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
Previous Research on Tunneling
QT: The Standard Presentation
- Why teach quantum tunneling?
- Unique result of QM, surprising, applications to alpha decay, field emission, scanning tunneling microscope.
- Learning goals – Students should be able to…
- Calculate or discuss qualitatively (depending on level of course) the probability of tunneling for various physical situations
- Describe the meaning of the potential energy and wave function graphs.
- Visualize how these graphs would change if physical situation altered.
- Relate mathematical formalism of tunneling to tunneling in the real world.
- The Study:
- Research Questions:
- Does our reformed curriculum help to address common student difficulties in learning about tunneling?
- Are our students achieving the learning goals described above?
- What practices support or hinder the achievement of these goals?
- Method: Collect qualitative data in reformed & traditional courses:
- Observations of lectures and problem-solving sessions
- Student interviews
- Student responses on homework and exams
- Student difficulties1-6:
- 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:
- Putting models in proper context7:
- Students don’t know what potential energy diagram represents.
- 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.
- Students mix up energy and wave function.
- Students think of classical objects physically going through walls, dissipation.
- Does not include…
- Discussion of what physical situation this potential energy represents.
- Discussion of how a plane wave is related to a physical particle.
- Square potential barrier
- Plane wave approaching from the left.
- Calculate reflection and transmission coefficients.
- Wave function and energy often on same graph.
- A few applications mentioned at the end: alpha decay, STM, field emission
Conclusion: artificially abstract model constrained by what is calculable
Suggestions for Improving Learning
- Interactive engagement techniques – peer instruction and collaborative homework sessions.
- Address common student difficulties – e.g. Energy Loss:
- Draw wave function and energy on separate graphs; concept questions designed to elicit confusion between wave function and energy:
- 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:
- Design interactive computer simulation11 to help students build better physical models and reduce reliance on calculable problems:
Students still struggle to put potential energy in proper context:
- 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!)
Student questions illustrate how abstract potential models are: Even with physical context, simple square barrier too abstract:
- “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?”
Plane waves are easier mathematically, but harder conceptually. Students have trouble relating plane waves to particles moving through space and time:
- “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?”
Issue of energy loss still comes up, but reasons are more subtle:
- “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)?”
- Particle splits into transmitted part and reflected part. Transmitted part has less energy because it’s only part of electron:
- “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.”
- 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.”
- Students know energy ≠ wave function, but think there must be some relationship between them:
- “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…”
- Previous focus on student difficulties is useful, but not enough.
- Must help students build models explicitly:
- How to relate potential energy diagrams to physical systems:
- 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!
- After introducing this physical justification for wire as square well, much less common to see students mixing up well and barrier.
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.
- Alpha Decay:
- “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?”
- “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?”
Other Issues Raised by Simulation
Determining each of these potentials requires many subtle approximations. In textbooks they are simply given.
- In interviews, students can easily make sense of real and imaginary parts of wave function in simulation:
- …but they can’t make sense of “phase color” representation:
- This is the ONLY representation used in most QM simulations!
- Simulation illustrates counterintuitive aspects of plane waves:
- Probability is not just proportional to ||2.
- Transmitted wave can have higher amplitude than incident wave:
- 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!
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- S. B. McKagan and C. E. Wieman, 2005 PERC Proceedings (2006).
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- 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.
- 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).
↑ you can access the “Quantum Tunneling” simulation from http://phet.colorado.edu