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Richard Feynman once famously stated that nobody un-derstands quantum mechanics. He was, of course, referringto the many strange, unintuitive foundational aspects ofquantum theory such as its inherent indeterminism and statereduction during measurement according to the Copenhageninterpretation. But despite its underlying fundamental mys-teries, the theory has remained a cornerstone of modernphysics. Most physicists, as students, are introduced to quan-tum mechanics in a modern-physics course, take quantummechanics as advanced undergraduates, and then take itagain in their first year of graduate school. One might thinkthat after all this instruction, students would have becomecertified quantum mechanics, able to solve the Schrdingerequation, manipulate Dirac bras and kets, calculate expecta-

tion values, and, most importantly, interpret their results interms of real or thought experiments. That sort of functionalunderstanding of quantum mechanics is quite distinct fromthe foundational issues alluded to by Feynman.

Extensive testing and interviews demonstrate that a sig-nificant fraction of advanced undergraduate and beginninggraduate students, even after one or two full years of in-struction in quantum mechanics, still are not proficient atthose functional skills. They often possess deep-rooted mis-conceptions about such features as the meaning and signifi-cance of stationary states, the meaning of an expectationvalue, properties of wavefunctions, and quantum dynamics.Even students who excel at solving technically difficult ques-tions are often unable to answer qualitative versions of thesame questions.

A growing number of physics education researchers,borrowing heavily from tools and methods developed forintroductory-level physics, have begun to investigate and ad-dress issues related to students understanding of quantummechanics.13 One interesting result is that most of the stu-dents difficulties are universal.3,4 That is, they are indepen-dent of teaching styles, textbooks, institutions, and studentsbackgrounds. And patterns of incorrect notions of quantummechanics are analogous to those that have been well-documented for introductory physics courses.

In this article we underscore the need for physics edu-cation research in quantum mechanics by illustrating some

of the most pervasive difficulties students have. We then sur-vey tools that are being developed to help improve anddeepen students understanding of this important subject.

Students misconceptionsSeveral studies have investigated misconceptions of studentswho were exposed to quantum mechanics in modern-physicscourses. The difficulties illustrated in the following exampleswere identified with the help of research-validated surveysadministered to 89 advanced undergraduates and more than200 first-year physics graduate students from seven universi-ties.3 Topics covered included general properties of the wave-function, time dependence of the wavefunction, probabilitiesfor measurement outcomes, and expectation values and dy-

namics for energy and other observables. Numerous in-depthinterviews followed the administration of the survey.In one survey question, students were asked to write

down the most fundamental equation in quantum mechan-ics. Only 32% of students indicated the time-dependentSchrdinger equation, H = i /t, or equivalent state-ments involving commutation relations. In contrast, 48% ofstudents wrote down the time-independent Schrdingerequation, H = . We note that Erwin Schrdinger re-ceived his Nobel Prize for the fundamental, time-dependentequation. As evidenced by the following two questions, thestudents sense that the time-independent equation is funda-mental is closely linked to a number of their difficulties andmisconceptions about quantum mechanics.

One of the survey questions asked students to

Consider the following statement: By defini-tion, the Hamiltonian acting on any allowed stateof the system will give the same state back, i.e.,H = E, where E is the energy of the system.Explain why you agree or disagree with thisstatement.

Figure 1 summarizes the responses that students gave.The correct answer was given by 29% of students: The state-ment is true only if is a stationary state. Among the incor-rect answers, a full 39% of students wrote that the statementis true unconditionally. Typically, those students were

2006 American Institute of Physics, S-0031-9228-0608-030-6 August 2006 Physics Today 43

Improving studentsunderstanding of

quantum mechanicsChandralekha Singh, Mario Belloni, and Wolfgang Christian

Chandralekha Singh is an associate professor of physics and astronomy at the University of Pittsburgh in Pennsylvania. Mario Belloniis an associate professor of physics andWolfgang Christian the Brown Professor of Physics at Davidson College in Davidson, NorthCarolina.

To address the misconceptions that students typically hold concerning quantum mechanics, instructorsshould couple computer-based visualizations with research-based pedagogical strategies.


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supremely confident of their answers. For example, onewrote: Agree. This is what 80 years of experiment hasproven. If future experiments prove this statement wrong,then Ill update my opinion on this subject.

A total of 11% of the students believed that any state-ment involving a Hamiltonian Hacting on a state describesa measurement of the states energy. Armed with that phi-losophy, some agreed with the statement they were asked toconsider. Others disagreed, stating that the wavefunction col-

lapsed after the measurement so that H = Enn, where n isthe stationary state into which the wavefunction collapsed.

Another 10% of students claimed that the statement istrue only if His not explicitly time dependent. In that case,they argued, the energy of the system is conserved and henceH = E. The argument is incorrect since need not be aneigenstate of H, even if His time independent.

A second question illustrates students misconceptionsabout expectation values and quantum dynamics for an elec-tron in an infinite square-well potential.

The wavefunction of an electron in a one-dimensional infinite square well of widtha, x(0, a), at time t = 0 is given by

(x, 0) = 2/7 1(x) + 5/7 2(x), where 1(x)and 2(x) are the ground state and first ex-cited stationary state of the system. (n(x) =2/a sin(nx/a), En= n

222/(2ma2), wheren = 1, 2, 3, . . . .) [The original question includeda potential energy diagram and slightly differentnotation.](a) Write down the wavefunction (x, t) at timet in terms of 1(x) and 2(x).(b) You measure the energy of an electron at timet = 0. Write down the possible values of the en-ergy and the probability of measuring each.(c) Calculate the expectation value of the energyin the state (x, t) above.

Only 43% of the students correctly responded to part aof the question. The most common mistake, made by 31% ofthe students, was to write a common phase factor for bothterms along the lines of (x, t) = (x, 0)eiEt/.

Follow-up interviews confirmed that students had trou-ble distinguishing the properties of stationary and nonsta-tionary states. Several students thought that the time de-pendence always took the form of a decaying exponential.During interviews, some of them explained their choice byinsisting that the wavefunction must decay with time becausethat is what happens for all physical systems. Interestingly,9% of the students wrote that (x, t) should not have any

Figure 1. Only 29% of studentswho have taken upper-division undergraduate quantum mechanics courses recognize the con-ditions under which the time-independent Schrdinger equation is valid. The main text breaks down the incorrect answers.

44 August 2006 Physics Today www.physicstoday.org

The ability to disseminate information on the Web throughpowerful search engines, databases, and digital libraries

has facilitated the wide availability of effective educational

tools. The ComPADRE Pathway project (http://www

.compadre.org) is acting as a steward for physics and

astronomy educational resources in the NSF-sponsored

National Science Digital Library (http://nsdl.org).

ComPADRE stands for Communities for Physics and

Astronomy Digital Resources in Education. Its goal is to aid

teachers and learners in finding, using, and sharing high-

quality resources tailored to their specific needs. The library

contains several collections focused on specific topics or user

groups. The quantum mechanics collection, the QuantumExchange, is accessible at http://thequantumexchange.org.

Several features make ComPADRE particularly effective.

The material is reviewed by editors, and in some cases peer

reviewed, before it is added to the collection. Resources are

cross-referenced to materials from other ComPADRE collec-

tions such as the one for physics education research. More-

over, users can search many databases at the same time and

receive integrated results with a single search request.

The ComPADRE digital library


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time dependence whatsoever; during interviews, some triedto justify their claim by pointing out that the Hamiltonian istime independent.

Part b of the question got more correct responses, 67%,than any other segment of the survey; that success makes acomparison with the responses for part c particularly reveal-ing. It shows that students could calculate probabilities forthe outcome of measurements, but many were unable to usethat information to determine an expectation value.

Many students who answered part b correctly, includingthose who also answered c correctly, calculated the expecta-

tion value of the energy Eby brute-force methods: They firstwrote E = +

*Hdx, then expressed (x, t) in terms ofthe two energy eigenstates, then acted on the eigenstates withthe Hamiltonian, and finally used orthogonality to obtaintheir answer. Some of those who missed part b got lost earlyin the process. Others did not remember such mechanicalsteps as taking the complex conjugate of the wavefunction,exploiting the orthogonality of stationary states, or using theproper integration limits.

Interviews revealed that many students did not know orrecall the interpretation of the expectation value as an en-semble average, and did not realize that the expectation value

asked for in part c could be calculated more simply by takingadvantage of their part-b answer. Some, including studentswho correctly evaluated (x, t) in part a, believed that the ex-pectation value of energy should depend on time. Other ques-tions posed to students confirmed that many of the difficul-ties they have are both conceptual and deep rooted.

Instructors are often surprised at how students who dowell on quantitative problems have difficulty when essen-tially the same problems are posed qualitatively. Often, qual-itative understanding of quantum mechanics is much morechallenging than facility with the technical aspects. Similar to

plug and chug problems in introductory physics, strictquantitative exercises in quantum mechanics often fail to pro-vide adequate opportunity for reflection on the problem-solving process and for drawing meaningful inferences.Based on our ongoing research, we believe that learning toolsthat combine quantitative and qualitative problem solvingcan be effective in helping students learn quantum mechanics.

Inappropriate generalizationThe universal nature of students misconceptions aboutquantum mechanics is somewhat surprising, given the ab-stract nature of the subject. Misconceptions in introductory-

www.physicstoday.org August 2006 Physics Today 45

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Figure 2. The dynamics of a wavepacketare visualized in many different ways in this simulation, for which the potential is aone-dimensional infinite square well of unit length. The initial wavepacket is localized at the wells center and has nonzeroinitial momentum. After a certain time, called the revival time, the wavefunction ( in position space, in momentum space)

will reform to its init ial configuration. So-called fractional revivals occur at rational fractions of the revival time. In this simula-tion the units are chosen so that = 2m= Trev= 1; the initial momentum is 80. (a)After 1/4 the revival time, the wavepackethas already spread to encompass the entire well, only to reform into two highly correlated copies at the center of the well but(b)with the copies having the opposite momentum distribution. In panels a and b, color indicates phase. (c) The normalizedprobability density, that is, the square of the magnitude shown in a, is plotted as a function of position after 1/4 of the revivaltime. Although not evident in the figure, the probability density oscillates rapidly within its Gaussian envelope. Other simulatedfeatures of the wavefunction evolution are (d) the quantum carpet, on which color indicates phase and brightness indicatesamplitude, and (e) the expectation values of position and of (f) momentum as a function of time.


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level physics are often viewed as originating from incorrectworldviews. But it is hard to explain conceptual difficultiesin quantum mechanics in that fashion.

Shared misconceptions in quantum mechanics can betraced in large part to incorrect generalizations of conceptslearned earlier, either in classical mechanics or in quantummechanics. For example, many students believe that an ob-ject with a label x is orthogonal to or cannot influence an ob-ject with a labely. According to this belief, eigenstates of thespin operators Sx and Sy are orthogonal. The improper gen-eralization originates from properties of classical vectors.

Heres another example: Many students believe that ifthe expectation value of a physical observable is zero in the

initial state, its expectation value cannot have any time de-pendence. That misconception is related to a similar one inintroductory physics: If the velocity of an object is zero, itsacceleration must be zero as well.

Sometimes students fail to distinguish between closelyrelated concepts and make inappropriate generalizations ina purely quantum context. We have already noted that manystudents claim that the time-independent Schrdinger equa-tion is always true. Other examples of inappropriate gener-alizations are more subtle.

Many students, for instance, believe that the expectationvalue of any time-independent Hermitian operator is alsotime-independent if the initial state is an eigenstate of thatoperator. This belief can be regarded as an incorrect general-ization from two true statements; namely, the expectation

value of a time-independent Hermitian operator in a station-ary state doesnt change with time, nor does the expectationvalue of a Hermitian operator, in any state, that commuteswith the Hamiltonian.

A related false belief is that measurement of any physicalobservable causes the system to get stuck in the measuredeigenstate forever unless an external perturbation is applied.Again, the statement is true only for observables whose opera-tors commute with the Hamiltonian, but students seem to gen-eralize that property of measurement to include all observables.

How can instruction in quantum mechanics be modifiedto reduce the difficulties that students have? One option is

that instructors can take advantage ofcomputer-based learning tools. (Thebox on page 44 describes ComPADRE,a digital library designed to help in-structors find the resources they need.)But teaching with technology, withouta sound pedagogy, does not automati-cally yield significant educationalgains (see the article by Carl Wieman

and Katherine Perkins, PHYSICS TODAY, November 2005, page36). An important first step in developing strategies thatcan help students is to be aware of the difficulties they typi-cally face while learning quantum mechanics. Later in thisarticle, we discuss how tutorials embedded in a coherent cur-riculum are well suited to direct targeting of the misconcep-tions that have been identified through research. But first weturn to simulations available for quantum mechanics classesand say a few words about recently developed laboratoryexperiments.

Experiments, virtual and realFast and realistic computer-based visualization tools can play

a key role in various advanced-level topics, including quan-tum mechanics. The computational power of desktop andlaptop computers is more than sufficient to run such simula-tions, and many heretofore inaccessible topics in quantummechanics can now be discussed and exemplified in concreteterms. Most simulations can be run either as standalone ap-plications or within Web browsers and are written usingmetalanguages and virtual machines that ensure that thesoftware is platform independent and adaptable.

One of the first quantum simulations for instructionalpurposes was created in 1967 by Abraham Goldberg and col-leagues, who simulated scattering of Gaussian wavepacketsoff different wells and barriers.5 Computer-generated proba-bility densities were displayed on a cathode-ray tube andphotographed, then the successive frames were turned into

a movie. Numerous textbooks, even to this day, have refer-enced or reprinted those images, and other texts have ex-tended the work by including more scenarios and depictingtime development with successive images,6 QuickTimemovies,7 or even interactive simulations.8

The Open Source Physics (OSP) project, with supportfrom NSF, has developed a freely available, open-source Javalibrary, along with documentation, that helps instructors todevelop physics software and distribute Web-based curricu-lar material.9 The OSP technology was originally designed forthe teaching of computational physics, but it has been usedto write and organize curricular material for classical me-


Figure 3. Students can explore one-dimensional scattering in this simula-tion from Kansas State Universitys

Visual Quantum Mechanics softwaresuite. The program allows the studentto customize the scattering potentialand scattered particle and then view

various wavefunction properties.


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chanics, electromagnetism, statistical physics, and quantummechanics as well. One attractive feature is that it allowsusers who are developing curricular materials to change andstore exercises, tutorials, text that describes problems, and soforth in a single, easy-to-distribute package.

All the quantum mechanics materials developed by theOSP project are accessible via ComPADRE; just go to theQuantum Exchange collection and search under osp oropen source physics. Those resources span the curriculumfrom introductory subjects to more advanced research-basedtopics such as the quantum mechanical revivals and frac-tional revivals that are of current interest to theorists and ex-perimentalists.10 Figure 2 shows a fractional revival. It was

generated with the OSP QMSuperposition program, whichtakes the expansion coefficients cn of a superposition of en-ergy eigenstates n with energies En and evolves the state ac-cording to (x, t)= n cnn(x) exp(iEnt)/. That analytictime evolution allows for a fast and accurate depiction of thestate even at time scales beyond which numerical solutionsoften fail.

In addition to classic simulations of quantum mechani-cal evolution, many computer-based simulations have beendeveloped that cover more contemporary topics, such asquantum information, or foundational issues that are usuallyskipped in standard sequences. One example is the OSPSpins program, which may be used for focused tutorials suchas simulations of single or multiple measurements on spin-1/2 particles and even the quantum-mechanical double-slit

and other which-way experiments. The original Java ver-sion of the program was written by David McIntyre,11basedon an earlier version by Daniel Schroeder and ThomasMoore. The OSP version extends previous work by allowingsimulated experiments to be stored and run easily.

Students can gain exposure to various contemporarytopics through laboratory experiments. Enrique Galvez andcolleagues, for example, have described a set of five experi-ments in quantum optics that can be set up for about$35 000.12 Topics include single-photon self-interference,quantum erasure, and projective quantum measurements.Timothy Havel and coworkers have discussed quantum-

information-processing experiments thatcan be performed using standard andwidely available nuclear magnetic reso-nance spectrometers.13 Laboratory experi-ences can cement student knowledge andunderstanding of quantum mechanics andcan stimulate students to relate quantummechanics to things they observe. GeorgeGreenstein and Arthur Zajoncs fascinat-ing book, The Quantum Challenge: Modern

Research on the Foundations of Quantum Me-chanics (Jones and Bartlett, 1997), suitablefor both undergraduates and practicing scientists, is full ofdescriptions of experiments related to foundations of quan-tum mechanics and their possible interpretations.

Quantum tutorialsEvidence suggests that the same types of interventions thathave proved successful at introductory levels (see, for exam-ple, the article by Edward Redish and Richard Steinberg,PHYSICS TODAY, January 1999, page 24) can benefit studentsof quantum mechanics. In particular, activities that engagestudents and force them to challenge their beliefs and un-derstandings have the greatest chance of success. Tutorialsdeveloped by physics education researchers attempt to

bridge the gap between the abstract quantitative formalismof quantum mechanics and the qualitative understandingnecessary to explain and predict diverse physical phenom-ena. They take into account likely student difficulties andprovide activities that do not require a total revamping of in-structional methods to have an impact. Rather, tutorials canbe effective supplements to traditional instruction.

The tutorial approach consists of three main compo-nents. First, carefully designed tasks elicit difficulties stu-dents have. Then the tutorials guide students through tasksthat help them overcome those difficulties and organize theirknowledge. The third component is a gradual reduction intutorial support as students develop self-reliance.

Several features of tutorials make them particularlysuited for teaching quantum mechanics.

They are based on research in physics education and payparticular attention to cognitive issues. Visualization tools help students build physical intuitionabout quantum phenomena. Students remain actively engaged as they are asked to pre-dict what should happen in a particular situation and thenreceive appropriate feedback. Group tutorial activities can supplement lectures, and tuto-rials can be used outside the class for homework or self-study. Self-sufficient modular units can be used in any order.

Daniel Styer14 and Dean Zollman and colleagues1 haveproposed that quantum concepts be introduced much earlier

www.physicstoday.org August 2006 Physics Today 47

Figure 4. Energy bands emerge frompotentials with many wells. In this simula-tion from the Visual Quantum Mechanicssoftware suite, students can vary the num-ber of wells, adjust well parameters, andadd impurities. This image, with 10 of 31regular wells displayed and no impurities,shows the energy bands emerging as hori-zontal stripes.


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in physics course sequences than is traditional. To assist in-structors, Zollmans group has developed the Visual Quan-tum Mechanics (VQM) software suite, which is targeted tohigh-school and college students.15 VQM integrates interac-tive visualizations with inexpensive materials and tutorialworksheets in an activity-based environment. Instructionalunits include quantum tunneling, luminescence, lasers, andpotential energy diagrams. Figure 3 shows an example fromVQM in which students explore scattering problems in onedimension. Students can calculate transmission coefficientsfor 1D scattering potentials, view scattering states, and com-pute probability densities. One interesting feature of the vi-sualization is that students can change the particle mass oreven Plancks constant and observe the consequences.

Redish and colleagues have developed a collection of re-sources incorporating tutorials and visualization tools (in-cluding VQM tools and interactive programs calledPhyslets8) to teach modern physics concepts to science andengineering students.16 Examples of instructional units in-clude the photoelectric effect, waveparticle duality, spec-troscopy, the shape of the wavefunction, light-emittingdiodes and quantum mechanical bands, and quantum mod-

els of conductivity. In the tutorial dealing with band struc-ture, students are first asked to use a VQM program calledthe Energy Band Creator to solve the time-independentSchrdinger equation for a single finite square-well potential.They then add a second well and observe hybridization orbondingantibonding states. Finally, they consider a largernumber of wells and discover that the allowed energy statesform bands. Figure 4 shows an example.

QuILTsWith support from NSF, one of us (Singh) has been develop-ing and evaluating tutorials for upper-level undergraduate

and beginning graduate students. The Quantum InteractiveLearning Tutorials (QuILTs) have been designed to engagestudents in the learning process and to help them build linksbetween the abstract formalism and conceptual aspects ofquantum mechanics. Topics covered include areas of fun-damental and contemporary interest such as the time-dependent and time-independent Schrdinger equations,time development of the wavefunction, the double-slit ex-periment and which-way information, Larmor precession ofspin, and quantum key distribution. QuILTs, which combineconceptual and quantitative problem solving but dont com-promise technical content, are based on the exemplary intro-ductory physics tutorials developed by Lillian McDermottsgroup at the University of Washington.17

The QuILT dealing with time development begins byasking students to predict the time dependence of two states,one stationary and the other nonstationary, for an electron inan infinite square well.18 As discussed earlier, more than halfof the students will choose an incorrect answer. The tutorialthen asks students to compare their predictions against acomputer simulation, adapted from OSP simulations, thatgives the evolution of the probability densities for both the

stationary and nonstationary cases.As students see that the probability density does not

vary with time for the stationary state (figure 5a) but doesvary for the nonstationary state (figure 5b), they are forced toconfront inconsistencies in their thinking. Research on learn-ing suggests that when students reach such a juncture, theywant to resolve the discrepancy between their prediction andobservation and are highly receptive to guidance that willhelp them analyze their conceptual difficulties and build amore robust knowledge structure. Working in an interactiveenvironment, students learn by example that the Hamilton-ian governs the time evolution of the wavefunction so that


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Figure 5. Students visualize quantum evolution as part of a Quantum Interactive Learning Tutorial. In these simulations a parti-cle evolves in a box of unit length and time advances from left to right. (a) The probability density for a stationary state doesnot depend on time, but (b) the probability density for a combination of two stationary states does, to the surprise of many stu-dents. When students see that their reasoning is wrong, they are eager to learn where they went awry.


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the eigenstates of the Hamiltonian are special with regard totime evolution. Moreover, not all allowed states are energyeigenstates. Students must answer questions that relate toconcrete examples to help organize their knowledge, andthey receive prompt feedback. Then students work on pairedproblems that use concepts similar to the tutorial problemsbut for which the contexts are different and help is not pro-vided. The paired problems serve a dual purpose: They helpstudents develop self reliance and provide instructors feed-back on the effectiveness of the QuILTs.

The teaching and learning of quantum mechanics cur-rently stand at the fortuitous crossroads where advances inexperimental, theoretical, computational, and educationalresearch meet. Physics education research has matured to ad-dress courses beyond the introductory level, and in particu-lar researchers have begun to investigate and improve stu-dents understanding of quantum mechanics. The guidanceprovided by research-based learning tools has the potentialto increase the number and proficiency of students who pur-sue advanced degrees and careers in physical sciences andengineering.

References1. See, for example, the theme issue of Am. J. Phys.70(3) (March

2002) and the papers presented at the 1999 annual meeting ofthe National Association for Research in Science Teaching, avail-able at http://perg.phys.ksu.edu/papers/narst.

2. H. Fischler, M. Lichtfeldt, Int. J. Sci. Educ.14, 181 (1992); P. Jollyet al.,Am. J. Phys.66, 57 (1998).

3. C. Singh,Am. J. Phys.69, 885 (2001); C. Singh, in Proceedings of the2004 Physics Education Research Conference, J. Marx, P. Heron,S. Franklin, eds., American Institute of Physics, Melville, NY(2005), p. 23; C. Singh, in Proceedings of the 2005 Physics EducationResearch Conference, P. Heron, L. McCullough, J. Marx, eds.,American Institute of Physics, Melville, NY (2006), p. 69.

4. D. Styer,Am. J. Phys.64, 31 (1996).5. A. Goldberg, H. M. Schey, J. L. Schwartz, Am. J. Phys.35, 177

(1967).6. S. Brandt, H. Dahmen, The Picture Book of Quantum Mechanics,

Springer, New York (2001).

7. B. Thaller, Visual Quantum Mechanics, Springer, New York (2000).8. J. Hiller, I. Johnston, D. Styer, Quantum Mechanics Simulations:The Consortium for Upper-Level Physics Software, Wiley, New York(1995); M. Belloni, W. Christian, A. Cox, Physlet Quantum Physics:An Interactive Introduction, Pearson Prentice Hall, Upper SaddleRiver, NJ (2006).

9. H. Gould, J. Tobochnik, W. Christian, An Introduction to Com-puter Simulation Methods: Applications to Physical Systems, 3rded., Benjamin Cummings, Upper Saddle River, NJ (2006); theOpen Source Physics project code library, documentation, andcurricular materials can be downloaded from http://www.opensourcephysics.org.

10. H. Katsuki et al., Science311, 1589 (2006).11. See http://www.physics.oregonstate.edu/~mcintyre/ph425/

spins/spinsapplet.html.12. E. J. Galvez et al.,Am. J. Phys.73, 127 (2005).

13. T. F. Havel et al.,Am. J. Phys.70, 345 (2002).14. D. Styer, The Strange World of Quantum Mechanics, Cambridge U.

Press, New York (2000).15. See the Visual Quantum Mechanics website at http://perg

.phys.ksu.edu/vqm.16. E. F. Redish, R. N. Steinberg, M. C. Wittmann, A New Model

Course in Applied Quantum Physics, available at http://www.physics.umd.edu/perg/qm/qmcourse/NewModel.

17. L. McDermott, P. Shaffer, Physics Education Group, Tutorials inIntroductory Physics, Prentice Hall, Upper Saddle River, NJ(2002).

18. The tutorial can be accessed from http://www.opensourcephysics.org/projects/collections/quilt.html.

August 2006 Physics Today 49 See www.pt.ims.ca/9467-13

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