The Formal Reasoning of Quantum Mechanics: Can we make it concrete? Should we?, Dean Zollman.

Dean Zollman, Kansas State University, USA


The teaching and learning of quantum mechanics is very frequently postponed until relatively late in a student’s academic career. In U.S. universities students typically receive a quick introduction to some aspects of one-dimensional quantum mechanics from the end of the second or beginning of the third year of their university studies and then do not study quantum mechanics in any depth until the fourth year. Thus, the major concepts which have driven much of the development of physics and of modern technology during the 20thcentury are delayed until the end of a physicist’s academic career and are frequently not studied by other students at any time during their careers.


One reason for this delay is the rather abstract nature of quantum mechanics itself. We can easily argue that, for the way in which quantum mechanics is traditionally taught, students need to have generally developed their formal reasoning skills. For example, formal operations, in the Piagetian sense, include hypothetical and deductive reasoning, abstract thought, use of symbolic representation, and the use of transformations. Quantum mechanics is a hypothetical system for understanding very small objects. It relies heavily on the use of symbolic representations and deduction to apply quantum mechanics to a variety of situations. Symmetry arguments, and therefore transformations, are a significant part of many presentations of quantum mechanics. Therefore overall, we can assume that the traditional mode in which quantum mechanics is taught is very abstract and requires rather sophisticated formal operational procedures.


Significant research dating back to the 1970s has shown that many university students have not yet developed formal operations (McKinnon & Renner 1971). In fact, the traditional way of teaching classical physics is a significant mismatch for many of these concrete operational students. Thus, it is not surprising that many physicists conclude that quantum mechanics is not understandable by students who are not studying physics very carefully and are in their third or fourth year at a university. Many people have concluded that learning quantum mechanics at a lower level is not possible and thus should not even be attempted (Arons 1990). They argue that the students will only be able to memorize isolated facts and repeat things without true understanding. Thus, the students are better served if we spend all of our time on classical physics where concrete learning experiences can more easily be constructed rather than attempting to teach them something that they could learn only with great difficulty, if at all.


  1. Why teach quantum mechanics to non-physicists?


The discussion about the abstract nature of the normal presentations of quantum mechanics seems rather valid. The simplest response to these conclusions is to avoid teaching this topic at any but the most advanced levels. However, some arguments favour attempting to find ways to teach the topic to students who have not yet reached full formal operations. For example, quantum mechanics was the most important development in 20thCentury physics, and it has dominated physics and technology for well over a half a century. Thus, at the beginning of the 21stCentury it is time to allow all interested people access to these ideas. Further, many experts predict that within the next 10 years miniaturization of electronics will reach the quantum mechanics limit. It would be nice if people who are trying to take the next step – development or business – understood what that meant. Finally, many other very complex and abstract processes – the election of an American President, for example – fill our lives. Perhaps an appreciation of quantum physics can help us understand the role of measurement in these events.


  1. Making quantum mechanics concrete


Our group at Kansas State University has been convinced that we should make the teaching of quantum mechanics more concrete than it normally is. We have worked to develop both the pedagogical style and the presentation of content so that students who are still developing their formal operational skills can appreciate and understand some of the features of contemporary quantum physics. For a pedagogical strategy we have adopted the basic Learning Cycle. The Learning Cycle was developed by Robert Karplus about 30 years ago and has been successfully used in almost all levels of teaching (Karplus 1977, Karplus et al 1975, Zollman 1990). While many people have adapted or changed the basic Learning Cycle, we find that the one that Karplus originally introduced works quite well for our teaching situation. Each Learning Cycle begins with an Exploration where students complete activities prior to the introduction of a new concept. These activities prepare them for the introduction of new concepts which can explain their observations during the Exploration. The concept introduction provides the new principles on which the students will build and frequently includes the development of models that can help explain the observations. Once the students have the new concepts and models they complete an Application in which they apply the newly learned information to situations that are similar but not identical to the ones they have already studied.


With the addition of model building in some cases our Learning Cycle comes very close to the Modelling Cycle that has been developed by Hestenes and his co-workers (Wells et al 1995). We also emphasize collaboration among students as they are learning. This cooperative effort is also an important part of the Modelling Cycle.


We have created Learning Cycles for a variety of different types of students. Our basic approach is that all students, even those who are more advanced in their reasoning skills and academic careers, can profit from a more concrete or intuitive approach to the abstract ideas of quantum physics. We began by developing materials for secondary school students and those university students who would complete a physics course but not study physics beyond one year. As these materials were developed, they were used by faculty who were teaching higher-level courses to physics students. We then created a set of materials for that group and have recently expanded to include materials specifically aimed at medical students and physics students in the last year of their undergraduate university careers. Each set of materials has a somewhat different approach and a different level of mathematical sophistication. However, all of them follow a basic Learning Cycle and focus on concrete visualization rather than abstract mathematical deduction.


  1. Device orientation


One way to make abstract ideas concrete is to connect the concept directly to something in the students’ experiences. In one way such a connection is easy for quantum physics. Almost every contemporary technological device could not exist if a designer of that device did not have an understanding of quantum science. At the same time the connection between quantum science and something as ubiquitous as the television remote control is not immediately obvious. Thus, we have combined hands-on experiences, visualizations, and traditional instruction to help the students see these connections.


Students should recognize these objects and see them in their everyday life. Light emitting diodes (LEDs), for example, are everywhere. Although many students do not know the name, they have seen them in their computers, remote controls, etc. By examining the properties of LEDs the students learn that LEDs are different from other light sources. Then, with the help of computer visualizations they understand how the light emitting properties are related to the quantization of energy in atoms. We occasionally use devices that students may have heard about and may have seen pictures of, but they have probably not encountered. The scanning tunnelling microscope is the best example. We do not expect students to use a scanning tunnelling microscope although it is possible for students to build one. But most students will not be able to build such a device. So, in this case, we use a combination of a simulation and an interactive program (Rebello et al 1997).


We also use a variety of solid light sources. Infrared detector cards are a rather interesting example. They are a fairly recent development – at least fairly recent for inexpensive versions. TV repair people need to know if a television remote control is emitting infrared. How can they do that? It is rather simple if they have a video camera. The camera responds to IR and shows a bright spot where the IR is emitted. So, every TV repairperson needs a video camera, and he/she can find out whether there is light coming out of the remote control. But that is rather expensive. Another way to detect IR is with rattlesnakes, which are sensitive to infrared. So, every TV repairperson could have a rattlesnake. But that is rather expensive in a different way. However, one can buy a little card that responds to IR by emitting visible light. Thus, it absorbs low energy light and emits higher energy light.


The Star Trek Transporter is also a quantum mechanical device. If one reads the Star Trek Users’ Manual, one finds that the Transporter has a component called a Heisenberg Compensator (Sternbach 1991). When one of the writers for Star Trek was asked, “How does the Heisenberg Compensator work,” he responded, “Very well” (Time 1994). Because Werner Heisenberg is one of the founders of quantum science, we must assume that this Compensator is related to his Uncertainty Principle. In one of our units we ask students to address the fantasy device in terms of basic quantum mechanics principles. These and several other devices are introduced to students. In each case we show how the devices are related to quantum mechanics. Further, the students learn how the devices work at the atomic level.


  1. Using visualization & model building


In the Visual Quantum Mechanics instructional materials we provide as concrete a description as possible about how we know about atoms and how we use that knowledge to build models. One of our learning units focuses on spectroscopy and its role as evidence for energy quantization. This unit begin with a study of the light emitting diode (LED). We can convince students that the LED is related to contemporary physics because they can read statements such as “A genuine White Light Super Bright LED … utilizes an advanced Quantum Well technology…” (Electronics 2000) After observing how different colours of LEDs respond to changes in voltage and observing the spectra from both LEDs and gas spectral tubes, the students are ready to build an energy level model of the atom. A visualization program, Spectroscopy Lab Suite, provides a set of simulated experiments, similar to the ones that they have just done, that are coupled to building energy models of atoms. The activities include the emission and absorption of light by gases, the emission by solids – particularly LEDs, several types of lasers, and common emission processes such as fluorescence and phosphorescence (Rebello et al 1998).


Students generally use the Gas Emission program after observing the spectra emitted by gas discharge tubes. The design of this component was motivated by the results from a preliminary field test. We found that students related the spectral lines for a gas to the discrete energy levels, rather than transitions between these energy levels. The Emission module was created to alleviate this misconception. Students create a trial spectrum for a gas by manipulating the energy level diagram of a gas, indicating the transitions on it. They compare their trial spectrum with the real spectrum for the gas.


The component screen for the Emission module shows an array of simulated gas lamps and a power supply on the left. To create the feel of the real experiment that the students have already completed, they must drag one of the gas lamps into the power supply. This action causes the lamp to emit light and its spectrum appears at the top of the screen. There are five known gases (hydrogen, helium, neon, lithium and mercury) available to the student, and an unknown gas. In the case of the unknown gas, the student can change the spectral lines to create any hypothetical spectrum.


A scale, which represents energy in the atom, is displayed on the lower right side of the screen. The students’ task is to manipulate energy levels and transitions and reproduce the spectrum of the gas. This procedure addresses our research about students’ understanding of atoms. Students can move the energy levels and observe the corresponding changes in the energy of the spectral line. To make the spectral line in the trial spectrum coincide with one in the real spectrum, the students must create a transition between two energy levels whose difference in energies is equal to the energy of the emitted light.


By using this program students learn that the energy of the emitted light is equal to the change in energy within the atom. More importantly they see that only certain discrete energy levels are needed to explain the observed spectrum. From knowledge of energy conservation and the data presented by the spectrum of a gas, students can discover that energy states in atoms are quantized. This critical discovery of 20thCentury physics follows from empirical results and an explanation in terms of energy – no knowledge of wave functions or the Bohr Atom is needed.


The Gas Lamps Emission component does not enable students to determine the exact energy levels of a given gas, but rather construct a model based on energy differences. When this component is used in a classroom environment with students working in small groups, different groups of students may arrive at different energy levels within the models to explain the spectrum of the same gas. Rather than tell some students that they are wrong, a teacher can use this situation to discuss the nature of scientific models and limitations based on the models by available data. In creating their models the students had available to them only the observed spectrum and the conservation of energy. With no further information they could create several different sets of energy levels which match the data. By having students compare their results with others in the class, they can begin to understand how more than one solution to a problem can be “right” when it is based on limited information. However, while they cannot create a complete picture of the atom, all students agree that discrete energy levels are necessary.


Other modelling in Spectroscopy Lab Suite is similarly connected to experiments. For example, students create energy band and gap models for the observation that LEDs which emit different colours of light have different threshold voltages. They also interpret the behavior of electrons in conduction, valence and impurity bands to explain why a glow-in-the-dark toothbrush stops glowing if it is placed in liquid nitrogen. In all cases the energy model building is connected directly to an observation that the students can make.


In building the instruction that led to these programs, we expected the students to be interacting with each other and with the teacher. For example, the observation that different energy levels can give the same result is effective because two groups of students obtain different but equally correct answers. The teacher and the students’ peers are, thus, important to our teaching-learning process.


  1. Conceptual approaches to wave functions


Students with concrete reasoning skills can move beyond spectra and energy models of the atom to learning activities involving wave functions. Developing experiments with real equipment to explore and apply wave functions is rather difficult. However, we can create visualizations which help the students explore. For the students who are not science or engineering majors we avoid the mathematics of quantum mechanics and rely heavily on visualization in which the students manipulate variables, and the computer solves Schrödinger’s Equation. The students must then interpret the results in terms of the conceptual knowledge.


For both the secondary students and beginning physics students we begin the study of the wave nature of matter with an experimental observation – electrons can behave as waves. After the students have discussed how interference patterns indicate wave behavior and have observed the interference of light, we turn their attention to electrons. They can observe a real experiment if the equipment is available, use video simulations (Kirstein 1999) or see pictures in books. To investigate the wave nature of electrons further, the students use a simulation program which enables them to control variables in electron, two-slit experiments. Using results such as those shown in Figure 3, the students can discover a qualitative relation between the wavelength of the electron and its energy. They compare the changes in the pattern for changes in energy of electrons with similar changes when one observes the interference of light at different wavelength. They can easily conclude that the wavelength of electrons decreases as the energy increases.


An issue that students will sometimes raise is the relation between the particle’s charge and its wave behavior. Their reasoning is, “Diffraction is the spreading out of a wave. Like charges repel. In a beam the charge must cause the electrons to spread out.” We test this hypothesis by comparing the diffraction pattern of simulated proton and neutron, two-slit experiments. The patterns are identical, so charge must not be a factor.


After a few more experiments, including a variation in mass, we introduce the de Broglie equation.

We have not actually derived the equation experimentally but have given a feasibility argument for it. While this approach is not historically accurate, it seems to provide students with a somewhat more concrete introduction to an abstract concept than stating de Broglie’s hypothesis and then using interference experiments to verify it.


To connect the matter waves to probability we employ a simulated electron interference experiment. Setting the particle flux to a few per second the students watch the pattern develop. After a few particles have hit the screen, we have the students stop the “experiment.” Now, we ask them to predict where the next electron coming from our electron gun will appear on the screen. The students very quickly fall into discussing the location in terms of probability. They can indicate some location where the electron will rather definitely not appear and several where it is very likely to appear. However, they cannot give a definitive answer. Thus, we can introduce the wave function and its probabilistic interpretation based on the students’ experience with indeterminacy.


With wave functions we emphasize conceptual understanding by having students manipulate graphic images in accordance with their knowledge. For example, we ask the students at all levels to sketch wave functions qualitatively. Following procedures that appeared in French and Taylor (French & Taylor 1978), some sketching is done with paper and pencil. However, we find that students can easily be very inexact with paper and pencil, and sometimes exactness is needed. So, we have created a program that does very little except that it allows the students to vary the wave function and match boundary conditions.


We have discovered some interesting ways in which the students use this program. First, if we tell the students that the wave function is smooth, they will make it smooth to many derivatives. The idea that two functions just stick together does not occur to them. Second, we use of the word “decreasing” for exponential decay. When we use “decay,” the students immediately think of radioactive decay. They interpret that to mean that the electrons are radioactively decaying in the region where the total energy is less than the potential energy. So, we use the phrase “decreasing wave function.”


The third and fourth year university physics students still use a basic Learning Cycle style approach. However, the Explorations require a little bit of formal operations. These students are still asked to match boundary conditions graphically. However, the Wave Function Sketcher program for these students also includes the common mathematical language of physicists.


In addition, the students are expected to work with both the wave function and its derivative when they are matching the boundary conditions. Using the Wave Function Sketcher for the advanced students provides an intuitive, and somewhat concrete, approach to understanding the process of matching boundary conditions. After the students have completed these activities they are ready to use the mathematics involved in boundary value problems to complete the solutions for wave functions in various one-dimensional situations. While this type of Learning Cycle primarily focuses on formal operations, it does provide visualizations that are more concrete than typical mathematical symbols and, we hope, helps the students build their intuition about wave functions.


  1. Does visual quantum mechanics work?


The units have been used in secondary schools and in universities throughout the U.S. and in a few other places. Actually, we do not know all the places that it is being used because, during the field test phase, all the material was on the web and people download it. We have given materials to people in Southeast Asia and throughout various parts of Europe as well as the U.S. Most of the original units have now been translated into Hebrew. (Arieli 2001) Thus, the materials, except for the new Advanced Visual Quantum Mechanics, have been thoroughly field-tested.


Most of our reports, however, have come from the U.S. Approximately 175 different teachers in 160 different schools have used the materials in classes and reported results back to us. Students’ attitudes toward these materials are very positive. They frequently make comments like, “I really like this better than our regular physics. Can we keep doing it?” (We don’t tell the instructors that.) Our staff hass observed teachers using the materials in a variety of different schools. The students interact with the materials and each other; and they seem to be learning. Most of the teachers also have positive attitudes; a few do not. We certainly have the problem that many teachers in the U.S. do not have a very strong background in quantum mechanics. Even though we are approaching quantum mechanics in a much different way than it is normally taught, some teachers still feel uncomfortable. Building the teachers’ confidence is very important. We are working on that aspect now by building a Web-based course for secondary science teachers. (Connect to


Student learning was also rather good. During our observations of the teaching, we noticed that the hands-on component for both the real experiments and the visualizations was important. Some teachers decided that it was too much trouble to have the students work in a hands-on mode with all of these programs. So, they just demonstrated the programs to the students. In these cases learning went down; attitudes went down; everything went down. Hands-on activities make a difference. Of course we should not be surprised because we built the material for the students to use; not for the teacher to talk about.


With the second year physics students we have done some testing, but it has not been as extensive as for the secondary students. These units are somewhat shorter and built to be in one to two-hour units within a traditional “modern physics” course. We have found that the students’ attitudes are generally very positive toward this type of learning and material. However, occasionally a student would feel that he or she was not getting all of the material that he or she would need for advanced level courses. Some of the first students with whom we tested the material are now taking a fourth-year quantum mechanics course. We will be investigating with them how well the materials that they learned in our course are serving them in the more advanced course. In terms of the student questions we found that these materials motivated students to ask very high-level conceptual questions. Rather than most of the questions about wave functions being concerned with the procedural efforts of manipulating equations, the students were focused on what the wave function means and how it can be interpreted. We asked the students questions on examinations which were very similar to those that they might find in a higher-level course. In general the performance on such questions was really quite good. So, overall even though we have not tested the materials as carefully as the materials for lower level students, we feel quite confident that the materiaals are teaching well and are providing conceptual understanding through concrete hands-on and visualized activities.


We do not at this time have similar information about the materials for the fourth year students. These materials are very new and have yet to have a significant amount of classroom tests performed on them.


  1. Conclusions


Concerning the questions which we posed in the title of this talk, we believe that the Visual Quantum Mechanics project has shown that we can make quantum mechanics accessible to students who are at the concrete operational stage. Further, we believe that we must provide ways to allow students who are not formally operational to begin to understand some of the features of the most important scientific advances during the 20thcentury. Based on a large number of field tests and a rather careful evaluation of student attitudes and learning, we have concluded that the Visual Quantum Mechanics materials have been successful in teaching some abstract concepts to students who have limited science and mathematics background and who probably use concrete or transitional operations. Our materials are also successful in teaching the conceptual ideas of quantum mechanics to students who have stronger science and engineering backgrounds by providing them with some concrete experiences. The combination of hands-on activities, pencil and-paper exercises, and interactive computer visualizations seem to work well in a classroom environment where student-student and student-teacher interactions are taking place. Thus, we feel that we have built a foundation for providing instruction in the most important aspects of 20thCentury physics to a broad range of 21stCentury students.




The work described here has been supported primarily by the U.S. National Science Foundation. Additional funding has come from the U.S. Department of Education, the Eisenhower Professional Development Program and the Howard Hughes Medical Institute. The development of the original Visual Quantum Mechanics teaching materials profited from significant work by N. Sanjay Rebello, Lawrence Escalada, and Michael Thoresen. Kirsten Hogg and Lei Bao were instrumental in the development of materials for second year physics students, while Waldemar Axmann is the primary author and programmer for Advanced Visual Quantum Mechanics. Dr. Hogg has also been the primary author of the Web-based materials while Kevin Zollman is the primary programmer for the on-line course. Chandima Cumaranatunge programmed the visualizations described in this paper. Rami Arieli has provided valuable feedback and suggestions for improvement while he has been creating the Hebrew translation. We have worked with Manfred Euler, IPN – Kiel, and Hartmut Wiesner, LMU – Munich, on some aspects of Visual Quantum Mechanics. We have profited greatly from input from undergraduate students and teachers at many other universities and high schools where the Visual Quantum Mechanics materials have been tested.




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This paper was presented to the First International GIREP Seminar, 2nd – 6th September, 2001, University of Udine, Italy.