Developing and Researching PhET simulations for Teaching

Developing and Researching PhET simulations for Teaching Quantum Mechanics

S. B. McKagan,1 K. K. Perkins,1 M. Dubson,1 C. Malley,2 S. Reid,1 R. LeMaster,1 and C. E. Wieman3, 1

1University of Colorado, Boulder, CO 80309, USA2PixelZoom, Inc., Boulder, CO 80303, USA

3University of British Columbia, Vancouver, BC V6T 1Z1, CANADA(Dated: January 17, 2008)

Quantum mechanics is difficult to learn because it is counterintuitive, hard to visualize, mathe-matically challenging, and abstract. The Physics Education Technology (PhET) Project, known forits interactive computer simulations for teaching and learning physics, now includes 18 simulationson quantum mechanics designed to improve learning of this difficult subject. Our simulations includeseveral key features to help students build mental models and intuitions about quantum mechanics:visual representations of abstract concepts and microscopic processes that cannot be directly ob-served, interactive environments that directly couple students’ actions to animations, connectionsto everyday life, and efficient calculations so students can focus on the concepts rather than themath. Like all PhET simulations, these are developed using the results of education research andfeedback from educators, and are tested in student interviews and classroom studies. This articleprovides an overview of the PhET quantum simulations and their development. We also describeresearch demonstrating their effectiveness and share some insights about student thinking that wehave gained from our research on quantum simulations.

PACS numbers: 01.40.Fk,01.40.G-,01.40.gb,01.50.ht

I. INTRODUCTION

Quantum mechanics has challenged many of the great-est minds in physics, so it is no surprise that it is a diffi-cult subject for students to learn. In addition to the stan-dard rigors associated with any topic in physics, quan-tum mechanics presents many of its own unique chal-lenges that conspire to make it extraordinarily difficultand frustrating for most students to build mental models.It is counterintuitive and surprising to find that the mi-croscopic world does not behave at all the way we wouldexpect, as the intuitions we have built up from interact-ing with our daily environment do not hold up. Becausemost of the phenomena we study in quantum mechanicscannot be observed directly, it is often difficult to con-struct mental models by which to visualize such elusivephenomena. It is also mathematically challenging, in-volving lengthy calculations to analyze the simplest phe-nomena, with most real-world phenomena falling outsidethe realm of our ability to calculate. Finally, at least inthe form it is often taught, quantum mechanics is discon-nected from everyday life, focusing on simplified abstractmodels at worst, and phenomena with which we have nodirect experience at best.

Extensive research in quantum mechanics educationshows that students often do not learn what instructorswould like them to learn in high school modern physicscourses1–5, sophomore level modern physics courses6–11,junior level quantum mechanics courses10–15, and evengraduate courses16. Research on the development oftransformed modern physics courses17–19, as well as ontutorials targeting specific student difficulties20–23, sug-gests that improved student learning is possible. How-ever, most research that has been done so far has focusedon only a few key topics such as atomic models, the pho-

toelectric effect, and the properties of wave functions.This is only the tip of the iceberg, and much remains tobe discovered regarding student learning of other topics,including non-traditional topics such as real-world ap-plications and interpretations of quantum mechanics, aswell as student beliefs about quantum mechanics and thenature of science, and how to best improve student under-standing of all aspects of quantum mechanics. There is aneed for further research and development of techniquesand tools for effectively teaching quantum mechanics.

Educational computer simulations are promising toolsthat have been shown to be effective in helping studentslearn many topics in introductory physics24–26. Becauseof the added problems of visualizing and building an in-tuition for the abstract principles of quantum mechanics,the power of simulations to provide interaction, visual-ization, and context has the potential to be even morehelpful in this subject than in introductory physics.

Many teachers and researchers have developed com-puter simulations to assist students in learning quantummechanics.27–34 While many of these simulations maybe useful for providing visual models of quantum phe-nomena, research on their user interface and effective-ness for learning has been limited. Many of the userinterfaces or representations of physics are not consistentwith research on user-interface design26 and how studentslearn35, potentially limiting their effectiveness.

The Physics Education Technology (PhET) Projectcreates research-based interactive computer sim-ulations for teaching and learning physics andmakes them freely available from the PhET web-site (http://phet.colorado.edu). The simulations areanimated, interactive, and game-like environmentswhere students learn through exploration. We empha-size the connections between real-life phenomena and

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the underlying science, and seek to make the visualand conceptual models of expert physicists accessible tostudents. We have attempted to address the problem ofstudent learning of quantum mechanics by developingPhET simulations in this subject using our research-based design principles25,26, and conducting research ontheir effectiveness in various contexts.

In this paper we present an overview of the PhETquantum simulations (Section II), how they are devel-oped (Section III), examples of classroom use and studiesof their effectiveness (Section IV), and insights into stu-dent thinking we have gained from conducting studentinterviews on these simulations (Section V).

II. QUANTUM MECHANICS SIMULATIONS

We have two main goals for PhET simulations: in-creased student engagement and improved learning. Sim-ulations are specifically designed to support students inconstructing a robust conceptual understanding of thephysics through exploration. Their design is groundedin research. We draw from existing research literatureon how students learn, conceptual difficulties in physics,and educational technology design. We also make exten-sive use of student interviews and classroom testing toexplore usability, interpretation, and learning issues, andto develop general simulation design principles.

PhET is best known for our simulations on topics inintroductory physics, such as Circuit Construction Kit,Masses and Springs, and The Moving Man.36–38 How-ever, the features that make these simulations effectivefor learning introductory physics are even more impor-tant for learning quantum mechanics.

We now have a suite of 18 simulations on various as-pects of quantum phenomena. These fall into three broadcategories, illustrated in Table 1: fundamental princi-ples of quantum mechanics, key ideas in historical ex-periments, and quantum principles underlying everydaylife applications. The PhET quantum simulations areavailable from http://www.phet.colorado.edu/quantum.Each simulation has a web page that includes a briefdescription, learning goals, sample classroom activities,and “Tips for Teachers,” which discuss approximationsmade in the simulation, common student difficulties, andsuggestions for classroom use.

Fundamental Principles Historical Experiments ApplicationsQuantum Tunneling Photoelectric Effect Lasers

and Wave Packets Davisson Germer: Neon Lights &Quantum Wave Interference Electron Diffraction Other DischargeQuantum Bound States Stern-Gerlach Experiment LampsDouble Wells and Rutherford Scattering Simplified MRI

Covalent Bonds Models of the ConductivityBand Structure Hydrogen Atom SemiconductorsFourier: Making Waves Blackbody Spectrum Nuclear Physics

TABLE I: PhET simulations on quantum mechanics

A. Visualization

FIG. 1: Photoelectric Effect simulation.

Simulations are powerful tools for helping students vi-sualize electrons, photons, atoms, wave interference, andother quantum phenomena that they cannot observe di-rectly. While students can conduct experiments on topicssuch as the photoelectric effect and double slit interfer-ence in many physics labs, there is much going on insidethese experiments that they cannot observe. Photoelec-tric Effect (Fig. 1) allows students to watch electronstravel between the plates, helping them to build a modelof why the current increases when you increase the in-tensity (they can see that more electrons leave the plate)but does not increase when you increase the voltage (theycan see that the electrons travel faster between the platesbut the number of electrons stays the same). QuantumWave Interference (Fig. 2) allows students to follow alight wave from the source and through the slits, observ-ing it interfering with itself and collapsing into a dot onthe screen. Models of the Hydrogen Atom (not shown)allows students to “see” inside atoms.

B. Interactivity

PhET simulations are highly interactive, directly cou-pling students’ actions with the animation. Adjustmentof controls results in an immediate animated responsein the visual representations. Our research with studentinterviews shows that this interactivity helps students en-gage with the content and establish cause-and-effect rela-tionships.25 Further, interactivity that allows students toswitch between representations enhances students’ abili-ties to connect multiple representations.25 This interac-tion appears to be particularly effective for helping stu-dents construct understanding and intuition for abstractand unfamiliar quantum phenomena. For example, inQuantum Bound States (Fig. 9), students can learn aboutthe relationship between potential energy and wave func-tion by clicking and dragging directly on the potential

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FIG. 2: Quantum Wave Interference simulation.

energy diagram to change the offset, height, and widthof potential wells, and immediately see the effect on theshape of the wave function.

C. Context

The focus on real-world contexts and applications thatis a characteristic of nearly all PhET simulations is par-ticularly helpful in grounding quantum mechanics in stu-dents’ everyday experiences. For example, SimplifiedMRI (Fig. 3) enables students to learn about nuclear spinand energy splitting in the context of MRI. Neon Lightsand other Discharge Lamps (not shown) enables studentsto see how neon lights work based on the concepts ofatomic energy levels, energy transfer from electrons toatoms, photon emission, and atomic spectra. Puttingbasic concepts in these real-world contexts helps stu-dents appreciate the relevance of the physics, and work-ing through how MRIs and discharge lamps work alsohas the potential to help them understand the physicsbehind these applications.

D. Taking advantage of the Computer

Many of the quantum simulations take advantage ofthe power of computers to quickly do complex calcula-tions without exposing the user to the details. Thus, stu-dents can explore quantum tunneling and quantum waveinterference qualitatively and focus on understanding theconcepts without getting bogged down in the math. Thishas the potential to radically transform the way quan-tum mechanics is taught because it allows the instruc-tor to focus on the problems that are most importantfor students to understand rather than on the problemsthat are easiest to calculate. For example, while plane

FIG. 3: Simplified MRI simulation.

waves are certainly easier to calculate than wave pack-ets, we have found that plane waves are actually muchmore difficult conceptually for students to understand.39Quantum Tunneling and Wave Packets (Fig. 4) allows usto begin our instruction on tunneling with wave packets,so that students can visualize an electron as a slightly-but-not-completely delocalized object that approaches abarrier, interacts with it, and then partially reflects andpartially transmits. This is not only much easier to vi-sualize and understand than a wave packet spread overinfinite space interacting with a barrier for all time, butalso more physically accurate.

FIG. 4: Quantum Tunneling and Wave Packets simulation.

Simulations provide a unique tool for exploring timedependence in a way that is impossible in print media,helping students to see how quantum phenomena evolveand change in time. In Models of the Hydrogen Atom,Neon Lights and Other Discharge Lamps, and Lasers,students can observe atoms absorbing and emitting pho-

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tons. In Quantum Tunneling and Wave Packets andQuantum Bound States, students can observe how wavefunctions change in time, exploring, for instance, the in-terchange between real and imaginary parts, the oscilla-tion of superposition states, and the collapse of the wavefunction when a position measurement is made.

III. DEVELOPING RESEARCH-BASED SIMS

Fig. 5 illustrates the design process for creating PhETsimulations. The design cycle starts with content andstudent interface experts creating a detailed initial lay-out based on the learning goals of the simulation and theresearch base, including education and cognitive scienceresearch and the PhET design guidelines40. After cre-ating an initial version of the simulation that all teammembers feel is clear, accurate and engaging, we con-duct student interviews to determine whether studentscan understand how to use the simulation and achievethe learning goals. These interviews always reveal in-terface weaknesses, resolve interface questions that werenot agreed upon by the team, and often reveal pedagogi-cally undesirable (and occasionally unexpected desirable)features and subtle programming bugs. Subsequent revi-sions are made, and if they are extensive, a further set ofinterviews are conducted. These interviews are not onlyused to improve the particular simulation but continueto improve our research base. After interviews estab-lish that the desired engagement and learning is beingachieved, the simulation is used in a classroom settingwhere student use is observed and informally evaluated.

Research Base

Learning Goals

Initial Design~Final Design

Redesign

Interviews

Interviews

ClassroomUse

��

FIG. 5: The PhET Design Process

A. Building on Previous Research

Research on how people learn35 demonstrates that stu-dents learn by actively constructing their own under-standing, building on their prior knowledge. Further,experts and novices think about subjects differently. Ex-perts build an organized structure of knowledge that al-

lows them to monitor and reflect on their own under-standing and focus on the underlying concepts. Novicesoften don’t know what to focus on and get caught up indetails that experts view as irrelevant. Because workingmemory is limited, education should focus on essentialfeatures to reduce cognitive load.

PhET simulation design incorporates this research inmany ways. Visualization and interactivity help studentsconstruct mental models. Putting physics in familiarreal-world contexts helps students relate new concepts toprior knowledge. Simulations eliminate extraneous de-tails that are unavoidable when working with real equip-ment, such as the color of the wires or details of how thevariable voltage supply works in the photoelectric effectexperiment.

We reduce cognitive load and help students constructtheir own understanding by starting simulations in simplestates, allowing students to gradually work up to explor-ing more advanced features. For example, many simu-lations include several tabs (e.g. Figs. 2-3), where thefirst tab focuses on the basic ideas, and later tabs includemore complex ideas. In Neon Lights and Other DischargeLamps and Lasers, the first tab allows students to explorethe behavior of a single atom before exploring a gas ofmany atoms. Fourier: Making Waves starts up with onlya single non-zero Fourier component and an invitation toadd more, so that students can build up complex patternsat their own pace rather than trying to make sense of apre-existing pattern. These designs are based on botheducation research about how students learn35, and ourown research showing that when we start simulations inmore complex states, students become overwhelmed25,26.

Research on faculty adoption of research-based cur-riculum41 demonstrates that instructors rarely adopt acurriculum as is, but tend to adapt it to suit their lo-cal circumstances. At the same time, instructors needguidance on the essential features of a curriculum tohelp them adapt it effectively. To enable adaptation,we design the simulations to be open-ended and general-purpose, so that each can be used in many different waysto achieve many different learning goals. To assist in-structors in using the simulations effectively, we provideguidelines for developing guided inquiry activities42 and“Tips for Teachers” with guidance on the use of indi-vidual simulations. In addition, we provide a database ofactivities including lesson plans, lecture notes, and home-work.43 The database includes activities developed by thePhET team as well as those contributed by teachers.

Simulation design is also based on research into stu-dent understanding of the specific content area of thesimulation. There has been some previous research onstudent understanding of quantum mechanics, which wehave incorporated into the design of the quantum sim-ulations. For example, research on student learning ofthe photoelectric effect shows that students often havedifficulty interpreting the circuit diagram, drawing qual-itatively correct I-V graphs, distinguishing the effects ofchanging intensity and changing wavelength, and recog-

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nizing that electrons are ejected by the light rather thanby the voltage.6,44 We designed the Photoelectric Effectsimulation to address each of these difficulties. The cir-cuit is shown as a cartoon-like physical picture, ratherthan as an abstract diagram, and the variable voltagesupply is illustrated as a battery with a slider. Studentscan interactively create I-V graphs. They can change theintensity, wavelength, and voltage, and immediately seethe effects of each. In our modern physics class, we use in-teractive lecture demonstrations and homework designedto use these features of the simulation to address knownstudent difficulties. This curriculum has been shown tobe very effective at helping students understand the pho-toelectric effect.44

Research on student learning about quantum tunnel-ing and wave functions shows that students often mixup wave function and energy.8,9,45 One possible causeof this confusion is that instructors and textbooks oftendraw both on the same graph. In Quantum Tunnelingand Wave Packets and Quantum Bound States, we showthe two quantities on separate graphs. Quantum BoundStates has also been designed to address research show-ing that students often have difficulty relating the shapeof the wave function to the shape of the potential45 by al-lowing students to interactively explore the relationshipbetween the two for a wide variety of potentials.

B. Observations of Students

In addition to addressing student difficulties seen in theliterature, we often design simulations to address difficul-ties we see during observations of students when takingfield notes in lecture and problem-solving sessions.

While there are many existing simulations on doubleslit interference of electrons, none address what we ob-served to be the biggest problem for students in under-standing this phenomenon: visualizing the behavior ofthe electrons in between the slits and the screen. Stan-dard instruction often shows the pattern on the screenand assumes that students will know how to interpretthis pattern, filling in the gap of the wave interferencethat must have created it. Our experience interactingwith students in a variety of contexts indicates that stu-dents need help constructing a model of how the electronscreate this pattern. Thus, in Quantum Wave Interfer-ence, we show an electron as a particle-like wave packetapproaching the slits and interfering with itself beforecollapsing to a dot on the screen.

When teaching the Davisson Germer experiment, weobserved in problem-solving sessions that many studentsdid not understand the main point of the experimentafter instruction. When instructors asked them to ex-plain the purpose of this experiment, students remem-bered that electrons were only detected at certain angles,but could not explain why. They viewed the electrons asparticles that happened to bounce off at certain anglesfor some reason they could not understand, rather than

recognizing how the observations could be explained bythe wave nature of electrons. Although Quantum WaveInterference was not designed to address this difficulty,we found that it could be used to do so. In working withstudents one-on-one, we found that if we set up an ar-ray of barriers to represent atoms and demonstrated howelectron wave packets aimed at this array reflected andinterfered such that there were intensity maxima at cer-tain angles and minima at other angles, students immedi-ately responded with expressions like, “Oh, it’s interfer-ing like a wave!” and were then able to correctly explainthe purpose of the experiment. While this method waseffective in helping students develop a correct explana-tion, the array of barriers was tedious to construct anddifficult to change. Students often attempted to explorehow changing the spacing and size of the barriers wouldchange the pattern, but gave up quickly when they real-ized how hard it was to modify each barrier. To facilitatesuch exploration, we developed a new simulation, Davis-son Germer: Electron Diffraction, in which an array ofatoms is set up automatically and the spacing and sizecan be changed by moving a slider. A classroom studydemonstrating the effectiveness of this simulation will bepresented in Section IVB.

C. Student Interviews

After developing an initial version of a PhET simula-tion, we test it in interviews in which students are di-rected to “think out loud” as they explore a simulation,either with no directions or with a simple guiding ques-tion. These interviews help us refine the user interfaceand pedagogical effectiveness of simulations. As we willdiscuss in Section V, interviews also provide new insightsinto student thinking and simulation effectiveness. Inthis section, we illustrate some examples of how we haveused interviews to refine simulations. Unless otherwisenoted, in all examples discussed in this article, studentswere engaged in undirected exploration of a simulation.

Sometimes we come up with ideas that just don’t work.In the initial version of Photoelectric Effect, we attemptedto reduce students’ cognitive load by starting with a“simple” model in which all electrons were ejected withthe same energy. We thought that as students becamemore comfortable with this simple model, we would thenintroduce the “realistic” model, in which electrons wereejected with a range of energies. The simulation allowedstudents to switch between models with radio buttonslabeled “simple” and “realistic.” In interviews every stu-dent got caught up trying to figure out the difference be-tween these two modes, and either gave up or developedan incorrect explanation. Further, when we used the sim-ulation in class, many of the student questions during lec-tures and problem-solving sessions revolved around try-ing to understand the difference between the simple andrealistic models. In response, we modified the simulationby replacing the “simple” and “realistic” radio buttons

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with a checkbox labeled “show only highest energy elec-trons.” It is unchecked by default, so that the simulationstarts in the “realistic” model where electrons are ejectedwith a range of energies. When we used the new versionin class the following semester, student questions focusedmuch more on the physics behind the simulation.

Interviews often reveal that seemingly small details canmake a big difference in student understanding. For ex-ample, ∆k and σk are two common labels for the widthof a wave packet in Fourier space. Since most physicistsare equally comfortable with both, the choice betweenthe two seems irrelevant. However, in interviews on anearly version of Fourier: Making Waves in which we usedthe label ∆k, students referred to it as “the change in k.”Because this incorrect interpretation of the label seemedso plausible, they were never able to determine what itactually meant. In interviews after we changed the la-bel to σk, students initially had no idea what the labelmeant. They were thus more willing to to explore it andwere able to determine the correct meaning.

Observations of what students attempt to do with sim-ulations often helps determine what additional featuresare needed. These features are often not obvious andcould not have been foreseen by the developers. For ex-ample, Neon Lights and other Discharge Lamps includesan option for configurable atoms, in which students canmodify the energy levels by clicking and dragging onthem. In the initial version, students tried to drag thepictures of atoms labeling the energy levels rather thanthe levels themselves, and got frustrated when this didn’twork. In response, we modified the simulation to allowstudents to click and drag the atoms as well as the levels.

Unfortunately, interviews sometimes reveal problemsthat we do not know how to solve. We have found thatstudents recognize when the scale is unrealistic and don’tattempt to attribute meaning to the relative size of ob-jects such as for the macroscopic images of electrons, pho-tons, and atoms in Photoelectric Effect, Neon Lights andother Discharge Lamps, and Lasers.25 However, we havenot found any method that communicates clearly to stu-dents when the scale changes within a simulation. Forexample, in Quantum Wave Interference, students canchoose to view interference of photons, electrons, neu-trons, or Helium atoms, all of which exhibit wave prop-erties at vastly different time and distance scales. Weinitially indicated the change in scale only by changingthe units on the ruler and stopwatch, but students ei-ther did not notice or did not know how to interpretthis change. We then tried adding a feature in which aclock with a note that says “slowing down time” and/ora magnifying glass with a note that says “zooming out”appears when the time/distance scale changes, but stu-dents did not know how to interpret this either. We leftthis feature in as a reminder, because while it didn’t helpstudents, it also didn’t hurt. Students can still learnmany other things from the simulation without recogniz-ing the change in scale, and if instructors want studentsto notice this change, they can point it out explicitly or

incorporate this idea into homework activities.The issue of changing scale also caused problems in

an early version of Models of the Hydrogen Atom, whichallowed students to explore both atomic spectra andRutherford scattering by shooting light and alpha par-ticles at an atom. However, because light interacts withthe electrons and alpha particles interact with the nu-cleus, these two processes occur at very different scales.Initially we tried to gloss over this fact by showing bothat an intermediate scale. This led to a great deal ofconfusion in interviews. Students thought the alpha par-ticles were interacting with the electrons and often drewincorrect conclusions, such as that alpha particles arenegatively charged. After these interviews we decidedthat we were trying to show too many different things ina single simulation, and split the alpha particle featureinto a separate simulation called Rutherford Scattering.

IV. CLASSROOM USE AND TESTING

A. Examples of Classroom Use

Most of the quantum simulations were developed foruse in a reformed large-lecture modern physics course forengineering majors.19 This course used the simulationsin a variety of contexts including: general visual aids; in-teractive lecture demonstrations, where we demonstratedkey phenomena and asked students to make predictionsabout the behavior of simulations using clickers; andhomework, which guided students through explorationof simulations. Our course material is available from thePhET activities database43 (search for author “McKa-gan”) or from our modern physics course archive46.

An example of a visual aid is our use of Quantum WaveInterference in lecture to demonstrate how the double slitexperiment shows that light must be both a wave thatgoes through both slits and a particle that hits the screenat a single location.50 This lecture led to an unexpectedonslaught of deep, fundamental questions that took upnearly an entire class period. Student questions included:

• How can it be such a huge blob and then be de-tected in one place?

• Is it that we just don’t know where it is or is itreally spread out in space?

• What does it take for a photon to collapse to asingle point? How does that happen?

• In real life can you really turn it down so low thatyou only have one photon coming out at a time?

• How big is a photon? Can it be a meter wide?

These questions are similar to those asked by thefounders of quantum mechanics as they worked out themeaning of this new theory. Student difficulties often donot reflect the historical questions of scientists becausestudents struggle with much more basic questions. Inthis case, we argue that the visualization provided by the

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simulation allowed students to see the heart of the issueand ask deep questions earlier in the learning process.

Fig. 6 illustrates the use of a simulation for an interac-tive lecture demonstration. This is a typical question inwhich students discuss with their neighbors and then se-lect an answer using clickers. After asking a few studentsto give explanations for their answers to the whole class,we use Quantum Tunneling and Wave Packets (Fig. 4)to demonstrate what actually happens. Students alwayspoint out that the simulation in fact shows the wave be-ing partially reflected and partially transmitted, ratherthan reflected OR transmitted, as in the correct answerD. After a class discussion of this discrepancy, we use the“Make Quantum Measurement” button in the simulationto demonstrate that after it is measured, the electron isalways reflected or transmitted, never both.

An electron is traveling through a very long wire, approaching the end of the wire:

e-

A. stop. B. be reflected back. C. exit the wire and keep moving to the right. D. either be reflected or transmitted with some probability. E. dance around and sing, “I love quantum mechanics!”

If the total energy E of the electron is GREATER than the work function of the metal, V0, when the electron reaches the end of the wire, it will…

E

V(x)=V0 V(x)=0

FIG. 6: Sample interactive lecture demo question with Quan-tum Tunneling and Wave Packets. The correct answer is D.

Another way to use simulations is in homework. Inour modern physics class, students work through a seriesof questions using Lasers to build up an understandingof how a laser works.51 The homework starts with basicquestions about absorption and spontaneous and stim-ulated emission, works through the steps of building alaser and troubleshooting a broken laser, and ends withessays on why a population inversion is necessary to builda laser and why this requires atoms with three energy lev-els instead of two. Most students are able to give correctand thorough explanations in these essays.

B. Classroom testing of simulation effectiveness

We have conducted several studies in our reformedmodern physics course to test the effectiveness of sim-ulations and other aspects of the course.

The most extensive testing of classroom use of a spe-cific quantum simulation has involved Photoelectric Ef-fect. In a recent study we showed that with our curricu-lum that included both interactive lectures and home-

work using the simulation, learning much greater thanwith either traditional or previous reformed instruc-tion.44 For example, on an exam question about whetherincreasing the voltage between the plates would lead toelectrons being ejected when the light frequency was toolow, an average of 83% of students answered correctlywith correct reasoning in the courses using the simula-tion, compared to 20% of students in a traditional courseand 40% of students in a traditional course supplementedby a research-based computer tutorial.

In the course as a whole, where simulations were usedextensively in all the ways discussed in Section IVA, wefound high learning gains (measured by the QuantumMechanics Conceptual Survey47) and a lack of shift inbeliefs about physics (measured by the Colorado Learn-ing Attitudes about Science Survey48). In contrast, in thecourse for engineering majors the semester before our re-forms and in the corresponding course for physics majors,there were low learning gains and large negative shift inbeliefs.19 While we made many reforms in this course,the simulations played a large role in all of them, andlikely contributed to the improved learning and beliefs.

Student perceptions provide a further indicator of sim-ulation effectiveness. On the end-of-term survey for ourmodern physics course (N = 173), the average studentranking of the usefulness of the simulations for theirlearning on a scale of 1 (not useful) to 5 (a great deal)was 4.0, close to the highest ranked aspect of the course.The usefulness rankings for other aspects of the courseranged from 3.2 (the textbook) to 4.3 (the posted lec-ture notes). Students also had the opportunity to makecomments about the simulations in the survey, and 35%(N = 61) chose to do so. Of these comments, 80% werepositive comments about the usefulness of the simula-tions, for example:

• Great sims, I can’t imagine QM without them.• The simulations were crucial in the learning

process.• The simulations were the best part of class, they

practically answer physics questions all by them-selves. I would recommend continuing to developthese and add more. Without these I think I wouldhave been lost in the course.

• I definitely not only enjoyed the simulations, butI’d go as far to say that the simulations taught methe most about the course because I could reallyvisualize the inner workings of the physics processesthat were going on.

• I thought the simulations were great. It helpedme to gain intuition about the topic. This is espe-cially useful in quantum mechanics where it is notnormally possible to directly observe the describedphenomena.

Other types of comments about the simulations includedpointing out: that the simulations need guidance to beuseful (13%); that the simulations were incorrect or notuseful (8%); specific technical problems (7%) (most of

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these have since been resolved); and that the simulationsare not experiments (3%).52

We also conducted qualitative observations of stu-dents in this course by taking field notes in lecture andproblem-solving sessions and conducting regular inter-views with six students. In all these observations, weconsistently saw that for topics where we used simula-tions, students developed extremely vivid mental models.For example, when we asked students in problem-solvingsessions and interviews about topics related to simula-tions, they gave animated responses easily and withoutmuch time for thought. On exam questions on topics suchas the photoelectric effect, discharge lamps, and lasers,students gave vivid, detailed responses, often referringto the simulations explicitly and correctly rememberingminute details. On the other hand, when we asked stu-dents about other topics not related to simulations, suchas models of the atom or infinite square wells (before wedeveloped simulations on these topics), students had tothink for a long time, attempting to retrieve memorizedfacts, and often mixed up important details.

In another classroom study, we set out to determine theprevalence of student difficulties with the Davisson Ger-mer experiment discussed in Section IIIB, and to evaluatethe effectiveness of the simulation in addressing these dif-ficulties. This study was conducted in a modern physicscourse for physics majors that used much of the samecurriculum as the course for engineering majors discussedelsewhere in the paper, but had a very different popula-tion of students. In general, the physics majors seemedto pick things up more quickly and had fewer of the dif-ficulties observed among the engineering majors.

(a) All students (N=59)

electrons are waves, explain with interference:

electrons are waves, no explanation:

non-wave explanation:

blank / no explanation:

(b) Students who read (N=38)

36%

29%

31%

8%

47%

32%

21%

��

��

��

��

waves, inter-ference

waves, inter-ference

non-wave explanation

waves, no explanation

non-wave expla-nation

waves, no explanation

other

FIG. 7: Student responses to reading quiz on Davisson Ger-mer experiment.

In the class before students were expected to read therelevant section of the textbook (Knight 24.449), theywere told that there would be a reading quiz on the Davis-son Germer experiment in the next class. This readingquiz was emphasized much more than usual, to ensurethat students would do the reading. In the quiz, studentswere asked to answer to the following question: “In theDavisson Germerexperiment, Davisson and Germer shota beam of electrons at a lattice of Nickel atoms and foundthat the electrons were only detected at certain angles.Explain the reason for this result and why it was impor-tant.” The correct answer, that the observed pattern wascharacteristic of an interference pattern and therefore itshowed that electrons behave as waves, is illustrated bythe first sample student response in Fig. 7. As shown inFig. 7a, only about a third of students gave this type ofanswer. Another third said that electrons are waves butdid not explain this result in terms of interference, as inthe second sample response. Most of the remaining thirdgave an incorrect explanation that did not involve wavebehavior at all, as in the third sample response.

Since it is possible that these poor results were the re-sult of students not doing the reading, a clicker questionin the next class asked students whether they did thereading on the Davisson Germer experiment. Studentswere promised that their instructor would not look attheir individual responses, and 35% admitted to not do-ing the reading or not remembering whether they did ornot. Fig. 7b shows the responses to the reading quiz forthe students who said they did the reading. (Studentswho may have done the reading but were not present forthe clicker question are not included in Fig. 7b.) Whencounting only students who did the reading, the percent-age of students who answered the reading quiz questioncorrectly goes up to nearly half. However, the remaininghalf either did not explain how the experiment leads to awave model, suggesting that they had simply memorizedthe answer without understanding it, or gave explana-tions of the experiment that did not involve waves at all.

The reading quiz was followed by an interactive lectureon the Davisson Germer experiment, in which the Davis-son Germer: Electron Diffraction simulation was used inone of the two sections, and a homework in which stu-dents in both sections were asked to use the simulation toexplain the Davisson Germer experiment and its appli-cation to understanding the structure of crystals.53 Ona midterm exam, students were again asked to explainthe inferences that could be drawn from this experiment,but were not told what was seen in the experiment. Onthe exam question, 92% of students correctly explainedthat there was an interference pattern that illustratedthe wave nature of electrons. These exam results indi-cate that the simulation, along with the accompanyinglecture and homework, was extremely effective in helpingstudents understand the Davisson Germer experiment.

9

V. LEARNING FROM STUDENT INTERVIEWS

The primary purpose of the think-aloud interviewsthat we conduct as part of our simulation design processis to find problems with the simulations in order to im-prove them. However, interviews are also valuable fordemonstrating the effectiveness of simulations and giv-ing general insights into student thinking. In this sectionwe present some examples of what we have learned frominterviews on quantum simulations.

Interviews help us determine what students can andcannot learn from each simulation. (Details of these in-sights are provided in the “Tips for Teachers” availablefrom the web page for each simulation.) We have foundthat students can usually learn some important conceptsfrom undirected exploration of simulations, but they canlearn much more from using the simulations in conjunc-tion with activities that guide their exploration. Withundirected exploration, students can often give correctexplanations of many of the concepts that the simulationis designed to teach, but do not necessarily recognize thatthey have learned, often because they do not understandthe significance or application of the content.

For example, students with no previous instruction onFourier analysis who explored Fourier: Making Wavesin interviews were able to give correct descriptions ofFourier analysis and explain everything in the first tab,but claimed that they did not understand the point ofthe simulation. Students with no background in mod-ern physics who explored Photoelectric Effect were ableto correctly explain how the experiment worked and theresults, but did not make connections to the greater im-plications for the nature of light.

The simulations can be greatly enhanced by a goodactivity (e.g. homework, lab, or interactive lecture) thatguides students’ exploration towards the learning goalsof the instructor and helps place the concepts within alarger context. Many activities are available for the quan-tum simulations on the PhET website43, some developedby PhET team members as part of our modern physicscourse transformation, and some contributed by users.While we have not done studies that directly compareundirected and guided exploration of the simulations, wehave measured learning gains from using guided activi-ties in the studies discussed in the previous section, andwe have seen improved student learning as a result ofguidance in interviews.

For example, in interviews on Quantum Tunneling andWave Packets, most students were able to explain the be-havior of the simulation after they hit the ‘Make Quan-tum Measurement” button in “wave packet” mode, butneeded help from the interviewer to explain the behav-ior in “plane wave” mode (see Fig. 8). In “wave packet”mode, this button causes the probability density to col-lapse to a narrow packet whose position is determinedrandomly according to the probability density immedi-ately before measurement. In “plane wave” mode, thisbutton causes the probability density to go to zero every-

FIG. 8: The “Make Quantum Measurement” button in Quan-tum Tunneling and Wave Packets measures the position of theelectron in the probability density vs. position graph in (a)“wave packet” mode and (b) “plane wave” mode.

where. Out of six students interviewed on this simula-tion, three were able to explain the behavior in “wavepacket” mode without help, two were able to explain itafter a hint from the interviewer, and one never madesense out of it. Three students who had successfully ex-plained the behavior in “wave packet” mode then triedthe button in “plane wave” mode. All three expressedconfusion over what they saw and none were able to ex-plain it on their own. The interviewer then asked, “Howfar does a plane wave extend in space?” All three stu-dents quickly answered that it extends over infinity, andtherefore the probability of measuring it in the regionshown on the screen is zero. The results of these inter-views suggest that a guided activity including the ques-tion asked by the interviewer could help students learnthe effects of measurement on plane waves much moreeffectively than unguided exploration.

An interview on Quantum Bound States provides anexample of a student learning an advanced topic fromundirected exploration of a simulation. This simulationcontains two advanced tabs that allow students to ex-plore double and multiple wells (Fig. 9). (These tabsare also available separately as the simulations DoubleWells and Covalent Bonds and Band Structure.) In mostof our interviews with Quantum Bound States, the stu-dents spent so long playing with single wells that theynever got to the advanced tabs, so we have only con-ducted one interview in which the student spent morethan a few minutes playing with two wells. This student,who had previous instruction on single wells but not dou-ble wells, was able to explain, based on his explorationof the simulation, the reason for the pairs of symmetricand anti-symmetric states for double wells: “...becausewe have two wells here, so... I want to think that oneis more centered around this one and the other is morearound this one, and I guess we don’t know which oneis which, which is why they’re both symmetrical aroundthese.” He was troubled, however, that he was unableto determine the physical interpretation of the difference

10

FIG. 9: The “Two Wells” tab of Quantum Bound States (alsoDouble Wells and Covalent Bonds), showing the symmetric(ψ1) and anti-symmetric (ψ2) states. The “Many Wells” tab(also Band Structure), not shown, allows users to create anarray of up to 10 wells.

between these two states. While it is possible that thisinterview result was idiosyncratic, it is a valuable exis-tence proof that it is possible for a student to learn a veryadvanced concept from undirected exploration of the sim-ulation. If one student can learn so much with so littleguidance, it is likely that many students can learn thisconcept with a guided activity.

Interviews also help us determine the range of levelsof students for which simulations are appropriate. Thereare a few simulations, such as Quantum Tunneling andWave Packets, Quantum Bound States, Double Wells andCovalent Bonds, Band Structure, and Davisson Germer:Electron Diffraction, that require a basic knowledge ofthe phenomena being illustrated and therefore do notappear to be effective for students who have not hadany instruction on the relevant topics. Other simulationsthat one might imagine are too advanced have provento be surprisingly effective for a wide range of students.For example, several simulations have been used success-fully in lecture demos and homework in courses for non-science majors, including Nuclear Physics, Conductivity,and Semiconductors in “The Physics of Everyday Life,”and Fourier: Making Waves in “Sound and Music.” Ininterviews, after a half hour of unguided exploration,students with no science background have been able togive good qualitative explanations of the physics behindsimulations such as Quantum Wave Interference, Lasers,Neon Lights and Other Discharge Lamps, PhotoelectricEffect, Nuclear Physics, and Semiconductors. Finally, aPhET team member’s 9-year-old son enjoys playing withLasers and Models of the Hydrogen Atom, and has figuredout much of the basic physics behind these simulations.For example, he can explain how the photons change the

energy levels and how to change the separation of energylevels to match energy of light to get lasing in Lasers.

In interviews with students who have had previous in-struction on the topics covered by the simulations, wefind that the visual representations can help students ad-dress incorrect models that would otherwise be difficultfor an instructor to detect. For example, one student,upon seeing the wave packet representing a photon inQuantum Wave Interference, said, “Until now, I thoughtthat, if I were to represent one particle, it would just beone thin line going up. I did not know that it would belike, all over here.” When the interviewer asked why hethought it would be one thin line, he described his in-structor drawing a series of thin lines [wave fronts] andreferring to a line as “this one wave.” Another studentinitially predicted that if you moved the slits further fromthe screen the separation between the interference fringeswould decrease, and was able to use the simulation to cor-rect his prediction and develop an explanation for whythe separation actually increases.

Interviews on Models of the Hydrogen Atom providea further example of a simulation uncovering an incor-rect model developed from previous instruction. In theseinterviews, some students described the Plum Puddingmodel as a cloud of negative charge filled with littlespecks of positive charge, rather than the other wayaround. The use of the word “cloud” suggests that thesestudents are mixing up the Plum Pudding model withthe Schrodinger model, in which the electrons are oftendescribed as a cloud of negative charge. These studentsinitially thought that the electron in the simulation wasa proton, but were eventually able to identify it correctlyby using the legend or by comparing it to the electronsin other models.

The development of Quantum Wave Interference,Quantum Tunneling and Wave Packets, and QuantumBound States illustrates what we have learned aboutwhat representations of wave functions are most con-ducive to student learning. Our team put a lot of thoughtinto how to represent quantum wave functions (Fig. 10)in these simulations. Most textbooks only show plotsof the real part of the wave function, but also discussthe imaginary part of the wave function. The magnitudeand phase may be discussed in a junior level quantum me-chanics course, but usually not in a sophomore level mod-ern physics course. Most non-PhET simulations of wavefunctions use a “phase color” representation in which acurve representing the magnitude of the wave function isfilled in with colors representing the phase (Fig. 10b).

In observations and interviews in many contexts, wenoticed that students often asked about the meaning ofthe imaginary part of the wave function, but never aboutthe real part. Further, students often forgot about theimaginary part entirely, or said that you only need toconsider the real part when squaring the wave function.We hypothesize that students overestimate the impor-tance of the real part and underestimate the importanceof the imaginary part for two reasons. First, the unfor-

11

FIG. 10: Representations of the wave function in QuantumTunneling and Wave Packets : (a) real and imaginary partsand (b) magnitude and phase. In interviews students canmake sense of (a) but struggle with (b).

tunate choice of words “real” and “imaginary” naturallyleads to the idea that one is more “real” than the otherin the common English sense of the word, when in factboth components are on equal footing mathematically.Second, the fact that many textbooks illustrate only thereal part (but label it as ψ) may encourage students tofocus only on this part of the wave function.

To address this problem, we illustrate both the realand imaginary parts on equal footing in the simulations(Fig. 10a). We suspected that the real and imaginaryparts of the wave function would be easier for studentsto understand than the magnitude and phase, becausethese representations relate more easily to what studentstypically calculate and to familiar sine and cosine waves.However, we also included options to show the magnitudeand phase color for completeness.

In interviews on Quantum Wave Interference, one stu-dent commented that he did not understand real andimaginary numbers, and one student who wondered whythe imaginary part didn’t look different from the realpart until he paused the simulation and could see thatthey were out of phase. Aside these two students, whoseconfusion stemmed more from their expectations thanfrom the simulation, students did not express any confu-sion over the real and imaginary representations of thewave function in interviews on Quantum Tunneling andWave Packets and Quantum Wave Interference. Sev-eral students also learned important concepts by playingwith the real and imaginary views. For example, studentslearned from the simulation that the real and imaginaryparts were 90 degrees out of phase, and that the real andimaginary parts add up to a constant probability den-sity in an energy eigenstate even though each individualcomponent changes in time.

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 interviewer

asked them what it was showing. One student said itwas “some sort of frequency type of thing” and specu-lated that teal would constructively interfere with tealand destructively interfere with the opposite of teal. An-other stared at the screen in confusion for 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, but we recommend caution in its use.

VI. CONCLUSION

In summary, PhET quantum simulations are designedto address previously-known student difficulties in quan-tum mechanics, as well as many new student difficultiesuncovered as a result of our research. The key features ofPhET simulations - visualization, interactivity, context,and effective use of computations - are particularly ef-fective for helping students understand the abstract andcounterintuitive concepts of quantum mechanics. Ourresearch has shown these simulations to be effective inhelping students learn, and has revealed new insights intohow students think about quantum mechanics.

VII. ACKNOWLEDGMENTS

We thank Noah Finkelstein, Wendy Adams, and therest of the PhET team and the Physics Education Re-search Group at the University of Colorado. We grate-fully acknowledge the NSF, the Hewlett Foundation, theKavli Foundation, and the University of Colorado for pro-viding the support to develop the simulations, and tomake them freely available to all educators and students.

12

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Developing and Researching PhET simulations for Teaching