
t-1 p re2 c ? dr- dr r(r) r-(r-) gr-(
r-) d3(r- - r)
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Method We simulate Ps in solids via Path
Integral Monte Carlo (PIMC) and Density
Functional Theory (DFT)
simulated Ps probability density within solid
Ps chains
Codes are written by students in Fortran 9X and
run both on Linux workstation and Appleseed
Macintosh cluster using Mac MPI libraries.
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Ideas and ingredients of PIMC method (1)
research
course
Positron has state Ygt
Quon has state A gt
The overlap of the state with another one, say
Ggt, is ltGAgt, the amplitude for measuring a
green positron. amplitude2 is observable the
probability
The overlap of the state with another one, say
ri gt, is ltri Y gt, the amplitude for measuring
positron located at ri . amplitude2 is
observable the probability
You can always insert a complete set of states
into any calculated quantity ?i rigt ltri
1 (Actually, ? rgt ltr dr 1)
Ggt ltG Mgt ltM 1
but ...
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Ideas and ingredients of PIMC method (2)
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Ideas and ingredients of PIMC method (2)
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Ideas and ingredients of PIMC method (2)
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Ideas and ingredients of PIMC method (2)
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Ideas and ingredients of PIMC method (2)
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Ideas and ingredients of PIMC method (2)
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Ideas and ingredients of PIMC method (2)
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Ideas and ingredients of PIMC method (2)
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Ideas and ingredients of PIMC method (2)
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Ideas and ingredients of PIMC method (2)
ri
is the probability (density) to be found at
location ri
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PIMC method yields the thermal probability density
ri
is the probability (density) to be found at
location ri
Two free, 1000-bead chains
e and e- 1000-bead chains
frequency
frequency
xi, yi, zi (a.u)
xi, yi, zi (a.u)
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Binning the relative coordinate gives you
density corresponding to the orbital (here,
ground state)
Number, ni, of beads in bin from r to r dr
Normalized so that Si ni dr 1
P (r)
P (r) /r2 a.k.a. R(r)2
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From a students writeup The harmonic oscillator
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Can calculate other observables
(there is some numerical art to this )
V(r) e E z
a 36 a.u.
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Research gets done, students get trained

• Student Posters
• Conference talks
• Theses
• Papers
• Grants
• Classroom-type understanding?

Ps electron in FCC Argon
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Unsolved problems

• Reading the literature (Help us, AJP!)
• Research that relies on Relativistic QM (or
  foundations of QM, quantum gravity which are
  not problems for the best students, but )
• Early preparation in statistical mechanics (MIT
  model Do it soph year. Radical idea Make
  room in first-year classical mechanics. E.g.
  Leave massive pulleys and rolling stuff for
  later. Why do stuff twice? Why not do
  Lagrangians instead?)