A Deeper Look at Student Learning of Quantum Mechanics:
the Case of Tunneling
S. B. McKagan, K. K. Perkins, and C. E. Wieman
University of Colorado at Boulder
http://per.colorado.edu
Introduction
Why teach quantum tunneling?
Student difficulties1-6:
• Unique result of QM, surprising, applications to alpha decay, field
emission, scanning tunneling microscope.
• Learning goals – Students should be able to…
• Belief that energy is lost in tunneling
• Belief that reflection and transmission
are due to particles having range of energies
• Confusion about wavelength and amplitude
• Offset in wave function:
• Etc…
1. Calculate or discuss qualitatively (depending on level of course) the
probability of tunneling for various physical situations
2. Describe the meaning of the potential energy and wave function graphs.
3. Visualize how these graphs would change if physical situation altered.
4. Relate mathematical formalism of tunneling to tunneling in the real world.
The Study:
Research Questions:
1. Does our reformed curriculum help to address common student difficulties
in learning about tunneling?
2. Are our students achieving the learning goals described above?
3. What practices support or hinder the achievement of these goals?
Method: Collect qualitative data in reformed & traditional courses:
1. Observations of lectures and problem-solving sessions
2. Student interviews
3. Student responses on homework and exams
Research-based
Previous Research on Tunneling
QT: The Standard Presentation
• Discussion of what
• Square potential barrier
physical situation this
• Plane wave approaching from the left.
potential energy
• Calculate reflection and transmission coefficients.
represents.
• Wave function and energy often on same graph.
• Discussion of how a
• A few applications mentioned at the end: alpha decay, plane wave is related to
STM, field emission
a physical particle.
Conclusion: artificially abstract model constrained by what is calculable
10
Curriculum
• Interactive engagement techniques – peer instruction and
collaborative homework sessions.
• Address common student difficulties – e.g. Energy Loss:
1. Draw wave function and energy on separate graphs;
concept questions designed to elicit confusion between wave
function and energy:
This question
generates a large
amount of discussion
in class. While most
students (~80%)
eventually answer the
it correctly, listening
in on student
discussions reveals
that most don’t know
the answer right
away, and only figure
it out through
vigorous debate with
their neighbors.
2. Emphasize energy conservation and lack of dissipation in
Schrodinger equation; Tunneling Tutorial where students work
out KE, PE, TE in each region.
• Give physical context for potential energy diagrams:
Square Well
Does not include…
Includes…
Putting models in proper context7:
The role of language and metaphor in teaching8-9:
• “…when physicists speak or write, they refer to these analogical
models by using systems of conceptual metaphors. They tend to
say ‘X is Y,’ rather than ‘X is like Y in certain respects.’”
→ leads to difficulties for students.
Suggestions for Improving Learning
• Putting quantum tunneling in context leads to deep hard questions
about how to relate course material to reality.
(Experts can’t answer many of these questions!)
• Previous focus on student difficulties is useful, but not enough.
• Must help students build models explicitly:
1. How to relate potential energy diagrams to physical systems:
•After
introducing
this physical
justification
for wire as
square well,
much less
common to
see students
mixing up
well and
barrier.
• Students still struggle to put potential energy in proper context:
•
•
“I have trouble understanding what the potential is when we are looking at
models of an electron in a wire, free space, finite square well, infinite square
well. I am sort of getting this idea of it being similar to a work function in
that once the potential (V) is less than the potential energy, the electron is
out of the wire. I can usually follow the math/calc that follows the examples
okay, but the overall concept of this potential (V) still confuses me, and so I
still don't have a firm grasp of [what] the square well models
mean/represent/whatever.”
“I cant find a general description of an infinite well, i understand what it
does but not what it is or where its used.“
“Voltage is used when we talk about electromagnetic forces, like the
coulomb force. What I'm confused about is that we used a voltage well to
show the strong force in effect. Is it accurate to show the strong force as
a very deep voltage well?”
• Student questions illustrate how abstract potential models are:
Alpha Decay:
V(r)
•“How do the Coulomb force and the strong
force relate to each other?”
•“Is the potential really square like that?”
•“How do you find the distance where the
strong force takes over?”
•“Do alpha particles already exist in the
nucleus or are they created upon radioactive
decay?”
2. Simulation allows instruction to focus on more realistic wave
packets, rather than plane waves.
• Real world examples are important, not just to help students see
relevance but for them to make sense of the models they are using!
Other Issues Raised by Simulation
r
Determining each of
these potentials
requires many subtle
approximations.
In textbooks they are
simply given.
STM:
•“As the electrons tunnel through, isn’t the
sample potential energy going to drop?”
•“The quantum tunneling microscope can be
used on any material even though not every
material has a “sea” of electrons? Wouldn’t
losing an electron in a crucial covalent bond
break the molecule apart?”
Sample
• In interviews, students can easily make sense of real and imaginary
parts of wave function in simulation:
applied voltage
tip
…but they can’t make sense of “phase color” representation:
• Even with physical context, simple square barrier too abstract:
•
•
•
Square Barrier
“Why do electrons flow if there is no potential difference between the
left and right sides of the barrier?”
“Can we really follow the behavior of a single electron?”
“What about interactions with atoms?”
• Plane waves are easier mathematically, but harder conceptually.
Students have trouble relating plane waves to particles moving
through space and time:
This is the ONLY representation used in most QM simulations!
• Simulation illustrates counterintuitive aspects of plane waves:
•
•
“Say you have two finite lengths of wire very close together. I don't
really see how we assume the electron is in one wire, get a solution, then
use that to determine psi across the gap, and then use that to determine
the probability that the electron is in the other wire. Over time don’t
the probabilities even out (i.e. we have no clue which wire the electron's
in)?”
• Design interactive computer simulation11 to help students build
better physical models and reduce reliance on calculable problems:
Transmitted wave
• Issue of energy loss still comes up, but reasons are more subtle:
1. Particle splits into transmitted part and reflected part. Transmitted part has
less energy because it’s only part of electron:
2. Students understand that KE is lost when electron escapes first wire, don’t
understand mechanism by which it regains KE when it enters second wire:
“Yeah, because it takes energy to get out of metal, the work function.
And it takes the amount of the potential energy--the barrier, this is the
barrier’s, so it uses that energy up and then it has a much slower--so it’s
going much slower. And then once it hits the other metal, hey, it’s going
fast again... It’s just weird, a little bit.”
3. Students know energy ≠ wave function, but think there must be some
relationship between them:
from http://phet.colorado.edu
Probability is not just proportional to ||2.
Transmitted wave can have higher amplitude than incident wave:
Incoming wave
Reflected wave
•
•
•
“The total energy is constant… [but] the energy of the wave function on
this side… is decreasing. I want to make the energy of the wave
function on this side decrease. But I’m also wary about that because…
‘the energy of the wave function on this side’? You know, the wave
function is a wave function, and it has like parts to it, but it doesn’t have
like… No, it does… You can have a wave function like that… and it has a
different energy here than it has here.”
↑ you can access the “Quantum Tunneling” simulation
1. Students mix up
energy and wave
function.
2. Students think of
classical objects
physically going
through walls,
dissipation.
• Students don’t know what potential energy diagram represents.
Results
•
Reasons:
“So if we are to evaluate these [energy] diagrams, put our total energy
line in, evaluate how that corresponds with our potential energy, you--it
sort of-- maybe this forces me to think too much about energies. …that’s
more classical physics, is it not? If the particle has sufficient energy to
get to the other side. Quantum’s a whole other story where we’re not
talking about so much energies. We are, but we’re also talking about
probabilities, correct? So there’s sort of two ways to think about this,
and maybe that’s why I’m a little confused still, at this late date…”
Wave speed (phase velocity) ≠ particle speed (group velocity).
Note that these are surprising even to experts!
If we want students to be able to generalize knowledge, it’s dangerous to
sweep this stuff under the rug!
References:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
End Notes
B. Ambrose, Ph.D. thesis, University of Washington (1999).
L. Bao, Ph.D. thesis, University of Maryland (1999).
J. T. Morgan, M. C. Wittmann, and J. R. Thompson, 2003 PERC Proceedings (2003).
M. C. Wittmann, J. T. Morgan, and L. Bao, Eur. J. Phys. 26, 939 (2005).
J. Falk, Master’s thesis, Uppsala University (2004).
D. Domert, C. Linder, and A. Ingerman, Eur. J. Phys. 26, 47 (2005).
S. B. McKagan and C. E. Wieman, 2005 PERC Proceedings (2006).
D. T. Brookes and E. Etkina, 2005 PERC Proceedings (2006).
D. T. Brookes, Ph.D. thesis, Rutgers University (2006).
S. B. McKagan, K. K. Perkins, and C. E. Wieman, 2006 PERC Proceedings (in press).
Physics Education Technology Project, http://phet.colorado.edu.
Acknowledgements:
The authors thank the NSF for providing the support for this project. We also
thank all the members of the PhET Team and the Physics Education Research at
Colorado group (PER@C).