Lincoln D. Carr

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Physics Education ResearchGraduate Quantum
Mechanics

• Lincoln D. Carr
• Department of Physics
• Colorado School of Mines
• in collaboration with
• Sarah McKagan, University of Colorado, Boulder

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Outline

• Introduction
• I. Textbook
• II. Course Content
• III. Teaching Methods
• IV. Assessment
• Conclusions and Outlook

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Colorado School of Mines
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Carr Theoretical Physics Research Groupat the
Colorado School of Mines
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Carr Theoretical Physics Group Research Areas

• Quantum Many Body Physics
• Far-from-equilibrium quantum dynamics
• Macroscopic superposition and quantum tunneling
• Artificial Materials
• Graphene nano-engineering
• Anderson localization in mm-waves
• Ultracold quantum gases in optical lattices
• Nonlinear Dynamics
• Fractals and chaos in spin waves in magnetic
  films
• Solitons and vortices
• Nonlinear Dirac Equation

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Why is Reform Needed?

• Four periods of Quantum Mechanics
• I. 1925 - 1935
• II. 1936 1963
• III. 1964 - 1981
• IV. 1982 Present
• ?E.g. Sakurai covers I and part of II
• Experimental Advances, New Applications/Technology
  
• Foundations of Quantum Mechanics
• Quantum Information Processing
• USA behind in comparison to EU, Canada
• E.g. density matrix formalism
• Success of Physics Education Research (PER)in
  undergraduate arena
• Teaching Methods
• Assessment Tools
• PER Students already dont learn what theyre
  supposed to in junior level QM courses!

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What are our goals in the class?
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Overview of Physics Education Research

• Far more to our classes than what has been
  traditionally evaluated
• Our students are not learning what we believe
  them to
• They are learning some things we would not expect
• Sub-field of physics education research has
  something to say about this
• Tools for assessment
• Models of student learning
• Suggestions for curricula / approaches in class

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E.g. Hard Data with Force Concept Inventory
Engagement Improves Learning
interactive engagement
traditional lecture
R. Hake, A six-thousand-student survey Am. J.
Phys. 66, 64-74 (98).
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Traditional Model of Education
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Built in to our classes?
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Context of Growing Upper Division Physics
Education Research Efforts

• Electromagnetism
• Ohio State, Oregon, U. Colorado
• Quantum Mechanics
• U. Washington, Oregon, U. Colorado, Pittsburgh,
  Maine, Ohio State
• Thermodynamics
• U. Washington, Cal State Fullerton, Maine
• Classical Mechanics
• Grand Valley State (Michigan), U. Maine

What about graduate courses?
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Our Strategy for Reform in Graduate QM

• Benchmark course
• Standard CU Sakurai-based
• Two different new courses
• Cumulative use of best of from previous
• Same assessment tools throughout
• 7 forms of evaluation
• Gradually increase use of PER teaching techniques

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Systematics

• Students
• Technical, engineering, application-oriented
  university
• Non-traditional students 25 to 33
• Half Masters students in 1st semester only
• Small class size 20 in Fall, 10 in Spring
• Instructor
• First time teaching course
• Researcher in quantum many body theory
• Assistant professor
• Bubble form evaluations on par with department
  average

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I. Textbook

• Went through over 50 textbooks
• Classics Landau Lifshitz, Schiff, Sakurai,
• Specialized/Applied Peres, Cohen-Tannoudji,
  Levi,
• New QM Basdevant Dalibard, Rae, Gottfried
  and Yan 2nd Ed.,
• Chose 3
• Year 1 Sakurai, Modern Quantum Mechanics
• Year 2 Le Bellac, Quantum Physics
• Year 3 Gottfried and Yan, 2nd Ed., Quantum
  Mechanics Fundamentals
• Evaluate each book

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Sakurai

• Strengths
• Pedagogy readable by students (Ch. 1-2 only)
• Numerous physical examples
• Doesnt start with review or historical
  perspective
• Approx. Methods couched in terms of applications
• Weaknesses
• Covers only Periods I-II of QM
• Imbalances angular momentum for 1/3 of text
  symmetry chapter short not profound
• Misleading treatment of mixed states

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Le Bellac

• Strengths
• Ok pedagogy somewhat readable for students
• Many Physical Examples a whole chapter on
  applications
• Seamlessly integrated chapter on entanglement
• Clear statement of postulates of QM
• Weaknesses
• EU US educational mismatch
• Scattering theory, perturbation theory low level
• No path integration
• Spiral teaching method

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Gottfried and Yan

• Strengths
• Treats modern QM through all stages (I-IV) from
  the beginning
• E.g. pure and mixed states
• Very rigorous and careful on all points
• Breadth of material
• Quantum fluctuations, Wigner theorem, etc.
• Weaknesses
• Poor pedagogy no physical examples till Chapter
  4, page 165!
• NB Chapter 1 Fundamental Concepts
  inaccessible to student
• Spiral method of teaching no time
• Occasional extremely atypical treatment of topics
• E.g. degenerate perturbation theory

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II. Course Content

• Starting premise? Must integrate QM period
  III-IV material ?
• What will students need to conduct their research
  at our university?
• Condensed Matter, Optics, Nuclear, Renewable
  Energy, Theoretical and Computational Physics
• E.g. graduate QM often taught as a course in AMO
  applications
• Sacrifices necessary

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Material Cut (in years 2 and 3)

• Full-blown treatment of Wigner-Eckart Theorem and
  irreducible tensor operators
• Inelastic scattering, some scattering
  applications
• Undergraduate QM review
• Some AMO applications
• Full-blown Clebsch-Gordan Young tableaux etc.
• Scattering examples and details (e.g. Eikonal
  approximation, low energy s-wave limit)
• Math physics review (e.g. classical rotations)

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Material Added (in years 2 and 3)

• Postulates of QM
• Density matrix formalism, partial traces,
  entanglement
• Baby quantum field theory
• Quantum fluctuations
• Wigner theorem, more advanced symmetry treatment
• Polarization states (year 2 only)
• Unbounded operators, formal treatment of infinite
  dimensional Hilbert spaces (year 2 only)
• Breadth of applications nuclear, optics, solid
  state

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Final Syllabus Fall, QM I

• Stern-Gerlach/Qubit full immersion
• Formal framework
• Hilbert space Matrix, Ket, and Functional
  Repns Mixed vs. Pure States, Entropy
  Uncertainty Principle Quantum-Classical
  connections Postulates of QM
• Symmetries and Conservation Laws
• Translation, parity, Vector and Tensor operators,
  Rotation, Spherical Harmonics
• Basic Applications
• Benzene, NMR, a little Solid State Theory
• Angular Momentum
• Orbital, Spin, Addition, Baby Wigner-Eckart
• Basic Approximation Methods
• Perturbation theory (x3), variational method,
  minimal examples
• NB Harm. Osc. In year 2 only, could be
  re-introduced

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Final Syllabus Spring, QM II

• Propagators and Path Integration
• Harmonic Oscillator (year 3 only)
• Creation/Destruction Operators, Coherent States,
  Classical-Quantum connections, Eqs of motion
• Baby Quantum Field Theory
• Scalar 1D QFT, Quantum fluctuations of EM field
• Gauge Transforms, Ahranov-Bohm
• WKB (year 3 only in Fall in year 2)
• Hydrogenic Atoms
• Stark, Zeeman, Spin-orbit, van der Waals
• Identical Particles, Helium Atom
• Advanced Applications
• Photoelectric effect, Resonance states, 2nd
  treatment of t-dep. Pert. Theory
• Symmetry
• Wigners Theorem, Time-reversal, Overview of all
  symmetries
• Scattering Theory
• Calculating cross sections, Lippman-Schwinger
  eqn, Born approx, Partial waves, T and S
  matrices, t-dep. formulation via propagators,
  inelastic

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III. Teaching Methods

• What undergraduate Physics Education Research
  methods can be folded in to graduate instruction?
• Student to Instructor
• Elicit responses from every student, every
  lecture
• Student to student
• Individuals
• Students explain understanding to each other
• Groups
• Break into groups, 2-3 minutes to work on problem
• Clicker with fingers
• Small class size makes tech unnecessary
• Focus on concepts
• 25 conceptual component to each exam

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Other additions

• Conversation sitting during lecture
• Partial use of Socratic method
• Provide all notes online in advance
• Provide open questions to guide thinking in notes
• Get students to predict lecture
• Loooooooong pauses
• 10-2 rule

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IV. Assessment 7 Evaluation Methods

• Quantum Mechanics Conceptual Survey (QMCS)
• Graduate Quantum Mechanics Conceptual Survey
  (GQMCS)
• University Wide Bubble Forms
• Written Self-Evaluations
• Student interviews end of 3rd week of classes
• Department Head evaluation
• Physics Education Researcher evaluation

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QMCS Sample Questions

• The electron in a hydrogen atom is in its ground
  state. You measure the distance of the electron
  from the nucleus. What are the possible results
  of this measurement?
• A. You will definitely measure the distance to be
  the Bohr radius.
• B. You could measure any distance between zero
  and infinity with equal probability.
• C. The most likely distance is the Bohr radius,
  but it is possible to measure nearly any
  distance, with the probability falling off
  exponentially as the distance increases beyond
  the Bohr radius.
• D. There is a small range of possible distances,
  from a little bit less than the Bohr radius to a
  little bit more than the Bohr radius, with the
  minimum and maximum possible distances given by
  the minimum and maximum of the deBroglie wave.
• E. I have no idea how to answer this question.
• A particle with the spatial wave function ?(x)
  eikx can be thought of as a plane wave travelling
  along the x-axis. Its real part is a cosine
  wave, as shown in the figure at right. Which of
  the following statements most accurately
  describes the probability of finding the particle
  at any location along the x-axis?
• A. It is equally likely to find the particle
  anywhere along the x-axis.
• B. It is most likely to be found in the peaks of
  the wave.
• C. It is most likely to be found in the peaks or
  the troughs of the wave.
• D. The particle is actually located in one
  particular place, independent of the wave
  function, and that is the only place you can
  find it.
• E. I have no idea how to answer this question.

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GQMCS Sample Questions

• 6. Write the general form of a translation
  operator. What are the essential mathematical
  properties of such an operator? What is a
  particular instance of such an operator? How can
  one determine if the Hamiltonian is symmetric
  under such a translation?
• 7. What is an entangled state? Give an example of
  two particle and three particle entangled states
  based on direct products. What is a cat (NOON)
  state? Give an explicit example.
• 8. Explain the physical meaning of SU(2) and
  SO(3). How are they connected?
• 9. Explain the difference between a coherent
  state and a number state (here, number of energy
  quanta) for a harmonic oscillator.
• 10. Define the density matrix. What are its zero
  and infinite temperature limits? Define entropy
  in the density matrix formalism.

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Undergraduate QM Conceptual Knowledge Does not
Increase During a Graduate Course
Pearson correlation coefficient of -0.21 for
post-test and -0.0056 and pre-test.
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Graduate Conceptual Knowledge is a good indicator
of Final Exam Performance
Pearson correlation coefficient 0.74.
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Summary of Results of Evaluations

• No marked improvement in undergraduate conceptual
  survey
• Graduate survey has improved results with new
  syllabus (years 2 and 3)
• Self-evaluations superior tool for year to year
  improvement
• Student interviews give students a hand in
  setting course structure, goals
• Dept. Head / PER evaluations complementary

None of these methods took any extra
time!Besides regularly scheduled office hours
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Conclusions

• I. Textbook
• Sakurai now dated
• No pedagogically great replacement yet
• II. Course Content
• New syllabus combines best of QM periods I
  through IV
• QM Period I should be treated in undergraduate
  quantum mechanics
• III. Teaching Methods
• Student to student and student to instructor
  useful in graduate context
• IV. Assessment
• Undergraduate skills separate from graduate
  skills
• First graduate quantum mechanics conceptual
  survey produced

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Outlook

• Use of simulations for demonstrations
• Graduate-level adaptions needed, e.g. mixed
  states
• Larger sample
• Now studying undergraduate upper division
  classical mechanics with same methods

L. D. Carr and S. A. McKagan, Graduate Quantum
Mechanics Reform, American Journal of Physics,
in press, e-print http//arxiv.org/abs/0806.2628
(2009).
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The End
The End