A Deeper Look at Student Learning of Quantum Mechanics: the Case of Tunneling

S. B. McKagan,1 K. K. Perkins,2 and C. E. Wieman3, 1, 2

1JILA, University of Colorado and NIST, Boulder, CO, 80309, USA2Department of Physics, University of Colorado, Boulder, CO, 80309, USA

3Department of Physics, University of British Columbia, Vancouver, BC V6T 1Z1, CANADA(Dated: June 24, 2008)

We report on a large-scale study of student learning of quantum tunneling in 4 traditional and 4transformed modern physics courses. In the transformed courses, which were designed to addressstudent difficulties found in previous research, students still struggle with many of the same issuesfound in other courses. However, the reasons for these difficulties are more subtle, and many newissues are brought to the surface. By explicitly addressing how to build models of wave functionsand energy and how to relate these models to real physical systems, we have opened up a floodgateof deep and difficult questions as students struggle to make sense of these models. We conclude thatthe difficulties found in previous research are the tip of the iceberg, and the real issue at the heart ofstudent difficulties in learning quantum tunneling is the struggle to build the complex models thatare implicit in experts’ understanding but often not explicitly addressed in instruction.

PACS numbers: 01.40.Fk,01.50.ht,03.65.Xp

I. INTRODUCTION

Tunneling is a surprising result that has served to val-idate the theory of quantum mechanics by explainingmany real world phenomena such as alpha decay, mole-cular bonding, and field emission, and has resulted inapplications such as scanning tunneling microscopes. Asa case study in the counterintuitive yet applicable natureof quantum mechanics, tunneling is an important part ofany introductory course in modern physics or quantummechanics.

An examination of modern physics and quantum me-chanics textbooks, course syllabi, and interviews withfaculty who have taught such courses suggest that in-struction in tunneling should help students achieve thefollowing learning goals:

1. Calculate or discuss qualitatively (depending onthe level of the course) the probability of tunnel-ing for various physical situations

2. Describe the meaning of the potential energy andwave function graphs

3. Visualize how these graphs would change if thephysical situation were altered (e.g. changing bar-rier height and width)

4. Relate the mathematical formalism and graphicalrepresentation of tunneling to the phenomenon oftunneling in the real world

Tunneling has been a favorite topic of physics educa-tion researchers specializing in quantum mechanics, whohave found that many students have a great deal of trou-ble understanding even the most basic aspects of thistopic [1–7]. In designing a transformed course in modernphysics for engineering majors [8], we drew on the litera-ture of previous research to develop a curriculum aimedat addressing known student difficulties in understandingquantum tunneling [9]. Throughout the process of devel-oping and refining this course, we carried out a study to

answer the following research questions:

1. Does our transformed curriculum help to addresscommon student difficulties in learning tunneling?

2. Are our students achieving the learning goals de-scribed above?

3. What are the practices that support or hinder theachievement of these goals?

We find that our curriculum does help students over-come common difficulties and achieve our learning goals.While the common difficulties reported in the literaturedo arise in the transformed classes, they are less prevalentthan in comparable traditional classes, and they oftenarise for different reasons than discussed in the previousliterature. Further, we find new difficulties that have notbeen previously reported, associated with a struggle tomake sense of the models of quantum mechanics and re-late them to the real world.

The difficulties discussed in the literature are asso-ciated with the inability to apply the quantum modelto abstract model systems such as square barriers andsquare wells. Our transformed course focuses on relatingthese abstract model systems to reality, and our researchshows that the difficulties discussed in the literature aresurface features, masking a much more serious problem:In tunneling, as in other aspects of quantum mechan-ics, students fail to grasp the basic models that we areusing to describe the world as anything more than ab-stract model systems. These models include wave func-tions as descriptions of physical objects, potential energygraphs as descriptions of the interactions of those objectswith their environments, and total energy as a delocal-ized property of an entire wave function that is a functionof position. Thus, even when students can successfullyovercome problems that previous research has elucidated,such as relating wave functions to potentials, they maynot know what a wave function or a potential is.

Hestenes has pointed out that while “A physicist pos-

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sesses a battery of abstract models with ramifications al-ready worked out or easily generated,” standard physicsinstruction often treats these models implicitly ratherthan explicitly. [10] While this is true even in introduc-tory physics, the problem is more serious in quantum me-chanics, where the models are particularly abstract, andthe connection between the models and the real world ismore tenuous. Standard instruction in quantum mechan-ics, including tunneling, does not provide students withenough information to make sense of these models, to re-late them to anything real, or even recognize that theyexist. We have achieved a degree of success in teachingquantum tunneling by making these models more explicitand connecting them to real-world applications, and sug-gest further changes in this direction.

II. THE STANDARD PRESENTATION OFQUANTUM TUNNELING

Most textbooks on modern physics and quantum me-chanics have a discussion of quantum tunneling. Thediscussion is remarkably similar throughout these books,with the main difference being that modern physics text-books give less detail. Tunneling is defined as a wavefunction passing through a potential energy barrier thatis greater than its total energy. The typical presenta-tion includes an analysis of the plane wave solution tothe Schrodinger equation for a square potential energybarrier, as shown in Figure 1. Often the wave function,potential energy, and total energy are drawn on the samegraph, a practice which research has shown to lead to stu-dent confusion [3, 6], and which thoughtful authors haveavoided since the 1970’s [11]. Depending on the level ofthe textbook, the reflection and transmission coefficientsare either derived or given. This is typically followed bya discussion of some applications of quantum tunneling,such as alpha decay, scanning tunneling microscopes, andthe inversion of Ammonia molecules. Some textbooksalso include a discussion of tunneling wave packets, oc-casionally showing pictures of a tunneling wave packettaken from a numerical simulation such as in Ref. [12].Wave packets and applications are nearly always rele-gated to the end of the discussion of tunneling.

FIG. 1: The standard presentation of quantum tunneling:a plane wave tunneling through a square potential barrier.Total energy, potential energy, and the real part of the wavefunction are all drawn on the same graph, and the real partof the wave function is labeled as simply “wave function.”

In examining the standard presentation of tunneling,

one may ask how it aligns with the learning goals in Sec-tion I. The standard presentation certainly gives studentspractice in calculating relevant quantities for the case ofa plane wave and square barrier, but it does not givestudents the tools to extend these calculations to morerealistic systems. It also includes both a mathematicalmodel and a discussion of physical applications of thismodel. However, we argue that it does not provide suffi-cient links between the two. For example, there is almostnever a discussion of what physical system could producethe square barrier shown in Figure 1 or of how a planewave relates to a real particle. Further, when real appli-cations are discussed, their potential energy graphs areoften not discussed, making it harder for students to re-late the applications to the mathematical model. Thus,the standard presentation does not provide students withthe tools to extend the model of tunneling beyond squarebarriers to the more complicated potentials involved inreal physical systems, either quantitatively or qualita-tively.

III. PREVIOUS PHYSICS EDUCATIONRESEARCH ON QUANTUM TUNNELING

Many researchers have documented student difficultiesin learning quantum tunneling [1–7]. These researchers,working at many institutions in the United States andSweden, have found a fairly consistent list of student dif-ficulties.

The most common difficulty, discussed in all these ref-erences, is the belief that energy is lost in tunneling. Thecorrect description of energy in quantum tunneling is thatbecause there is no dissipation in the Schrodinger equa-tion, energy is conserved, as can be seen in Figure 1,where the total energy is constant throughout. The bar-rier itself represents the potential energy, which is zero onthe left and right, and some positive constant inside thebarrier.1 The kinetic energy is equal to the total energyon the left and right, and is negative inside the barrier.Ambrose [1] and Bao [2] report the student belief that ki-netic energy is lost in tunneling, although later researchshows that this difficulty is not limited to kinetic energy:Morgan et al. [3] quote students as saying that “energy”is lost, without specifying which kind of energy, and inour own work, we found that most students who thoughtthat energy is lost did not have a clear idea of which en-ergy is lost. When asked, they were just as likely to saypotential, kinetic, or total energy, and often used two oreven all three types of energy interchangeably within thesame explanation. [7]

1 It is linguistically awkward to speak of the potential energy “in-side the barrier”, since the potential energy is the barrier, butit is important to be explicit, as many students do not recognizethe equivalence of potential energy and barrier.

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There are two common explanations in the literaturefor the belief that energy is lost in tunneling. The first ex-planation (reason 1), attributable to the fact that mosttextbooks and lecturers draw the energy and the wavefunction on the same graph, is that students confuse thetwo, believing that the energy, like the wave function,decays exponentially during tunneling [2, 3, 6]. This ex-planation is reminiscent of the classic confusion betweenvelocity and acceleration in introductory physics [13–18];while students can correctly recite definitions and formu-lae for wave function and energy, they fail to distinguishbetween the two when solving problems. The second ex-planation (reason 2) is that students think that “‘work’is done on or by the particles while inside the potentialbarrier” [1] or that energy is “dissipated” as in a physi-cal, macroscopic tunnel [3]. Many researchers report onstudent interviews showing that both these explanationsare common among students. [1–7]

A third possible explanation (reason 3) suggested byBao is that students may be thinking of mechanical orelectromagnetic waves, in which the energy of the waveis related to the amplitude. [2] However, no evidence ispresented to support this explanation of student think-ing. In our observations and interviews in traditionalmodern physics courses, few students have sufficient un-derstanding of mechanical or electromagnetic waves tocause problems in their interpretation of the amplitudeof matter waves, and none have used such an explanation.As discussed in the Section VIB, we do see some evidenceof students using this explanation for energy loss in ourtransformed modern physics course, in which the depen-dence of amplitude on energy in electromagnetic wavesis heavily stressed.

Other common student difficulties reported in the lit-erature are: the belief that reflection at a barrier is dueto particles having a range of energies [1]; incorrectlydrawing the wave function with an offset between thehorizontal axes of the wave function on the left and rightside of the barrier, as in Figure 2a [3]; incorrectly draw-ing the wave function with a smaller wavelength on theright than on the left, as in Figure 2b [1, 3]; and misin-terpreting the meaning of the wavelength and amplitudeof the wave function.

In addition to these common student difficulties, in ourown previous research we found that many students donot know what the potential energy graph represents [7].Our results from student interviews are supported bymany conversations with practicing physicists who re-port having successfully completed quantum mechanicscourses as students without realizing what a potentialwell was until much later. We believe that this problemis due to the lack of physical context for potential energygraphs in the standard treatment discussed in Section II.We will return to this issue later.

Brookes and Etkina [20, 21] argue that physicists talkabout potential using a metaphor of a physical object, asillustrated by the terms “potential well,” “potential bar-rier,” and “potential step.” Because these metaphors are

FIG. 2: Common student difficulties reported in the litera-ture: incorrectly drawing the real part of the wave functionwith (a) an offset between the horizontal axes on the left andright side of the barrier and (b) a smaller wavelength on theright than on the left. These drawings are taken from stu-dent responses to an exam question asking students to drawthe real part of the wave function, as discussed in SectionVI. We have observed physics faculty making drawings simi-lar to both (a) and (b), and a popular introductory quantummechanics textbook contains a figure similar to (b) [19].

implicit and their limitations are not discussed, studentshave a tendency to overextend them, leading to many ofthe student difficulties that other researchers have doc-umented. Brookes and Etkina’s analysis overlaps withours, in that they also point out that physics professorsare not explicit in discussing the limitations of models.

IV. AN IMPROVED CURRICULUM FORTEACHING QUANTUM TUNNELING

As part of the transformation of a modern physicscourse for engineering majors [8], we developed a curricu-lum for teaching quantum tunneling. The course designwas based on PER, using interactive engagement tech-niques such as peer instruction and collaborative home-work sessions, focusing on real-world applications, andaddressing common student difficulties. The curriculumon quantum tunneling was designed to address commonstudent difficulties with this topic, which were knownfrom previous research (see section III). Throughout thecourse, we emphasized building models and relating themto the real world, asking students in lecture, homework,and exams both to construct their own models and toexplain models that had been presented to them. [9]

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A. Addressing student difficulties with energy loss

Several aspects of the instruction were designed toaddress the belief that energy is lost in tunneling. Asdiscussed in section III, previous research cites two rea-sons that students believe energy is lost in tunneling: (1)treating energy and wave function interchangeably, and(2) invoking dissipation.

To address reason 1, we were careful to draw energyand wave function on separate graphs. However, sincethe representation in Figure 1, in which they are plot-ted on the same graph, is ubiquitous in textbooks andother literature, it is impossible to avoid students beingexposed to it. This representation has been so ingrainedin us by our own education that we had to be on guard tokeep from drawing graphs this way ourselves! Therefore,we also used concept questions (multiple choice questionsposed in class that students discuss in small groups andanswer using a personal response system) and homeworkquestions to elicit student confusion between energy andwave function and address it directly. Figure 3 showsan example of a concept question used to address thisconfusion.

FIG. 3: A concept question designed to elicit student confu-sion between energy and wave function. The correct answer isC, but students who don’t understand the meaning of super-imposing a wave function graph on an energy graph may beinclined to answer A. When we ask this question in class, itgenerates a large amount of discussion. While most students(73-88%, depending on the semester) eventually answer thequestion correctly, listening in on student discussions revealsthat most don’t know the answer right away, and only figureit out through vigorous debate with their neighbors. Evenafter discussion, 9-19% give answer A.

To address reason 2, we emphasized energy conserva-tion and the lack of dissipation in the Schrodinger equa-

tion. One key feature of our curriculum was a tutor-ial [9] adapted from the Quantum Tunneling Tutorial inthe Activity-Based Tutorials Volume 2 [22], developedby Wittmann, Steinberg, and Redish. This tutorial wasdesigned to address the belief that energy is lost in tun-neling by asking students to work out the total, kinetic,and potential energy in each region and answer questionsabout energy conservation.

B. Giving potential energy a physical context

We also designed our curriculum to address our previ-ous finding that students are often confused by the mean-ing of the potential energy function [7]. We consistentlygave a physical context for potential energy functions,presenting square wells and barriers as illustrations ofreal physical systems, rather than mere abstractions. Itis worth noting that it was a great challenge for our teamof three expert physicists, including one Nobel Laureate,to think of even a single real physical system representedby a square well or a square barrier. This illustrates thatfor content that is outside of our area of research, evenphysicists sometimes do not know how an idealized text-book model can be applied to the real world.

The physical examples that we decided to use in ourcourse are illustrated in Figure 4: an electron in a shortwire as the context for a square well, and an electrontraveling through a long wire with a thin air gap as thecontext for a square barrier. Because the electrons arefree to move around within the wire, the potential en-ergy of an electron is constant anywhere inside the wire(and we can arbitrarily set the constant value to zero).Because the electrons are bound to the wire and requireenergy to escape, their potential energy outside the wirewill be a larger constant, so that the potential energy ofthe system is well approximated by a finite square well.In lecture, we ask students to predict the value of thepotential energy outside the wire by reminding them ofthe energy required to kick an electron out of a metalin the photoelectric effect, which they learned about ear-lier in the course. Students discuss this question in smallgroups, and most eventually recognize that the poten-tial energy outside the wire will be given by the workfunction of the metal. After the first semester, we alsoexplained the shape of the potential energy of a wire byproviding a microscopic model in which you add up allthe Coulomb potentials of the nuclei of the atoms in thewire, as illustrated in Figure 10.

We chose the physical context of an electron in a wirebecause it has practical applications for real circuits.While there are a few textbooks that provide physicalexamples of tunneling (an electron bouncing back andforth between two capacitors with tiny holes in them fora square well [23], and an electron traveling through a se-ries of metal tubes held at different voltages for a squarebarrier [24]), these examples are so artificial that no onewould ever create such a system for any reason other than

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FIG. 4: Physical contexts for (a) a square well and (b) asquare barrier. A square well with width L and height U0

represents a wire with length L and work function U0. Asquare barrier with width L and height U0 represents twolong wires with work function U0 separated by an air gapwith length L.

to demonstrate the abstract potentials used in introduc-tory quantum mechanics courses. We decided against us-ing the example of a charged bead moving along a wireheld at different potentials that was used in the orig-inal version of the Activities-Based Tutorials [22], alsobecause it seemed excessively artificial.

Our curriculum included many opportunities for stu-dents to practice building models of how potential energygraphs relate to physical systems. For example, in inter-active lectures, homework problems, and a tutorial weasked students to build up potential energy diagrams forsystems such as an electron in a wire, a scanning tunnel-ing microscope, and a nucleus undergoing alpha decay.We also asked students to reason through the physicalmeaning of the potential energy for various systems.

Further, we used the term “potential energy,” ratherthan the shorthand “potential,” to avoid confusion. 2 Al-though it would be preferable to use the symbol U , ratherthan the common convention V , for potential energy, tohelp students relate the potential energy in quantum me-chanics to the potential energy in other areas of physics,we used V in order to be consistent with the textbookwe chose for the first semester. However, we repeatedlyemphasized the meaning of this symbol, and explicitlypointed out the inconsistency in notation among differ-ent areas of physics.

2 The simulation discussed in the next section uses both labels tohelp students relate the more correct term to the term that ismore commonly used in textbooks.

C. The Quantum Tunneling Simulation

The standard presentation of quantum tunneling dis-cussed in Section II provides an abstract and decontex-tualized model that is difficult to visualize or connectto reality. The content of this presentation is artificiallyconstrained by what can be calculated. Students learnto calculate transmission coefficients for plane waves tun-neling through square barriers, not because this is a rel-evant problem, but because this is the only tunnelingproblem that can reasonably be calculated analytically.With modern computational techniques, however, it is nolonger necessary for the curriculum to be so constrained.

FIG. 5: The Quantum Tunneling and Wave Packets simula-tion provides interactive visual models of tunneling of wavepackets and plane waves in a variety of physical situations,and removes constraints imposed on curriculum by what prob-lems can be easily calculated.

We designed the Quantum Tunneling and Wave Pack-ets simulation [25] (see Figure 5) to provide easily ac-cessible interactive visual models of tunneling of wavepackets and plane waves in a variety of physical situ-ations, thus removing many constraints on curriculum.With the simulation, we can begin our instruction withwave packets, rather than plane waves, so that studentscan develop a visual model of what is happening in timeand space in quantum tunneling. This simulation wasdeveloped as part of the Physics Education TechnologyProject (PhET) [26], which provides free interactive com-puter simulations for teaching physics. Like other PhETsimulations, the Quantum Tunneling and Wave Pack-ets simulation is highly interactive, allowing students tochange the potential and total energies by dragging onthe graph, so that they can quickly explore a wide vari-ety of physical situations that would be cumbersome tocalculate. The simulation also provides a wide variety ofrepresentations, allowing students to view the real part,imaginary part, magnitude, and phase of the wave func-tion. To address the problem of students treating energy

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and wave function interchangeably, these quantities aredisplayed on separate graphs in the simulation.

We note that in the first semester of the reformedcourse, before developing the simulation, we attemptedto use existing simulations on quantum tunneling, asmany have already been developed by others [12, 27–32]. However, we found that students quickly becamefrustrated by the limitations of these simulations. Forexample, students wanted to be able to adjust the proper-ties of the wave packet and/or barrier, and to see the realpart of the wave function rather than just the magnitude.Further, all these simulations had features that researchhas demonstrated to be ineffective for student learning,such as plotting the wave function and the potential onthe same graph [3, 6], using a phase color representa-tion [33], and limited interactivity [34]. As a result, wedesigned our own simulation that we used in the coursestarting in Sp06.

V. THE STUDY

In order to answer the research questions in SectionI, we collected qualitative and quantitative data on stu-dent thinking about quantum tunneling in eight modernphysics courses over a five-semester period. Five of thesecourses were for engineering majors and three were forphysics majors. Four of the engineering majors’ courseswere taught using the transformed curriculum describedin the previous section. The first two semesters of thetransformed course were taught by the authors, and thenext two by another professor in the physics educationresearch (PER) group. 3 The remaining courses in thestudy were taught in a traditional manner along the linesof Section II, with minimal or no use of peer instruction,collaborative homework sessions, focusing on real-worldapplications, addressing common student difficulties, orinteractive simulations. Some of these courses used click-ers, but their use was less frequent and involved less dis-cussion than in the transformed courses.

The qualitative data we collected consist of observa-tions of students in lectures and problem-solving ses-sions, student responses to essay questions on home-work and exams, and student interviews. Interviews in-cluded students participating in a case study project, in-terviews on the Quantum Mechanics Conceptual Survey(QMCS) [35, 36], which includes questions on tunnel-ing [7], and interviews on the Quantum Tunneling and

3 While the transformed curriculum was designed specifically forengineering majors, it was used in the physics majors’ course inthe semester following this study. This course was not includedin the study due to lack of time. Anecdotal observations suggestthat the the physics majors tended to pick up on things faster,so that the prevalence of many of the difficulties cited here wassomewhat less for them, but the qualitative results of this studywere not significantly different for this population.

Wave Packets simulation described in the previous sec-tion. The quantitative data consist of student responseson the QMCS, homework, and exams.

Observations included approximately 200 lectures (20on tunneling) and 50 problem-solving sessions (5 on tun-neling). In lectures, a researcher (SBM) took detailedfield notes during and after class, writing down all ques-tions students asked of the lecturer and summaries of stu-dent discussions during clicker questions. The researcheralso took field notes immediately after problem-solvingsessions, writing summaries of her interactions with andobservations of students working on homework. Under-graduate learning assistants (LAs) who were hired to fa-cilitate student discussion during lecture and problem-solving sessions also took field notes, which provided anadditional perspective.

The case study interviews involved six students whowere taking the transformed course in Sp06. Each ofthese students participated in 11 interviews throughoutthe semester. Two of the interviews with each of thesestudents (12 interviews total) focused on quantum tun-neling. In the first interview on tunneling, students wereasked to go over the Tunneling Tutorial that they hadalready completed in class and homework. In the secondinterview, they were asked a series of questions aboutwhat happens to the energy and probability of tunnelingfor an electron approaching a barrier when the heightor width of the barrier is changed, culminating in thequestion in Figure 7.

We conducted interviews on the QMCS with 47 stu-dents, including 24 from transformed courses and 23from traditional courses. In these interviews, most ofwhich were conducted within two days after the QMCSpost-test was given in class, students were asked to statetheir answer to each question and explain their reason-ing out loud. Initially the interviewer intervened onlyto request that students talk more or to ask clarifyingquestions. After students had answered all questions,the interviewer asked more in-depth follow-up questionsabout issues raised in earlier responses, and if requested,helped students with the questions they had answeredincorrectly.

We conducted simulation interviews with six students:four from traditional courses and two who had taken thetransformed course the semester before we created thesimulation and were serving as LAs in the transformedcourse in the subsequent semester. In simulation inter-views, there was an initial period of free exploration, inwhich students were asked to play with the simulationand talk out loud with little interference from the in-terviewer, followed by a period of guided exploration, inwhich the interviewer asked students to explore aspects ofthe simulation they had missed or asked follow-up ques-tions about the concepts involved in the simulation.

All interview students were recruited by sending anemail to the entire class and paying volunteers $20 perhour for their time. The payment ensured a greater di-versity of volunteers than asking students to volunteer

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their time. (Some students stated that they were onlythere for the money and displayed little interest in thematerial.)

An electron with energy E is traveling through a conducting wire when it encounters a small gap in the wire of width w. The potential energy of the electron as a function of position is given by the plot at right, where U0 > E. Which of the following sketches most accurately describes a snapshot of the real part of the wave function of this electron?

w

e-

U(x)

xw0

U0>E E

A C B

D E

FIG. 6: A tunneling question from an early version of theQMCS. This question was developed to test students under-standing of tunneling wave functions. It has been removedfrom the QMCS because interviews suggest that it tests mem-orization rather than conceptual understanding.

While our data includes students working on many dif-ferent problems in many different contexts, a large partof it is based on the questions shown in Figures 6-7. Inparticular, the question in Figure 7 was asked as a multi-ple choice question in some versions of the QMCS, whichwas given in class as an ungraded practice test beforethe final, as an essay question in which students wereasked to “explain your reasoning” on the final exam onesemester, as a multiple choice question on the final examanother semester, and in interviews for the QMCS andcase study project.

By drawing on multiple forms of data, we have beenable to track similar responses among many courses, aswell as looking at changes in student thinking as fur-ther transformations were introduced into the curricu-lum. When we noted interesting patterns in observationsor interviews, we looked for corroborating evidence by re-viewing videos and transcripts of interviews, field notes,and responses to online participation questions, home-work, and exams.

VI. RESULTS

Even in our transformed courses, we saw some evidenceof most of the difficulties reported in previous researchon student understanding of quantum tunneling. How-ever, many of the previously reported difficulties wereless prevalent or appeared in more subtle forms than wesaw in traditional courses. We also saw many issues in

Suppose that in the experiment described in the previous question, you would like to decrease the speed of the electron coming out on the right side. Which of the following changes to the experimental set-up would decrease this speed?

A. Increase the width w of the gap:

B. Increase U0, the potential energy of the gap:

C. Increase the potential energy to the right of the gap:

D. Decrease the potential energy to the right of the gap:

E. More than one of the changes above would decrease the speed of the electron.

U(x)

xU0>E

E

U(x)

xU0

Ebecomes:

U1 w w

U(x)

xU0>E

E

U(x)

xU0

Ebecomes:

U1<E w w

U(x)

xw

U0>E E

U(x)

xU0>E

Ebecomes:

w

U(x)

xU0>E

E

U(x)

x

U0>E Ebecomes:

w w

FIG. 7: A tunneling question from an early version of theQMCS. This question was developed to explore the beliefthat energy is lost in tunneling. It has been removed fromthe QMCS because interviews suggest that responses fromstudents in transformed courses are not necessarily indicativeof whether students think energy is lost in tunneling.

our transformed courses that have not been previouslyreported.

A. Difficulties drawing wave functions

In a final exam question in the transformed course inSp06, we asked students to draw the real part of thewave function for an electron tunneling through an airgap in a wire (a square barrier of potential energy) andexplain their drawings. We were looking for a drawinglike that shown in Figure 6A or a similar drawing of atunneling wave packet. On this question, 18% of studentsdrew the wave function with an offset as in Figure 2a orFigure 6D, and 23% drew a shorter wavelength on theright than on the left as in Figure 2b. While these diffi-culties, both of which have been reported on extensivelyin the previous literature, were fairly prevalent, we no-ticed in exam responses and in interviews with studentsin both traditional and transformed courses that the rea-sons for these two difficulties were not particularly deep.Students never volunteered any reasons for drawing thewave function in either of these ways, and when asked byinterviewers to explain their drawings, they usually re-

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sponded that they just remembered it looking like this.Figure 6 shows a QMCS question designed to test stu-dents’ understanding of the wave function for a planewave tunneling through a square barrier. We note thatthe percentage of students who answered this questioncorrectly varied from semester to semester much morethan other questions, and in interviews, students rarelygave any reasoning for either correct or incorrect answers,but simply picked the picture that looked the most famil-iar or stated that they did not remember these pictures.In general, students’ explanations of their answers to thisquestion indicated that they were not reasoning throughit at all, but scanning for the picture that most closelymatched their memories of figures from the lecture ortextbook. Because this question appears to test mem-ory, rather than conceptual understanding, it has beenremoved from the QMCS.

B. Energy Loss: a new perspective

In the transformed courses, because there was such aheavy emphasis on energy conservation, students quicklylearned to say that energy is not lost in tunneling. Whenwe asked them directly on exams whether “the total en-ergy of an electron after it tunnels through a potentialbarrier is a) greater than, b) equal to, or c) less thanits energy before tunneling,” between 70% and 93% an-swered correctly that it is equal. In homework, whenstudents were asked “Does an electron lose energy whenit tunnels?” between 95% and 96% answered correctlythat it does not and gave clear explanations of their rea-soning, invoking the conservation of energy and the lackof dissipation. When the question in Figure 7 was askedas an essay question on the final exam, only 7% of allstudents (31% of those who answered incorrectly) explic-itly said that energy is lost in tunneling in their response,although a much larger percentage gave answers that im-plied energy loss.

In spite of students’ correct answers on direct ques-tions, energy loss in tunneling continued to be an issue.After the instruction described in Section IV, studentsasked repeatedly in lecture, problem-solving sessions, andonline participation homework, “Why is the total energythe same after tunneling?” (These were often the samestudents who had given correct and clear answers to thisquestion in earlier homework.) In questions about tun-neling that did not directly ask about energy loss, somestudents continued to give answers that implied that en-ergy is indeed lost in tunneling. Table I lists all the rea-sons we have seen students give for energy loss in inter-views and in responses to an essay question on an exam.As discussed in Section III, reasons 1 and 2 are the stan-dard reasons that have been given in most previous lit-erature and reason 3 was postulated in a previous study.Reasons 4-6 are new to the current study.

In this section, we will outline the general results fromthe question shown in Figure 7, and will then discuss

Reasons students may think energy is lost in tunneling(1) Treating energy and wave function interchangeably(2) Invoking dissipation(3) Using the energy-amplitude relation for EM waves(4) Confusion over how electron regains energy when it reen-ters wire(5) Treating total energy as a local characteristic of wavefunction(6) Difficulty connecting energy to wave functionrepresentation

TABLE I: Reasons students may think that energy is lost intunneling. Reasons 1-3 are discussed in previous literature,and reasons 4-6 are new to the current study.

the evidence for each of the reasons given for energy lossin Table I from interviews, observations, and student re-sponses to an essay version of the question in Figure 7asked on a final exam. Unless otherwise noted, all inter-view quotes are from students in transformed courses.

The question in Figure 7 was originally developed totest whether students think that energy is lost in tunnel-ing. In order to answer this question correctly, studentsmust recognize that total energy is constant and deter-mine that since potential and kinetic add up to total en-ergy, the way to decrease the kinetic energy on the rightmust be to increase the potential energy, as in the correctanswer, C. The distracters A and B are very effective ateliciting the belief that energy is lost in tunneling, sincestudents who think that energy is lost will usually thinkthat more energy is lost in one or both of these cases. Ininterviews with students in traditional modern physicscourses, we found that students’ answers to this questionwere good indicators of whether they believed that en-ergy is lost in tunneling; all students who did not choosethe correct answer expressed the belief that energy is lostin tunneling. [7]

However, in later interviews with students in our trans-formed modern physics class, we found that even stu-dents who explicitly said that energy is not lost in tun-neling sometimes chose incorrect answers, often for verysubtle reasons. (Occasionally, students even argued foranswers A and B by saying that no electrons will tunnelif you make the barrier sufficiently high or wide, and ifno electrons are coming out, you could say the speed iszero. While one could argue that these students were us-ing questionable logic, their incorrect answers were notdue to a misunderstanding of the physics.) We havealso found that this question is much more difficult thanother questions eliciting the idea of energy loss, with only37 − 58% of students answering correctly the first timethey see it. Students’ ability to answer this questionalso varies greatly depending on context. As shown inTable II, when the question was asked on the QMCS,an ungraded multiple choice conceptual survey that wasused as a review for the final exam, 37 − 41% of stu-dents in the transformed course for engineering majorsanswered correctly (higher than the scores in the tradi-

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Course A B C D E NTraditional Eng. Sp05 (QMCS) 18 10 24 15 33 68Traditional Phys. Sp05 (QMCS) 19 11 38 9 23 64Traditional Phys. Fa05 (QMCS) 12 11 38 15 24 54Traditional Phys. Fa06 (QMCS) 13 13 38 9 26 54Transformed Eng. Fa05 (QMCS) 12 10 37 5 37 162Transformed Eng. Fa06 (QMCS) 20 6 41 8 25 73Transformed Eng. Sp07 (QMCS) 13 10 40 7 31 120Transformed Eng. Sp06 (Exam essay) 2 3 58 5 31 177Transformed Eng. Sp07 (Exam mul-tiple choice, after QMCS)

1 2 90 6 0 147

TABLE II: Percentage of students who selected each answerto the question shown in Figure 7 in various courses. N is thenumber of students.

tional course for engineering majors and comparable tothe scores in the traditional course for physics majors).However, when we gave it instead as an essay questionon the final exam, asking students to “explain your rea-soning,” 58% answered correctly. We hypothesize thatthe process of explaining their reasoning led more stu-dents to figure out the correct answer; in interviews, wesaw that many students initially answered with a varietyof incorrect reasoning, but in the process of attemptingto explain their reasoning to the interviewer, eventuallycame to the correct explanation. When we gave it as amultiple choice question on a final exam after asking iton the QMCS and reviewing it in class, so that studentswere already familiar with it, 90% answered correctly.

Because of the subtlety of the reasons students givefor their answers and the context dependence of thescores, we no longer recommend the use of this questionin multiple choice format as a diagnostic. However, wehave found that it is extremely valuable for elicitingstudent thinking when used in an interview setting or asan essay question on exams.

Reasons 1-2: Treating energy and wave functioninterchangeably and Invoking dissipation

In the transformed courses, the two standard reasonsfor energy loss did arise, but were relatively infrequent.For example, when the question in Figure 7 was askedas an essay question on the final exam, only 10% of allstudents (43% of those who answered incorrectly) relatedthe decrease in speed to the exponential decay of the wavefunction, implying that they were treating energy andwave function interchangeably (reason 1), and only 16%of all students (67% of those who answered incorrectly)said that it requires more energy, or is harder, to tunnelthrough a wider or higher barrier, implying dissipation(reason 2).

Further, when students in interviews or problem-solving sessions seemed to have the common problemsof treating energy and wave function interchangeably orthinking of dissipation, they usually corrected themselveswithout intervention from the interviewer or instructor.An example can be seen in an interview with a studentwho is struggling to answer the question in Figure 7. The

student begins with a typical response in which she in-terprets the height of the wave function as the kineticenergy:

It’s either one of these two [A or B]. I’mjust trying to think about it. I think it’sthis one right here [B], because it would–thewave function would come up here and then itwould drop down a little bit but then it wouldkeep going, and the distance between it andthe potential energy would be the kinetic en-ergy, kind of ... Well, no, this is–scratch that.We’ll take it out. Because the energy has tobe the same on both sides...

After she answered the question correctly, the interviewerasked her to explain what she was thinking before. Shesaid:

Yeah. Um, I was thinking [pause] that a lotof times when I see these, I’m thinking of thewave function on top of it [draws wave func-tion on top of energy graph - see Figure 8] andI’m thinking of it dropping down a certain–dropping down, like, a certain rate depend-ing on the difference between the energy–theelectron’s energy and the potential energy orthe width. So I think about it that way. SoI was thinking, once it–if it’s coming up hereand it drops down a little bit, it’s gonna comeup here on this side. And I’m kind of think-ing, maybe, like, the amplitude of the wavefunction had to do with energy and so its dis-tance from this potential was the kinetic en-ergy, or kind of could represent the kinetic en-ergy. But then I wasn’t too sure about that,because I realized I was kind of thinking ofthe wave function instead of the energy, so Ihad to, like, re-evaluate how I was thinkingabout it, even though it kind of still worksthe same.

FIG. 8: A student drawing of a wave function on top of apotential energy graph.

It is interesting that this student instinctively thoughtof drawing the wave function on top of the energy graph,although this semester, aside from in the question shown

10

in Figure 3, there were no pictures of a wave function ontop of an energy graph in the lectures, textbook, or sim-ulations. This example illustrates that eliminating suchpictures, while helpful, is not sufficient to address theproblem of students treating energy and wave functioninterchangeably.

Thus, in interviews, students in the transformedcourse were usually able to let go of the typical ideasthat lead to belief in energy loss. However, thesestudents often did say in interviews that energy is lost intunneling. There are four further reasons they gave, allof which are distinct from the reasons most often givenin the literature.

Reason 3: Using energy-amplitude relationfor EM waves

In the transformed courses, the relationship betweenamplitude and energy for electromagnetic waves was veryheavily emphasized in the section on the photoelectriceffect in lecture, homework, and exams. Occasionallyin interviews and observations of students in the trans-formed courses, but never in interviews with students inthe traditional courses, students pointed out this rela-tionship, and asked why it was not the same for matterwaves, or assumed that it was the same. For example,in one interview, after a student had drawn a wave func-tion with the same amplitude on both sides of the barrier“because the kinetic energy’s the same, the total energy’sthe same, the potential energy’s the same on each side,”he corrected himself and explained, “I was thinking moreof the electromagnetic waves when I was thinking aboutthat.”

This reason was not very common and was onlyobserved in relatively strong students. We mention ithere mainly because it has been discussed in previousresearch. [2]

Reason 4: Confusion over how electron re-gains energy when it reenters wire

One reason that some students give for energy beinglost in tunneling appears to be associated with the partic-ular physical example we use in class, that is, an electrontraveling through a wire and tunneling through an airgap. Students can easily grasp the physical mechanismby which kinetic energy is lost when it goes from the firstwire into the air gap. Earlier in the course, in the con-text of the photoelectric effect, we discuss the energy re-quired to overcome the work function of the metal. Moststudents seem able to apply this concept to tunneling,recognizing that the electron loses kinetic energy when itescapes the wire into the air gap. However, students donot understand the mechanism by which the electron re-gains its kinetic energy when it goes into the second wire.Therefore, while they know that energy is conserved, theyexpress confusion over how the electron “gets back” theenergy that went into overcoming the work function.

For example, one student, who was sufficiently both-ered by this issue that she had asked her friends about

it, said in an interview:

The kinetic energy starts at E and then itdrops down, takes energy to get up, and thenit jumps back up to E. I talked to my friends.Why the hell, they don’t understand that ex-actly.

In another interview the following week, the same studentbrought up the issue again:

Yeah, because it takes energy to get out ofmetal, the work function. And it takes theamount of the potential energy–the barrier,this is the barrier’s, so it uses that energy upand then it has a much slower–so it’s goingmuch slower. And then once it hits the othermetal, hey, it’s going fast again... It’s justweird, a little bit. You’d think it would slowdown, but it is because the potential drops tozero again and conservation of energy, all theenergy goes to kinetic. But it is a weird ideafor me to think about.

We have found that presenting students with a grav-itational analogy of a ball rolling over a hill, in whichthe kinetic energy is lost as the ball rolls up the hill andregained as the ball rolls down the hill, helps students re-solve this confusion. In every case we have seen, the mainproblem is that students are not able to apply the con-cept of converting potential energy to kinetic energy tothis novel situation. However, they are quite comfortablewith this concept in the context of gravitational poten-tial energy, and once they are reminded of this familiarsituation, they can apply the new concept to the new sit-uation. For example, after the interviewer suggested theanalogy and asked why the ball regains its kinetic energy,the student quoted above said:

Because the gravitational force supplies itwith energy. Going back down, like, the po-tential increases as you get farther and fartherfrom the ground. When you’re lifting some-thing up, it gives it potential energy. Andthat’s what–and as the ball rolls back down,it gains kinetic energy, because it’s going backdown. So the force gives it energy, I guess.So maybe I’m just confused on what’s the–hmm. [pause] OK. Because I understand thisreally well. [points to picture of ball on hill]But why the electron is–or what the electron’sdoing and what–what’s the force here that isgiving it the energy. Here gravity and poten-tial, I understand that. Maybe here I don’tquite understand what’s giving it back its–like, what’s the force involved that’s giving itback its energy, in a sense... Yeah, I guessthat’s maybe my confusion. But when youlook at it that way, it’s really easy to under-stand. This is like a hill. It comes back down.

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It is worth noting that this student traced her confu-sion back to not knowing what the force was. In quan-tum mechanics instruction, we typically ignore forces al-together and speak only in terms of potentials.

Some researchers are reluctant to use gravitationalanalogies in teaching quantum tunneling, for fear thatthey may lead to the idea that tunneling involves a par-ticle traveling through a physical barrier like a hill andexacerbate the difficulty of thinking that energy is lostdue to dissipation. However, we have seen no evidenceof such a link. Further, as Brookes and Etkina [20] pointout, the gravitational analogy is already inherent in thelanguage we use, as seen in phrases such as “potentialstep” and “potential barrier,” and even in the word “tun-neling.” Even if we were careful to avoid such language,there is evidence that students have a tendency to in-terpret graphs too literally and think that higher on agraph means higher in space, regardless of context. [37]The gravitational analogy is an important aspect of theexpert model of tunneling. Therefore, we argue that itis preferable to address the strengths and limitations ofthe gravitational analogy directly, rather than to avoidits use.

Another possible concern about the gravitational anal-ogy is that the ball must have enough energy to get overthe hill, and thus it is not actually tunneling. However,confusion over how the electron gets back its kinetic en-ergy arose both in cases with the total energy less thanthe potential energy of the gap, as in tunneling, and incases with the total energy greater than the potentialenergy of the gap, as in the first example in the Tunnel-ing Tutorial. Thus, the difficulty does not seem to beparticular to the case of the total energy being less thanthe potential energy of the gap, and we have not seenany problems as a result of the gravitational analogy notcorresponding to tunneling.

Out of six students who were interviewed extensivelyon the relationship between the energy and wave func-tion in tunneling, two exhibited this difficulty, and it wasobserved in students working on homework in problem-solving sessions. There is also some indirect evidencefor this difficulty in some of the responses to the essayversion of the exam question shown in Figure 7: 11%of all students (48% of those who answered incorrectly)argued for option A and/or B by pointing out that theelectron would be slower in the gap in these cases thanin the original case. While it is possible that some ofthese students simply misread the question and thoughtit was asking how to slow the electron inside the gap,rather than to the right of the gap, it is clear from atleast some of these responses that this is not the case.For example, “want to decrease its KE coming out. Wecan only do this by increasing the PE in order to borrowmore KE from the system” and “a greater PE meansa decrease in KE the e- will have once it merges onright side. Both B & C would cause this result.” Theseresponses imply that students correctly understoodthat increasing potential energy reduces kinetic energy,

but did not recognize that the kinetic energy wouldincrease again when the potential energy goes back down.

Reason 5: Treating total energy as a localcharacteristic of wave function

Another reason students give for energy loss is that,since only part of the wave function is transmitted, onlypart of the energy is transmitted, with the rest being re-flected. For example, when a student working on home-work during a problem-solving session was asked to drawthe potential and total energy of a tunneling electron,he drew a picture like the one shown in Figure 9. Heexplained that the dotted line on the top left was the to-tal energy of the incoming particle, which was then splitinto the reflected part (bottom left) and transmitted part(right).

FIG. 9: A reproduction of a graph drawn by a student torepresent the potential (solid line) and total (dotted lines)energy of a particle tunneling through a barrier. This studentsaid the particle was losing energy because only a part of itwas transmitted.

Another student, who was attempting to answer thequestion in Figure 7 in an interview, after explicitly sta-ting that energy is not lost in tunneling, used a similarargument in an interview to justify her intuitive beliefthat the energy must be less on the right side of the bar-rier:

Interviewer: ...does that mean that the to-tal energy is going down when it goes throughthe barrier?

Student: The total energy is constant.I: Ah! OK, so what energy is decreasing

then, if it’s not the total energy?S: The energy— the energy— man—

Well, OK. What I’m saying, but what I’msaying with caution— is— the energy of thewave function on this side– is decreasing. Iwant to make the energy of the wave func-tion on this side decrease. But I’m also waryabout that because— ‘the energy of the wavefunction on this side’? You know, the wavefunction is a wave function, and it has likeparts to it, but it doesn’t have like— No,it does— You can have a wave function likethat... and it has a different energy here thanit has here.

I: Different total energy?S: No, total energy is of the entire wave

function. What is total energy then? Is it

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this plus this plus this? Yikes! I need tostudy this for the final.

This difficulty appears to be caused by a lack of under-standing of the fact that the total energy is a non-localproperty of the entire particle, rather than a local func-tion of position. This fact is not stressed in our class, norin any class we know of, but perhaps it should be.

We note that in both examples discussed above, theproblem involved a single electron tunneling, not an en-semble, and the students described it as such. Becauseour course did not discuss probability current, it is un-likely that this difficulty was due to students thinkingof probability current or ensembles of electrons splitting.We further note that in both cases, students had beenintroduced to tunneling in terms of both wave packetsand plane waves, so the difficulty cannot be attributeddirectly to either of these approaches.

While this difficulty has only been observed with thetwo students discussed above, these two examples arefrom courses in different semesters, taught by differentinstructors using different textbooks. Further, theproblems they were working on were quite different andthe two students had very different personalities.

Reason 6: Difficulty connecting energy towave function representation

Another reason students give for energy loss in tunnel-ing is related to the confusion between energy and wavefunction seen in reason 1, but is more subtle. In sev-eral interviews, we have observed students who explicitlystate that the energy is not the same as the wave func-tion, but that the two must be related somehow, so theexponential decay of the amplitude must imply a lossof energy. These students are careful to distinguish be-tween energy and wave function and do not use theminterchangeably, but believe that there must be some re-lationship between them. While reason 1 is simply afailure to distinguish between the two quantities, reason6 is a search for a simple causal relationship that doesnot exist.

For example, when asked whether the probability oftunneling would change if the width of the barrier in-creased, a student said some things made him thinkit would decrease, and other things made him think itwouldn’t change. When the the interviewer asked whatmade him think it wouldn’t change, he said:

Well, it would be diagrams like this. [pointsto energy graph] One thing that the textdoesn’t really have–doesn’t focus nearly asmuch as you guys do in the course, and Idon’t know if that’s good or bad, are thesediagrams. You guys use these diagrams a lot,which is great. The text doesn’t so much,it sort of approaches it in a little differentway. So if we are to evaluate these diagrams,put our total energy line in, evaluate howthat corresponds with our potential energy,

you–it sort of–[pause] maybe this forces me tothink too much about energies. For example–I mean, that’s more classical physics, is itnot? If the particle has sufficient energy toget to the other side. Quantum’s a wholeother story where we’re not talking about somuch energies. We are, but we’re also talkingabout probabilities, correct? So there’s sortof two ways to think about this, and maybethat’s why I’m a little confused still, at thislate date.

Another student, when asked whether the probability oftunneling would change if the initial energy of the incom-ing particle decreased, said:

The amplitude shouldn’t be affected by theenergy other than its exposition. Yeah. Ithink. And then–I believe it’s still gonna dothe exponential decay. [draws] OK. So now–OK, so, hmm, probability of the electron tun-neling through the barrier. The difference be-tween the total energy and the potential en-ergy of the gap is larger now, so I would say–Ifeel like, um, that would mean that it has lessof an opportunity, less chance, less probabil-ity of it tunneling through. What am I tryingto say here? When an electron has to converta certain amount of kinetic energy to comeout of a wire to potential energy, and in thiscase it has to convert this much [points] orthis much will be potential energy, that dif-ference there, which is more than the originalcase. So [pause] I don’t know. Um, I’m notquite figuring out how to connect it. But thelarger difference between the total energy andthe potential energy of the gap I think hassomething to do with the probability of theelectron tunneling through or not, comparedto the first.

This difficulty reveals why emphasizing that the wavefunction and energy are not the same thing is not suffi-cient to address the student belief that energy is lost intunneling. Even if students realize that energy and wavefunction are not literally the same thing, they struggleto make connections between these two quantities thatare emphasized in the study of quantum mechanics. Onequantity, the wave function, is wholly unfamiliar to stu-dents, and the other quantity, energy, is treated in anunfamiliar way: graphed as a function of position butapplied to a delocalized object.

Out of six students who were interviewed extensivelyon the relationship between the energy and wave functionin tunneling, four exhibited some form of this difficulty.It was also observed in students working on homework inproblem-solving sessions. It also came up in class whenstudents were working through the Tunneling Tutorial.Although the tutorial specifically said not to worry about

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the relative magnitudes when sketching wave functions,many students asked how to figure out the amplitudefrom the energy and expressed frustration that they couldnot find a connection.

Unlike the previous three difficulties, this was also ob-served in students in the traditional courses. For exam-ple, one student from a traditional course, when askedhow the wave function is related to the energy, replied, “Ican’t remember. I wish I did. But I can swear, well notswear, but I can almost remember my professor sayingthat the energy is encoded in the wave function, some-how, I can’t remember exactly now.”

C. Giving Potential Energy a Physical Context

One conclusion of our study is that understanding thecontext of potential energy graphs is a difficult task forstudents, and a great deal of instruction is needed to ad-dress this issue. In a previous study [7] we reported on in-terviews with students in the traditional modern physicsfor engineering majors course in Spring 2005. Students inthis course had no idea what the potential energy graphsmean. In the transformed courses, we observed that stu-dents still struggled with the basic meaning of potentialenergy graphs, but as we refined our curriculum, theirquestions about these graphs became more sophisticated,illustrating a struggle to relate the graphs to physical re-ality in a deep and meaningful way. The extent of ques-tioning from students in the transformed course indicateswhat a difficult subject this is, and how hopeless it isto expect students to build meaningful models of thesegraphs if the course does not explicitly help them do so.

In Fall 2005, the first semester of the transformedcourse, even after focusing on the physical context ofpotential and explicitly addressing possible confusionarising from sloppy language in the text using “poten-tial” and “potential energy” interchangeably, studentsexpressed a great deal of confusion over the meaning ofpotential energy. In weekly online extra credit, we askedstudents to submit any unanswered questions they hadabout the course material. Here is a sample of thesequestions regarding potential energy:

• “I get very confused by exactly what an infinite wellis. What is it, how is it infinite? do we just makeit that way?”

• “I have trouble understanding what the potentialis when we are looking at models of an electron ina wire, free space, finite square well, infinite squarewell. I am sort of getting this idea of it being similarto a work function in that once the potential (V) isless than the potential energy, the electron is out ofthe wire. I can usually follow the math/calc thatfollows the examples okay, but the overall conceptof this potential (V) still confuses me, and so I stilldon’t have a firm grasp of [what] the square wellmodels mean/represent/whatever.”

• “I cant find a general description of an infinite well,i understand what it does but not what it is orwhere its used.”

• “Voltage is used when we talk about electromag-netic forces, like the coulomb force. What I’m con-fused about is that we used a voltage well to showthe strong force in effect. Is it accurate to show thestrong force as a very deep voltage well?”

The first three students were struggling to make sense ofwhat the potential energy diagram for an infinite squarewell means and how it relates to a real physical system.The last student thought that the symbol “V ” that weused for potential energy represented voltage, althoughwe had pointed out repeatedly in class that it did not.

Further evidence for student confusion about poten-tial energy can be seen in our observations of students’responses to the first question of our Tunneling Tutorial,which asked students to draw the potential energy as afunction of position for an electron traveling through along copper wire and tunneling through an air gap (seeFigure 4b). At this point in the course, students hadworked extensively with a square well as a representa-tion of the potential energy of an electron in a wire, buthad not previously seen this example of a wire with an airgap. We expected them to use their knowledge of the po-tential energy of individual wires to draw a square barrierfor this new situation. While many students did draw thecorrect potential energy, we observed that many studentsgot it backwards, drawing a well instead of a barrier. Acommon student explanation for the well was that the airgap was a “hole” and therefore should be represented bya well. This response betrayed a lack of understandingof why a well represents the potential energy of the wire.In subsequent semesters, we added instruction before theTutorial on how to build up a square well by superim-posing the Coulomb potentials of all the individual atomsthat make up a wire. As illustrated in Figure 10, addingthe potentials of all the nuclei that make up a wire pro-duces a potential similar to a square well but with a dipat each nucleus. Since the electrons are mobile and at-tracted to these dips, adding in the potentials of the elec-trons tends to smooth out the potential to make it evenmore like a square well. After this instruction, anecdotalobservations indicated that fewer students drew a well in-stead of a barrier in the Tutorial. In later semesters, wealso demonstrated this concept with the Quantum BoundStates PhET simulation.

Students still struggled to relate the potential energygraph to reality, to the extent that some viewed the graphand the electron in the wire as describing two differentthings. Here is an example from an interview in whicha student was trying to figure out how the width of thebarrier affects the probability of tunneling:

But I don’t know if it explained as wellas it needed to or if I just didn’t understandas well as I needed to whether width [holdsout thumb and forefinger to indicate width of

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FIG. 10: An illustration of how to build up a square well byadding up the Coulomb wells of individual atoms (taken fromPowerPoint slides used in lecture in Sp06, Fa06, and Sp07).

space between them] meant actual real classi-cal physics width [holds hands out to indicatewidth of space between them] or more theo-retical width [points to potential barrier onsheet], which is like the–which might be morerepresented here.

Another example of a student struggling with potentialenergy graphs can be seen in a student who asked a ques-tion after class that revealed that he was misinterpretingthe pictures in Figure 4 as meaning that the wire wassitting on top of the potential energy.

Students also worried about the applicability, limita-tions, and relevance of the model of the square well foran electron in a wire. For example, students frequentlyasked about collisions with the atoms in the wire, andwhether these would constitute measurements of the elec-tron and localize it. In discussing tunneling, they strug-gled with the concept of infinitely long wires, and fre-quently discussed the reflection of the electron when itreached the end of the wire. While working through thetunneling tutorial, one student asked why the electronwould flow from one wire to the other if there was nopotential difference between the two wires. The answerto this question is that you would not have a net flowof electrons from left to right without a potential differ-ence, but that electrons would constantly flow back andforth due to thermal energy. This student also askedwhether you could really measure a single electron flow-ing through a wire and why we were studying it if youcouldn’t. He was satisfied only after a long explanationof how you could predict net current by adding up theeffects of single electrons. This example demonstratesthat even with a physical context, a square barrier withan equal potential energy on either side (the prototypicalsystem used in the standard presentation of tunneling) isstill artificial because in reality a net current does notflow without a voltage between the two sides of the bar-

rier. These questions further demonstrate that physicalcontext is important, not just for giving the material rel-evance, but for conceptual understanding of the materialitself.

V(r)

r sampletip

applied voltage

total e-

energy

fingerdoorknob

total e-

energy

(a) Alpha Decay (b) Scanning TunnelingMicroscope

(c) Getting shockedby a doorknob

FIG. 11: Potential energy graphs for (a) an alpha particle un-dergoing alpha decay, (b) an electron in a scanning tunnelingmicroscope, and (c) an electron in your finger when you getshocked by a doorknob. Determining how to draw each ofthese graphs requires many subtle approximations.

Student difficulties with potential energy can also beseen in the questions they asked during the section onthe applications of tunneling, which included alpha de-cay, scanning tunneling microscopes, and getting shockedwhen you rub your foot on the carpet and approach ametal doorknob. We asked students to figure out thepotential energy graphs for each of these applications,shown in Figure 11, using a series of concept questionsin lecture, as well as more detailed questions in home-work. Determining each of these potential energy graphsrequire many subtle approximations, which may not beapparent until one is faced with a barrage of student ques-tions. For example, to determine the potential energygraph for alpha decay, one must approximate the strongforce as a flat potential throughout the nucleus, althoughthere is no model in nuclear physics that predicts such apotential, one must recognize that the strong force dom-inates in the nucleus and the Coulomb force dominatesoutside, and one must treat the alpha particle that isgoing to be ejected as having an independent existenceand a well-defined energy prior to decay. Gurney andCondon [38] explicitly discussed all of these approxima-tions in their 1929 paper explaining radioactivity on thebasis of tunneling. Yet most textbooks simply give suchgraphs without explanation.

The following questions from students illustrate thatour students struggle with these approximations:

• “How do the Coulomb force and the strong forcerelate to each other?”

• “How do you find the distance where the strongforce takes over?”

• “Is the potential really square like that?”

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• “Do alpha particles already exist in the nucleus orare they created upon radioactive decay?”

In the first two questions, students are struggling withthe assumption that the strong nuclear force dominatesin the nucleus, and the Coulomb force dominates outsideof it. The last two questions illustrate the simplificationsrequired to come up with a solvable model.

Similar questions illustrated students’ struggles to un-derstand the potential energy graph for a scanning tun-neling microscope:

• “As the electrons tunnel through, isn’t the samplepotential energy going to drop?”

• “The quantum tunneling microscope can be usedon any material even though not every material hasa “sea” of electrons? Wouldn’t losing an electron ina crucial covalent bond break the molecule apart?”

The answer to the first question is that the potential en-ergy would drop if the sample were not hooked to a volt-age supply to keep the voltage constant. This studentmissed the function of the voltage supply, but the ques-tion illustrates that he was thinking carefully about thephysical system. He also recognized that the behaviorof the electrons could actually change the overall poten-tial energy, a fact which is never discussed in the stan-dard presentation, where the potential energy function istaken as a given. The answer to the second question isthat scanning tunneling microscopes do not work on insu-lators, an issue that is never discussed in modern physicscourses, but is the focus of a recent Nature article [39].

In spite of all these difficulties throughout the course,when we asked students to explain the physical mean-ing of the potential energy graph of a square barrier ona homework question towards the end of the last twosemesters, nearly all gave clear and correct explanationsand related the graphs to a real physical context.

Further, from interviews with 24 students in the trans-formed courses, there was only one case in which a stu-dent treated the potential energy graph as an externalthing unrelated to the potential energy of the electron,as we saw consistently in interviews with students in atraditional course in an earlier study [7]. This case wasso exceptional, especially because it was a particularlygood student (he received an A- in the course), that theinterviewer asked him afterwards if he had done the Tun-neling Tutorial. He said he had been busy that weekand skipped it, and jokingly commented, “In conclusion,that’s a good assignment, because you should listen tothis guy try to explain it!”.

D. Plane waves

Plane waves cause further barriers to student under-standing. While plane waves are mathematically simple,conceptually it is quite difficult to imagine a wave thatextends forever in space and time, especially when it is

tunneling. The language we use to describe tunnelingis time-dependent. For example, we say that a particleapproaches a barrier from the left, and then part of it istransmitted and part of it is reflected. This language isdifficult to reconcile with a picture of a particle that issimultaneously incident, transmitted, and reflected, forall time. The following student quote, from a homeworkquestion asking what questions students still had abouttunneling after instruction, illustrates the kind of confu-sion created by using plane wave solutions:

Say you have two finite lengths of wire veryclose together. I don’t really see how we as-sume the electron is in one wire, get a solu-tion, then use that to determine psi acrossthe gap, and then use that to determine theprobability that the electron is in the otherwire. Over time don’t the probabilities evenout (i.e. we have no clue which wire the elec-tron’s in)?

This student is actually struggling with two common is-sues for students: confusion over the physical meaningof plane waves, and concern over what happens whenthe electron gets to the end of the wire. Many studentshave trouble with the idea of wires extending to infinity,and talk about the electron waves reflecting off the endof the wire, interfering with themselves, and creating abig mess. This is physically accurate, but outside of therealm of standard treatment, which assumes that wiresdo not have ends.

In student interviews to test the usability and effective-ness of the Quantum Tunneling and Wave Packets sim-ulation, we saw that students were much more comfort-able with the wave packet representation than the planewave representation. We conducted interviews with sixstudents, all of whom had completed either a trans-formed (2) or traditional (4) course in modern physics.These students were asked to explore the simulation andthink out loud. The interviews started with free explo-ration, followed by questions from the interviewer aboutaspects of the simulation that the students had not ex-plored on their own. All students discovered plane wavemode on their own (the simulation starts in wave packetmode), but four out of six switched back to wave packetmode immediately and the other two only explored itfor a few minutes before switching back. Two studentsswitched back without comment, one commented thatthe plane wave was “too unrealistic,” one commentedthat he didn’t remember what a plane wave was andwas more familiar with a wave packet, and one com-mented, “That’s definitely a visualization I didn’t thinkof.” Only one student commented that plane wave modemade sense. Some students did eventually return to planewave mode in order to explore specific features, but allstudents spent most of the free exploration time in wavepacket mode, and quickly returned to wave packet modeafter answering the interviewer’s questions about planewave mode. One student, after trying it and switching

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back without comment, who didn’t use plane wave modeagain until the interviewer asked him to explore it, saidhe had forgotten about it.

It is possible that these students’ clear preference forwave packets over plane waves was merely due to thewave packet representation being more visually appeal-ing. However, we note that students were able to use thewave packet mode and learn new physics from it, in spiteof its complexity.

Because the Quantum Tunneling and Wave Packetssimulation provides such a compelling visual representa-tion, it immediately brings to the surface several trou-bling issues regarding plane waves that are swept underthe rug in standard treatments of tunneling because text-books focus on only a few special cases in which theseissues are not apparent.

FIG. 12: A case where the amplitude of the transmitted waveis higher than the amplitude of the incident wave.

According to the standard probability interpretation ofthe wave function, the reflection and transmission prob-abilities should be given by the area under the reflectedand transmitted parts of |ψ|2, respectively, divided bythe area under the incident part of |ψ|2. Since all theseareas are infinite, one can’t calculate the reflection andtransmission probabilities as one would naively expect.It is quite tempting (and quite wrong) to assume thatthe infinite widths simply cancel and that relative ampli-tudes should be a good indication of relative probabili-ties. However, this is not necessarily the case for planewaves, and the simulation reveals that there are evencases in which the amplitude of the transmitted waveis larger than the amplitude of the incident wave (seeFigure 12). This is such a surprising result that manyexperts, when they first see such a case, think there is abug in the simulation.

Students often cue off the amplitude of the plane waveas a measure of probability and draw incorrect conclu-sions. In observations of students attempting to calculatereflection and transmission coefficients during problem-solving sessions, we noticed that many students initiallyassumed that they were given by:

R = |B|2/|A|2 (1)

and

T = |C|2/|A|2 (2)

where A, B, and C are the amplitudes of the incident,reflected, and transmitted waves, respectively. Theseequations happen to be correct for plane waves tunnel-ing through a square barrier with the same potential onboth sides, since the particle speeds happen to cancel,but Equation 2 is wrong for a step potential or for anyother situation in which the potential is different for theincident and transmitted waves.

Thus, both faculty and students tend to assume thatthe amplitude alone is an accurate indicator of probabil-ity and make mistakes as a result. Yet most textbooksquickly gloss over this issue. Most quantum mechanicstextbooks simply state that the reflection and transmis-sion coefficients for plane waves are determined by theprobability current, without explaining why it is neces-sary to introduce this concept here and not elsewhere.(One textbook [24] justifies defining R and T in terms ofprobability fluxes by saying that it is done “by acceptedconvention” in order to ensure that R+T = 1!) In manymodern physics textbooks, this issue is not discussed atall, and the equation for the transmission coefficient issimply given, either in terms of particle velocities (v) orwave numbers (k),

T =vt|C|2vi|A|2 =

kt|C|2ki|A|2 (3)

(the subscripts i and t denote the incident and transmit-ted waves, respectively), with no explanation of wherethe factors of k or v come from. Some textbooks sim-ply give Equation 2 for the transmission coefficient, withno mention that this applies only for the special case inwhich vt = vi. [19, 40–44] While we understand that theauthors of these textbooks are attempting to avoid exces-sive mathematics that would obscure the basic concept oftunneling, presenting the transmission coefficient only forthis special case leads students to draw many incorrectconclusions when attempting to extend their knowledgeto other contexts.

We know of only two textbooks that give further jus-tification by deriving the probability for a wave packetin the limit that the width goes to infinity. [45, 46] How-ever, even in these books, it is not intuitively clear whyan infinitely wide wave packet should lead to a probabil-ity proportional to the particle speed. We recommend analternative treatment suggested by Lande et al. [47], inwhich reflection and transmission coefficients are derivedfrom wave packets, demonstrating that the factor of the vresults from the fact that the widths of the reflected andtransmitted wave packets are a function of the speed atwhich they move in their respective media. This deriva-tion is more intuitive than the derivation from probabilitycurrent, both because it relates more easily to the typicaldefinition of probability as it relates to the amplitude ofthe wave function, and because wave packets are morephysical than plane waves.

A second problematic issue that is often swept underthe rug is the issue of wave speed vs. particle speed.Because the treatment of waves is being pushed out of the

17

physics curriculum at many institutions, many studentsdo not know the difference between phase velocity (vφ =ω/k) and group velocity (vg = dω/dk). For a Schrodingerwave function, the phase and group velocity are given by:

vφ =~k2m

+V

~k(4)

vg =~km

(5)

While the velocity of a particle corresponds to the groupvelocity of its wave function, the only velocity appar-ent in the visual representation of a plane wave is thephase velocity. The distinction causes confusion whenthe potential energy changes. Students can see in thesimulation that if they increase the potential energy, the“wave speed” increases, which seems to contradict theirintuition that increasing the potential energy should de-crease the kinetic energy, and therefore the speed (sinceKE = E − V ). In fact, increasing the potential energyincreases the phase velocity, or wave speed, but decreasesthe group velocity, or particle speed. The only way weknow to gain any physical intuition for the group veloc-ity of a plane wave is again to imagine it as an infinitelywide wave packet, in which case the group velocity is thespeed at which that wave packet travels.

The distinction between wave speed and particle speedalso causes problems in trying to explain why the prob-ability is proportional to the current and not simply tothe square of the amplitude of the wave function. As dis-cussed above (see Figure 12), the transmitted amplitudecan be larger than the incident amplitude if the trans-mitted particle speed is smaller. However, in all suchcases, the wave speed is actually larger, so it appearsthat the transmitted wave has larger amplitude and ismoving faster, obscuring the correct explanation, that ithas a smaller particle speed to compensate for the largeramplitude.

We point out these issues so that instructors will beaware of the complexities inherent in discussing planewaves and consider the advantages of focusing on morerealistic wave packets. We do not have solutions for howto address the difficulties with plane waves (aside fromavoiding plane waves and focusing on wave packets), andwe hope that other researchers will pursue these ques-tions further.

E. Representations of complex wave functions

Students often have difficulty understanding the mean-ing of complex wave functions. This can perhaps best beillustrated by the observation that students frequentlyask, “What is the physical meaning of the imaginary partof the wave function?” but never ask about the physi-cal meaning of the real part, even though both have thesame physical significance.

Ambrose [1] found that some students believe that thewave function is only “real” in classically allowed regions,

so that the real part is zero inside the barrier. We sawthis problem in one interview.

All of the textbooks used in courses in this study reg-ularly plotted only the real part of the wave function,but referred to it as “the wave function,” as in Figure 1.In the QMCS and in the transformed courses, we alwayslabeled such pictures explicitly as the real part. How-ever, we found in interviews that even students in thetransformed courses who had seen explicit discussion ofboth the real and imaginary parts were often confusedby requests to draw “the real part of the wave function.”When asked to draw the real part of the wave function onthe exam question discussed at the beginning of SectionVI, 6 students (3%) said that the wave function is onlyreal inside the barrier and set it to zero everywhere else.

FIG. 13: (a) A representation showing the real and imaginaryparts of a wave function and (b) a representation showing themagnitude and phase of a wave function. In interviews wesee that students can make sense of representation (a) butnot representation (b).

To address these problems, we designed the Quan-tum Tunneling and Wave Packets simulation (as well astwo other PhET simulations on quantum wave functions,Quantum Bound States and Quantum Wave Interfer-ence) to include both the real and imaginary parts on anequal footing (see Figure 13a), and to include time depen-dence so that students could see how the wave functionalternates in time between the real and the imaginaryparts. For completeness, we also included the “phasecolor” representation used exclusively in most non-PhETsimulations of wave functions, in which a curve represent-ing the magnitude of the wave function is filled in withcolors representing the phase (Fig. 13b).

In interviews with five students on Quantum Wave In-terference [48], one student commented that he did notunderstand real and imaginary numbers, and one studentwondered why the imaginary part didn’t look differentfrom the real part until he paused the simulation andcould see that they were out of phase. Aside from thesetwo, whose confusion stemmed more from their expecta-tions than from the simulation, the students intervieweddid not express any confusion over the real and imagi-nary representations of the wave function in interviewson Quantum Tunneling and Wave Packets and QuantumWave Interference. Several students also learned impor-tant concepts by playing with the real and imaginary

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views. For example, students figured out from the simu-lation that the real and imaginary parts were 90 degreesout of phase, and that the real and imaginary parts addup to a constant probability density in an energy eigen-state even though each individual component changes intime.

On the other hand, the “phase color” representationcaused significant problems for most students. In in-terviews on Quantum Wave Interference, three out offive students interviewed explored this view. None of thethree made any comments on it on their own, aside fromone student who said it hurt his eyes, so the interviewerasked them what it was showing. One student said it was“some sort of frequency type of thing” and speculatedthat teal would constructively interfere with teal and de-structively interfere with the opposite of teal. Anotherstared at the screen in confusion for over a minute, andthen described it as “some sort of representation of boththe real part and the imaginary part” showing that “pinkis areas of high real part and low imaginary part or some-thing?” Another student was unable to give any expla-nation. When the same three students were interviewedlater on Quantum Tunneling and Wave Packets, the twowho had given explanations in earlier interviews did notcomment on phase view again. The student who hadbeen unable to give any explanation remembered thatthis view had been used in his quantum course, but stillcould not explain what it meant. Of three additional stu-dents who were interviewed on Quantum Tunneling andWave Packets but not Quantum Wave Interference, twoexpressed frustration over the phase view and were un-able to explain it, and the third, when asked to explain it,said only that it showed “something about wavelength.”When given a choice, none of the students spent muchtime in phase mode, returning quickly to real or magni-tude mode after answering the interviewer’s questions.

“Phase color” is still an option in the simulations forinstructors who would like to explicitly teach the use ofthis representation or use activities developed for othersimulations. However, based on our interviews, we do notrecommend the use of the “phase color” representationwith students.

F. “Hard Questions”

One striking result of our transformed instruction wasthe number of student questions probing the relationshipof the course material to reality, many of which weresufficiently difficult that most expert physicists could noteasily answer them. Many examples of these questionshave already been discussed in Section VIC. Below aresome further examples:

• What [happens if the electron is spread out] in thewire, and you cut the wire in half?

• How come we don’t count the position in the wire?How come we only count the energy?

• Wouldn’t there be a charge difference in the wire ifit were more likely to be found in the center?

• If everything’s got to be measured for it to be local-ized, how come everythings already localized? Imnot going around measuring things.

We hypothesize that these questions are a result of thecombination of interactive engagement techniques with afocus on real world applications. Our students are con-stantly engaged in a struggle to relate the material to re-ality. We regard the quantity of such questions as a signthat this struggle is very difficult. We question whetherthere is much learning in courses where students are notasking such questions.

VII. LESSONS FOR IMPROVING STUDENTLEARNING OF QUANTUM TUNNELING

Our research demonstrates that a focus on addressingcommon student difficulties is helpful, but not sufficient,for improving student learning of quantum tunneling. Byaddressing these difficulties and focusing on relating thematerial to reality, we have uncovered deeper problems instudents’ ability to use the basic models of quantum me-chanics, such as wave functions as descriptions of physicalobjects, potential energy graphs as descriptions of the in-teractions of those objects with their environments, andtotal energy as a delocalized property of an entire wavefunction that is a function of position. We have foundthat real world examples are useful not just to help stu-dents see the connection to their lives, but also to helpthem make sense of the models they are using.

Effective curriculum on quantum tunneling must ex-plicitly help students learn to build these models. Twopractices that we have found useful are focusing on howto relate potential energy graphs to physical systems andstarting with wave packets rather than plane waves.

There are several further practices that, although wehave not tested them on a large scale, our research sug-gests would be valuable. These include:

1. Tutorials to lead students through the process ofdrawing potential energy graphs for various physi-cal situations. 4

2. Explicit discussion of the strengths and weaknessesof gravitational analogies.

4 The Activity-Based Tutorials [22], one of which we adapted foruse in our curriculum, also include several exercises for buildingup the idea of potential energy graphs through lab activities withcarts on magnetic tracks. We did not use these activities due tolack of time and lack of a lab section in our course, and becausewe were more interested in helping students understand poten-tial energy diagrams for real quantum systems than for classicalsystems. However, our research indicates that such an approachcould be useful, if connections are made between these classicalexamples and examples of systems where quantum mechanicsapplies.

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3. Explicit discussion of the reasons for the focus inquantum mechanics on an energy representationrather than the force representation used in intro-ductory physics.

4. Explicit discussion of why total energy is quantized(for bound particles), but potential energy is not.

VIII. ACKNOWLEDGMENTS

We thank Chris Malley, the software engineer for theQuantum Tunneling and Wave Packets simulation, as

well as Mike Dubson and Sam Reid for help with thephysics and numerical methods involved in the simula-tion. We thank Michael Wittmann for convincing usto write this paper, and Travis Norsen, Mike Dubson,and Chandra Turpen for useful discussions and feedback.We also thank the PhET team and the Physics Educa-tion Research Group at the University of Colorado. Thiswork was supported by the NSF, The Kavli Institute, TheHewlett Foundation, and the University of Colorado.

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