Physics Education Research:
Graduate Quantum Mechanics
Lincoln D. Carr
Department of Physics
Colorado School of Mines
in collaboration with
Sarah McKagan, University of Colorado, Boulder
Outline
Introduction
I. Textbook
II. Course Content
III. Teaching Methods
IV. Assessment
Conclusions and Outlook
Colorado School of Mines
Carr Theoretical Physics Research Group
at the Colorado School of Mines
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
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 don’t learn what they’re supposed to in junior
level QM courses!
What are our goals in the class?
Expert
Novice
Formulas &
“plug ‘n chug”
Pieces
By Authority
content
Concepts &
Problem Solving
structure
Coherence
process
Independent
(experiment)
Drudgery
affect
Joy
think about science like a scientist
think about education like a scientist
Adapted from: Hammer (1997) COGNITION AND INSTRUCTION
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
E.g. Hard Data with Force Concept Inventory:
Engagement Improves Learning
traditional lecture interactive engagement
R. Hake, ”…A six-thousand-student survey…” Am. J. Phys. 66, 64-74 (‘98).
Traditional Model of Education
Individual
Instruction via
transmission
Content
Built in to our classes?
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?
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
Systematics
Students
Technical, engineering, application-oriented university
Non-traditional students 25% to 33%
Half Master’s 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
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
Sakurai
Strengths
Pedagogy; readable by students (Ch. 1-2 only)
Numerous physical examples
Doesn’t 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
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
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
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…
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)
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
Final Syllabus: Fall, QM I
Stern-Gerlach/Qubit – full immersion
Formal framework
Hilbert space; Matrix, Ket, and Functional Rep’ns; 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
Final Syllabus: Spring, QM II
Propagators and Path Integration
Harmonic Oscillator (year 3 only)
Creation/Destruction Operators, Coherent States, Classical-Quantum
connections, Eq’s 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
Wigner’s 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
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
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
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
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.
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.
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.
Graduate Conceptual Knowledge is a good
indicator of Final Exam Performance
Pearson correlation
coefficient 0.74.
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…
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
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).
The End
The End