10
Teaching quantum mechanics on an introductory level Rainer Mu ¨ ller a) and Hartmut Wiesner b) Lehrstuhl fu ¨r Didaktik der Physik, University of Munich, Schellingstrasse 4, D-80799 Mu ¨nchen, Germany ~Received 25 July 2001; accepted 9 November 2001! We present a new research-based course on quantum mechanics in which the conceptual issues of quantum mechanics are taught at an introductory level. In the context of virtual laboratories, the students discover from the very beginning how quantum phenomena deviate from our classical everyday experience. The results of the evaluation of the course show that most of the students acquired appropriate quantum mechanical conceptions, and that many of the common misconceptions encountered in traditional instruction have been avoided. © 2002 American Association of Physics Teachers. @DOI: 10.1119/1.1435346# I. INTRODUCTION Quantum mechanics has forever changed the physicists’ picture of the world. At the beginning of the 20th century, the advent of relativity and quantum mechanics marked not only the discovery of just another new theory, but an entirely new framework for all of physics. Relativity changed our ideas of space and time, and quantum mechanics introduced indeter- minism, probabilities, and nonlocality into the foundations of physics. Quantum mechanics shapes our view of nature in a funda- mentally new way. We think that not only physicists should have the privilege to understand how the world works. Edu- cated citizens should at least have the possibility to become acquainted with the strangeness and beauty of quantum phe- nomena. However, most students who do not major in phys- ics never have a chance to learn about the conceptual issues of quantum mechanics. For example, in the German Gymna- sium ~whose upper level is comparable to the first two years of college in the United States!, atoms and quanta are stan- dard parts of the curriculum. However, the emphasis is on aspects such as the photoelectric effect or Bohr’s atomic model, which do not really probe the classical conceptions prevalent in the students’ minds. Our new course on quantum mechanics deals with the conceptual questions of quantum mechanics. It is addressed mainly to nonphysicists. Special emphasis is placed on quali- tative reasoning. Physicists can rely on their knowledge of the quantum mechanical formalism to overcome conceptual difficulties, but nonphysicists do not possess such a strong supporting basis. Therefore, conceptual clarity is even more important. The strange and counterintuitive phenomena of quantum mechanics cannot be incorporated in a coherent cognitive picture without the aid of carefully chosen basic concepts that help to organize them. Our strategy is to let the students discover some of the exciting and bizarre quantum phenomena that deviate from our classical everyday experi- ence. At the same time, we want to provide a conceptual framework within which a solid understanding can be con- structed. There are several books that follow a semiqualitative ap- proach to quantum mechanics. First of all, there are the chap- ters on the double-slit experiment in Feynman’s famous lectures, 1 which have had a great influence on most of the subsequent attempts to teach the conceptual aspects of quan- tum mechanics. From the more recent approaches we men- tion the books by Rae, 2 Albert, 3 and Silverman 4 which dis- cuss the foundations of quantum mechanics at a moderate mathematical level. An example of a university course ex- plicitly devoted to nonphysics majors is the Visual Quantum Mechanics project, 5 where a hands-on approach to the appli- cations of quantum mechanics is pursued. II. RESEARCH ON STUDENTS’ CONCEPTIONS It is known that after traditional instruction, students are likely to show classical misconceptions and to confuse clas- sical and quantum notions. Given the counterintuitiveness of quantum mechanics, these misconceptions are not surprising. To avoid them and lead the students to a correct understand- ing of quantum mechanics, it is important to know the com- mon misconceptions that traditional instruction is likely to promote. There have been a number of investigations of students’ misconceptions and their difficulties in understanding quan- tum mechanics. Much of the early work came from the Frankfurt, Bremen, and Berlin groups in Germany ~for sur- veys in English see Refs. 6 –9, where the courses developed by the German groups are also discussed!. Further research has been carried out by Mashhadi, 10 Styer, 11 Johnston, Craw- ford, and Fletcher, 12 Bao, Redish, and Steinberg, 13 and Ireson. 14 Here we want to report on the results of our own investi- gations which up to now have been published only in the German literature. In our first research project, 15,16 523 Gym- nasium students answered a questionnaire on their concep- tions of quantum physics after instruction. In addition, 27 students were interviewed orally. The interviews lasted about 1 h. The questions ranged from fact reproduction ~‘‘How would you measure an atomic spectrum’’! to interpretational issues such as their view of determinism/indeterminism. In a second project, 17,18 37 university students ~future physics teachers! were interviewed in a similar manner. It was found that 52% of them had already heard about quantum physics in school, 79% had attended a theoretical quantum mechan- ics lecture. It is remarkable that both groups gave very simi- lar answers. This similarity indicates that the results can be considered to be typical. In the following, we give an over- view of the main misconceptions found in our investigations. The first three questions ~from Ref. 16! show how students distinguish classical and quantum objects. ~1! What are the essential properties of classical objects? The student responses can be categorized into the following items ~multiple replies possible!: ~i! mass, weight ~85%!, ~ii! 200 200 Am. J. Phys. 70 ~3!, March 2002 http://ojps.aip.org/ajp/ © 2002 American Association of Physics Teachers

Teaching quantum mechanics on an introductory level - Teaching quantum...Teaching quantum mechanics on an introductory level Rainer Mu¨llera) and Hartmut Wiesnerb) Lehrstuhl fu ¨r

  • Upload
    vudat

  • View
    238

  • Download
    4

Embed Size (px)

Citation preview

Teaching quantum mechanics on an introductory levelRainer Mullera) and Hartmut Wiesnerb)

Lehrstuhl fur Didaktik der Physik, University of Munich, Schellingstrasse 4, D-80799 Mu¨nchen, Germany

~Received 25 July 2001; accepted 9 November 2001!

We present a new research-based course on quantum mechanics in which the conceptual issues ofquantum mechanics are taught at an introductory level. In the context of virtual laboratories, thestudents discover from the very beginning how quantum phenomena deviate from our classicaleveryday experience. The results of the evaluation of the course show that most of the studentsacquired appropriate quantum mechanical conceptions, and that many of the commonmisconceptions encountered in traditional instruction have been avoided. ©2002 American Association

of Physics Teachers.

@DOI: 10.1119/1.1435346#

isheneoeto

dld

dumphysunarn-o

mn

thssao

tuono

resicthtue

tun

apauhequ

e

ratex-mli-

relas-s ofsing.and-m-to

nts’an-he

ped

ti-the

cep-27out

n a

dsicsan-

mi-ber-s.

ts?ing

I. INTRODUCTION

Quantum mechanics has forever changed the physicpicture of the world. At the beginning of the 20th century, tadvent of relativity and quantum mechanics marked not othe discovery of just another new theory, but an entirely nframework for all of physics. Relativity changed our ideasspace and time, and quantum mechanics introduced indminism, probabilities, and nonlocality into the foundationsphysics.

Quantum mechanics shapes our view of nature in a funmentally new way. We think that not only physicists shouhave the privilege to understand how the world works. Ecated citizens should at least have the possibility to becoacquainted with the strangeness and beauty of quantumnomena. However, most students who do not major in phics never have a chance to learn about the conceptual isof quantum mechanics. For example, in the German Gymsium ~whose upper level is comparable to the first two yeof college in the United States!, atoms and quanta are stadard parts of the curriculum. However, the emphasis isaspects such as the photoelectric effect or Bohr’s atomodel, which do not really probe the classical conceptioprevalent in the students’ minds.

Our new course on quantum mechanics deals withconceptual questions of quantum mechanics. It is addremainly to nonphysicists. Special emphasis is placed on qutative reasoning. Physicists can rely on their knowledgethe quantum mechanical formalism to overcome concepdifficulties, but nonphysicists do not possess such a strsupporting basis. Therefore, conceptual clarity is even mimportant. The strange and counterintuitive phenomenaquantum mechanics cannot be incorporated in a cohecognitive picture without the aid of carefully chosen baconcepts that help to organize them. Our strategy is to letstudents discover some of the exciting and bizarre quanphenomena that deviate from our classical everyday expence. At the same time, we want to provide a concepframework within which a solid understanding can be costructed.

There are several books that follow a semiqualitativeproach to quantum mechanics. First of all, there are the chters on the double-slit experiment in Feynman’s famolectures,1 which have had a great influence on most of tsubsequent attempts to teach the conceptual aspects oftum mechanics. From the more recent approaches we mtion the books by Rae,2 Albert,3 and Silverman4 which dis-

200 Am. J. Phys.70 ~3!, March 2002 http://ojps.aip.org/a

ts’

lywfer-f

a-

-ee-

s-es

a-s

nics

eedli-falg

reofnt

emri-al-

-p-s

an-n-

cuss the foundations of quantum mechanics at a modemathematical level. An example of a university course eplicitly devoted to nonphysics majors is the Visual QuantuMechanics project,5 where a hands-on approach to the appcations of quantum mechanics is pursued.

II. RESEARCH ON STUDENTS’ CONCEPTIONS

It is known that after traditional instruction, students alikely to show classical misconceptions and to confuse csical and quantum notions. Given the counterintuitivenesquantum mechanics, these misconceptions are not surpriTo avoid them and lead the students to a correct understing of quantum mechanics, it is important to know the comon misconceptions that traditional instruction is likelypromote.

There have been a number of investigations of studemisconceptions and their difficulties in understanding qutum mechanics. Much of the early work came from tFrankfurt, Bremen, and Berlin groups in Germany~for sur-veys in English see Refs. 6–9, where the courses develoby the German groups are also discussed!. Further researchhas been carried out by Mashhadi,10 Styer,11 Johnston, Craw-ford, and Fletcher,12 Bao, Redish, and Steinberg,13 andIreson.14

Here we want to report on the results of our own invesgations which up to now have been published only inGerman literature. In our first research project,15,16523 Gym-nasium students answered a questionnaire on their contions of quantum physics after instruction. In addition,students were interviewed orally. The interviews lasted ab1 h. The questions ranged from fact reproduction~‘‘Howwould you measure an atomic spectrum’’! to interpretationalissues such as their view of determinism/indeterminism. Isecond project,17,18 37 university students~future physicsteachers! were interviewed in a similar manner. It was founthat 52% of them had already heard about quantum phyin school, 79% had attended a theoretical quantum mechics lecture. It is remarkable that both groups gave very silar answers. This similarity indicates that the results canconsidered to be typical. In the following, we give an oveview of the main misconceptions found in our investigation

The first three questions~from Ref. 16! show how studentsdistinguish classical and quantum objects.

~1! What are the essential properties of classical objecThe student responses can be categorized into the followitems~multiple replies possible!: ~i! mass, weight~85%!, ~ii !

200jp/ © 2002 American Association of Physics Teachers

liod

ts

f

ts

d

.oo

-

onhe

s.

eisnonclee

nsysdth

ititstohe

fd

yyo

o

ad

stu-rsestoon-

i-vi-ytu-oall in

ofels’

niteiesons-leI

ry

yainn

umsi-um, I

n-

ce.m. I

nndthet

sissi-cor-sonsthat

ics.

iclean-

size, volume, shape~43%!, ~iii ! velocity, movement~38%!,~iv! momentum ~27%!, ~v! position ~15%!, ~vi! density~15%!, ~vii ! energy~12%!. It is remarkable that nearly alstudents mentioned mass, but only few mentioned positThe ~dynamical! property velocity/momentum is consideremore important than position or energy.

~2! What are the essential properties of quantum objec~i! Mass~37%!, ~ii ! charge~37%!, ~iii ! velocity/momentum~37%!, ~iv! energy~26%!, ~v! spin ~22%!, ~vi! energy levels/quanta~15%!, ~vii ! position not exactly determined~11%!,~viii ! no absolute mass~11%!, ~ix! de Broglie wavelength~7%!. For quantum objects, mass is not as dominant asclassical objects. Charge is mentioned often. Category~vii !indicates the conception of ‘‘smeared’’ quantum objecwhich is discussed in more detail below.

~3! What is the main difference between a classical anquantum object?~i! For many students~30%!, there is asmooth transition between quantum and classical physicsobjects become smaller, quantum behavior shows up mclearly. ~ii ! 26% of the students argue in terms of dualismthe necessity of model descriptions.~iii ! Quantization, especially energy quantization~19%!. ~iv! Large velocities arepossible for quantum objects~15%!. ~v! Quantum effectssuch as the Compton effect, spin, interference.~vi! Quantumobjects do not possess a position property~11%!. ~vii ! Otheranswers~7%!.

The following questions give an overview of the commconceptions and misconceptions of quantum objects sucphotons and atoms. To give a more vivid illustration, whave included typical student answers to most categorie

~4! What do you mean by ‘‘photon’’~from Ref. 16!? ~i!One third of the students described a photon as particllight that has wave as well as particle properties; 17% minterpreted the wavy line that symbolizes a photon in maCompton effect diagrams as the trajectory of the phot@Student P2: ‘‘Photon is denoted a light quantum, a partiand it moves in the form of a wave.’’ Interviewer: ‘‘Thphoton itself?’’ P2: ‘‘In the form of a wave forward~drawswavy arrow!.’’ #. ~ii ! 25% of the students remark that photodo not have a rest mass~‘‘they only have a mass when themove at the velocity of light’’!. ~iii ! 17% define a photon aan energy quantum, and~iv! 8% state that a photon is emittein the transition of an atom from an excited state toground state.

~5! How do you conceive electrons in an atom~from Ref.17!? ~i! Bohr’s atomic model or planetary model~17%!.~S31: ‘‘There are circles ... around the nucleus ... just orbThey are circles. And the electrons are on different orbThey move on them and they can jump from one orbitanother ... if they get more energy, they can jump to a higorbit.’’ ! ~ii ! Bohr’s model with cautionary remarks~24%!.~S18: ‘‘The orbits ... I still have that picture when I think oan atom. One is told that it’s not correct, but one is so useit and, after all, it is employed again and again.’’! ~iii ! Con-crete ideas of ‘‘clouds’’/smeared charge~14%!. ~iv! ‘‘Orbit-als’’ with probability distribution~38%!. ~S29: ‘‘It’s the wavefunction that represents the particles, there is the theororbitals, the orbitals can be represented in space. Thenknow where the electrons are approximately and the whthing works with the probability interpretation’’!.

The two dominant conceptions are the two variantsBohr’s model ~together 41%! and the picture of orbitals~38%!. It is remarkable that even if quantum mechanicideas are mentioned, Bohr’s model is almost always use

201 Am. J. Phys., Vol. 70, No. 3, March 2002

n.

?

or

,

a

Asrer

as

of-y.,

e

s..

r

to

ofoule

f

las

the starting point of the discussion. Because most of thedents interviewed for this study had quantum physics couin high school as well as in the university, it is legitimatesay that the Bohr model is a very dominant and stable cception.

Although it is not compatible with the quantum mechancal conception of the atom, the Bohr model may be inetable as an intermediate step.19 Possibly the lack of an easvisualization of the quantum mechanical model forces sdents to stick to this model. If this hypothesis is true, the gof our instruction should not be to erase the Bohr modethe students’ minds, but to convey the conscious usephysical models and let them have insight into the modlimitations.

~6! Permanent localization~from Ref. 17!. The studentswere asked, ‘‘Does an electron in an atom have a defiposition at each moment of time?’’ The answer categorwere ~i! the electron has a definite but unknown positi~21%!. ~S1: ‘‘Yes, it has to be somewhere, but it isn’t accesible through a measurement.’’ S2: ‘‘I would say in principit has a definite position, we just don’t know it. That’s howimagine.’’! ~ii ! The electron has a position but no trajecto~due to insufficient knowledge of initial conditions! ~7%!.~iii ! Localization in a region with some probability~25%!.~S8: ‘‘It’s like that, they have no definite position, to mmind, they are just located arbitrarily somewhere in a certregion.’’ S32: ‘‘You cannot localize it that precisely, you caonly give a probability of finding ...’’! ~iv! No definite posi-tion because of the uncertainty relation~18%!. ~v! Other~11%!, indifferent ~18%!.

~7! Heisenberg’s uncertainty relation~from Ref. 17!:Question: What is the meaning ofDx andDp? ~multiple re-sponses allowed!. ~i! Measurement uncertainties~15%!.~S15: ‘‘Suppose you know the errorDx. Then you can deter-mine the minimum error you have done in the momentmeasurement.’’! ~ii ! Disturbance during measurement: potion measurement influences the particle’s moment~21%!. ~S13: ‘‘When I measure the position very preciselyalter the momentum.’’! ~iii ! ‘‘Regions of localization’’~18%!, for example, spatial region where the particle is cofined; width of the wave function.~iv! Interval within whichthe exact value lies with some probability~18%!. ~S18: ‘‘Itis, so to speak, the probabilities of the momenta at this plaThis is the most precise statement about the momentucan only say the momentum lies in the interval betweenp6Dp.’’ ! ~v! Standard deviation of a statistical distributio~13%!. ~S21: ‘‘If I repeat an experiment several times ameasure position and momentum, I don’t get alwayssame, i.e., if I have identical initial conditions, I don’t gealways the samex and the samep, but it varies. If I graph itI get a standard deviation.’’!

III. OUTLINE OF THE COURSE

The results presented in Sec. II form the empirical bafor the development of our course. We want to avoid clascal misconceptions and help our students to construct arect quantum mechanical understanding. For these reawe concentrate on those features of quantum mechanicsare radically different in comparison to classical mechanWe therefore focus on the following aspects:

d Born’s probability interpretationis introduced early andused throughout. In introductory courses, wave–partduality is often characterized as the main mystery of qu

201R. Mu¨ller and H. Wiesner

thpee

an-

oic

ou

tetco.en

redu

–-de

o

eninal

e

h

arfongtio

nm

ilf

se-

thdi

.

itya

tu

t adte

h ation,

tail.

na-k’sher

jorra-stite

isn of

the-r-a

lys-testve

isiesvir-the

tum mechanics. In contrast, we point out that there is noing mysterious about wave–particle duality once a prounderstanding of the probability interpretation has beachieved.

d A major new feature of quantum mechanics is thatclassi-cally well-defined dynamic propertiessuch as position,momentum or energy cannot always be attributed to qutum objects.20,21If an electron is not in a momentum eigestate, it does not possess the ‘‘momentum’’ property~atleast according to the standard interpretation!. Similarly, anelectron in an atomic orbital~an energy eigenstate! doesnot possess the ‘‘position’’ property. We consider this ntable feature as a central element of quantum mechanIts discussion therefore takes a prominent place incourse.

d The measurement processhas perhaps led to more debathan any other topic in quantum mechanics. In contrasclassical mechanics, measurement can no longer besidered as a passive reading of pre-existing valuesquantum mechanics, measurement is an active procThere is a difference between ‘‘to possess a property’’ a‘‘to measure a property.’’ The special role of the measument process comes to light in the process of state retion and is illustrated, for example, by Schro¨dinger’s catparadox.

Although we consider nonlocality, the EinsteinPodolsky–Rosen~EPR! paradox, and Bell’s inequality as important as the subjects above, no attempt has been mainclude them in the course. The primary reason is time limtation.

Conceptual clarity is a vital condition for the successthe course. We therefore base our course on theensembleinterpretation of quantum mechanics,21–23 according towhich the predictions of quantum mechanics apply tosembles of identically prepared objects. In our view, thisterpretation provides a clear and comprehensible way of ting about quantum phenomena. Similarly, the idea ofstatepreparation20–25 helps one to construct a conceptual framwork that serves as a basis for a deeper discussion.

The course consists of two parts with different goals. Temphasis of the first part, orbasic course, is on purely quali-tative reasoning. Students explore the foundations of qutum mechanics without the difficulties introduced by the fomalism. Simulated laboratories provide an environmenttheir experiences. They are confronted early with the strabehavior of quanta and the central aspects of interpretaare discussed. In the second part, theadvanced course, anintroduction to the quantum mechanical formalism is giveIt is intimately linked with the discussion of the quantumechanical interpretation given in the basic course.

The division into two parts allows the course to be easadapted to various demands. For example, in a courseliberal arts majors, we would stop after the basic courEngineering science students~or, in the German school system, the Leistungskurse! would obtain a first introductioninto the more formal elements of quantum mechanics inadvanced course. The two parts of the course will becussed in detail in the following sections.

The structure of the basic course is summarized in FigWe proceed in a spiral fashion. First, we introducephotonsand give a qualitative discussion of wave–particle dualthe probability interpretation, and the nontrivial notion ofdynamic property in quantum mechanics. In the second

202 Am. J. Phys., Vol. 70, No. 3, March 2002

-rn

n-

-s.r

ton-

Inss.d-c-

toi-

f

--k-

-

e

n--ren

.

yor.

es-

1.

,

rn

of the spiral, we considerelectronsin the double-slit experi-ment. The insight gained with photons is deepened ahigher level. The probability interpretation is formulatequalitatively with wave functions, and the concept of stasuperposition is introduced. The basic course ends witdiscussion of more complex issues such as state reduccomplementarity, Schro¨dinger’s cat, and decoherence.

IV. PHOTONS

We now discuss the contents of the course in more de

A. Photoelectric effect

The course starts with the photoelectric effect, its explation in terms of photons, and the determination of Plancconstant. This topic is fairly standard and needs no furtexplanation.

B. Preparation of dynamical properties

As stated earlier, the notion of preparation plays a marole in our course. Already in classical physics, the prepation of initial conditions is important. For example, to tethe law of projectile motion, one needs to prepare definvalues of position and velocity for the projectile. In thstage, we define preparation as the systematic productioa dynamical property of a classical or a quantum object~thisdefinition is extended later tostatepreparation!.

Preparation of properties can be illustrated nicely withpolarization of light. A horizontally oriented polarization filter is a device that produces light with the ‘‘horizontal polaization’’ property. A second polarization filter can serve astest for this property. If it also is oriented horizontally, nearall of the light passes through it showing that the light posessed the property horizontal polarization. In contrast, awith a vertically oriented polarization filter shows a negatiresult: no light passes through.

C. Wave and particle behavior in a Mach–Zehnderinterferometer

The Mach–Zehnder interferometer sketched in Fig. 2the playground for the exploration of the quantum propertof light. We have developed a program that simulates atual laboratory in which all the experiments needed incourse can be performed by the students.26

Fig. 1. Structure of the basic course.

202R. Mu¨ller and H. Wiesner

soretuberaleticnetha

eeeca

inonthe

c-

.trudiou

holinr.nni

o-

er

oe

oretoter

Thationare

ec-d

t isthests

n isnnotes-ro-

fer-thesineditialthatisthe

-

re-ared

ted

ons.and

anoese dendth

ers.

The path lengths in the two arms are slightly differentthat a circular interference pattern appears on the scwhen laser light is sent through the interferometer. The sdents know that interference is a characteristic of wavehavior. They are next confronted with an experiment whsingle photons are sent in. The interference pattern gradubuilds up from the particle-like detection events of singphotons. For the first time, the students see wave and parbehavior in the same experiment. The result shows thatther wave nor particle model suffices alone to explainexperimental results. A satisfying model must incorporatepects of both.

The photon is detected as a localized object on the scrIt is natural to ask whether it is similarly localized within thinterferometer. Or, formulated in the language of dynamiproperties, does a photon in the interferometer possessposition property?

To answer this question, we place a polarization filtereach of the interferometer arms. If both are oriented horiztally, the interference pattern gradually builds up as inprevious experiment~see Fig. 3!. The same is true if both aroriented vertically.

However, if the filters are oriented in orthogonal diretions, a different situation emerges~see Figs. 3 and 4!. Asbefore, each photon leaves a localized trace at the screenfrom these traces, no interference pattern emerges. A stureless distribution develops instead. Photons are founplaces where there were interference minima in the prevexperiments.

What does this result mean for our conception of the pton? Imagine that a photon was a localized object traveon exactly one of the two arms through the interferometethis idea were correct, the photon could interact with just oof the polarization filters. But to determine whether it is etitled to land on the position of an interference minimum,needs information about the orientation ofboth filters. Thephoton has to ‘‘know’’ whether they are parallel or orthognal.

If we exclude action-at-a-distance arguments, we havgive up the picture of a photon as a localized object traveing the interferometer. In the interferometer, the photon dnot possess the position property.

Fig. 3. Parallel polarization filters do not prevent the emergence ofinterference pattern.

Fig. 2. Mach–Zehnder interferometer.

203 Am. J. Phys., Vol. 70, No. 3, March 2002

en-

e-elly

lei-

es-

n.

lthe

-e

Butc-ins

-gIfe-t

tos-s

The next experiment shows that the situation is even mweird. An obvious objection is that a photon could split intwo parts at the beam splitter, go through the interferomeseparately, and then recombine somehow at the end.there is no such splitting can be shown if the polarizatfilters are replaced by photon detectors. If single photonsincident on the first beam splitter, it turns out that the dettors neverclick simultaneously. A photon is always detecteas a single entity; parts of a photon are never found. Iremarkable that this experiment has been carried out inlaboratory.27 Although its outcome sounds fairly obvioufrom a photon point of view, it is one of the few experimenthat clearly contradict semiclassical theory~quantized atomsplus classical light!.

Together, these experiments show that a single photoas strange an object as one can imagine. The results cabe explained by any classical model. They hint to the necsity of exploring the quantum mechanical measurement pcess in more detail.

D. The probability interpretation of quantum mechanics

As we have mentioned, a basic observation in an interence experiment with single photons is that the pattern onscreen builds up from the ‘‘hits’’ of single photons. It ilegitimate to ask whether these positions are predetermas in classical physics and can be predicted from the inconditions. In this stage of the course, the students learnone cannot predict the position of a single hit, but that itnevertheless possible to make accurate predictions forstatistical distribution ofmanyhits. This observation is generalized to the following important statement:Quantum me-chanics makes statistical predictions about the results ofpeated measurements on an ensemble of identically prepquantum objects. This preliminary version of the probabilityinterpretation is later, in the context of electrons, formulamore precisely in terms of the wave function.

V. ELECTRONS

The second part of the basic course is devoted to electrHere, the insights gained with photons are broadeneddeepened at a higher conceptual level.

A. Wave behavior of electrons

The demonstration of electron wave behavior withelectron diffraction tube is a standard experiment that dnot need to be described here. The students obtain thBroglie relation from the analogous relation for photons aconfirm it experimentally with the diffraction tube.

e

Fig. 4. No interference pattern emerges with orthogonal polarization filt

203R. Mu¨ller and H. Wiesner

aW

at

g

ean

thes

pld

ain

ncalxbes

e

aracsavathdayu

ncrre

wo

x-

n-ingson-d inlyntityeenme

ws

n

aslo-ty.

B. Double-slit experiment and probability interpretation

Since Feynman’s lectures,1 the double-slit experiment isbasic ingredient in many quantum mechanics courses.have written a computer program to interactively simulthe double-slit experiment~see Fig. 5!.28 Electrons~or otherobjects! are emitted by the source on the right, pass throuthe double slit, and are detected on a screen~the light bulbshown in Fig. 5 is needed later in the discussion of the msurement process!. Many of the experimental parameters cbe varied by the students.

In our course, we first use the double-slit experimentresolve the problem of wave–particle duality in terms of tprobability interpretation. We consider single electrons paing through the apparatus. The students see that in comanalogy with the photon case, the interference pattern graally emerges from single electron detection events. Agonly statistical predictions are possible.

Next we go one step further. We introduce the wave fution to describe the state of electrons in a completely qutative way in analogy with water or sound waves. For eample, the wave function behind the double slit canvisualized as a superposition of two cylindrical wavemerging from the slits:c5(cA1cB!/A2, whereA and Blabel the slits.

An essential point for the interpretation of quantum mchanics is Born’s interpretation ofuc(x)u2 as theprobabilitydensityof finding an electron at the positionx in a measure-ment. With this interpretation, the duality of wave and pticle, which is often so much emphasized in popularcounts, is no longer a mystery: The wave function spreadspace much like a classical wave and shows typical wphenomena such as superposition and interference. Mematically, interference arises from the cross terms insquare of (cA1cB!. However, when an electron is detectethat is, when a position measurement is made, it is alwfound localized at a certain position. The statistical distribtion of detection events can be calculated if the wave fution is known. With the Born interpretation, wave and paticle behavior which seemed to be incompatible are captuin a single picture.

C. Position property of electrons

In the interferometer experiment discussed in Sec. IVdiscussed how the conception of a photon as a localizedject with a well-defined position leads to conflicts with e

Fig. 5. Simulation program for the double-slit experiment.

204 Am. J. Phys., Vol. 70, No. 3, March 2002

ee

h

a-

o

s-eteu-,

-i--e

-

--ine

th-e

,s--

-d

eb-

periment. This conflict is very counterintuitive, and we canot expect the students to accept such a far-reachconclusion at once. Therefore we present analogous reaing for electrons as well. The argument has been publishesimilar form many times in the literature so we will give ona brief sketch. Suppose the electron goes as a localized ethrough one definite slit. Then it cannot ‘‘know’’ whether thother slit is open or closed. Its final position on the screshould not be influenced by this fact. Therefore, the sapattern on the screen should appear if~a! both slits areopened together for a timet, or ~b! first one slit is opened forthe time t, then the other one. However, experiment shothat it does matter whether~a! or ~b! is realized. Double-slitinterference is observed in case~a!, whereas a superpositioof two single slit patterns is observed in case~b! ~see Fig. 6!.This result demonstrates that the initial assumption wwrong. In this experiment, electrons do not behave likecalized objects. They do not possess the position proper

Fig. 6. ~a! Double-slit interference;~b! superposition of two single-slit pat-terns.

204R. Mu¨ller and H. Wiesner

e

icouatalua

-tta

as

s

unwltVaa

atb

-n

nefsetuu

isure

e

aste

ctfbti

atte

rhefos.

oin

fr-ednc

hastheher-

ionsar-

ofsly

of-

anytisti-ard-ea-itfsul-ea-il-

on

se,osed at

het aon

andr

ita-uss

fixed

forki-

By

he

D. Position measurement and the measurement postulat

The next major discussion is on the quantum mechanmeasurement process. We start directly from the previexperiment and ask the seemingly innocent question: Whthe result of a position measurement if the electrons actudo not possess the property position? The experiment weto answer this question has been conceived by Feynmalthough we interpret it slightly differently in our course.

The light bulb in the double slit simulation program symbolizes a position measurement device. It emits light thascattered by the electrons that pass the slit. Eventually,scattered light ends in our eye or a detection apparatus,we can infer from the direction of incidence where it wscattered.

If we turn on the light bulb, we see little light flashebehind one of the slits~visible in Fig. 5!. They give the resultof the position measurement. The electron is always fobehind exactly one of the slits. For a single electron,never see the light flashes at two or more positions simuneously ~see the analogous result for photons in Sec. I!.This result seems to contradict our previous finding thatelectron does not possess the position property. But thisparent contradiction once more underlines the special stof a measurement in quantum mechanics. It is resolvedthe fundamentalmeasurement postulate: In each measurement, a definite value for the measured observable is fouLater, in the advanced part of the course, we make a contion to the idea of an eigenvalue: The possible results omeasurement are the eigenvalues of the measured obable. The reason why eigenvalues are interesting in quanmechanics is that they are the possible results of measments.

On the other hand, this experiment shows that theredifference between ‘‘to possess a property’’ and ‘‘to measa property.’’ If we find an electron behind one of the slits, wcannot assume that it did possess the position propertyready before the measurement. A definite position valuonly realized in the measurement.

E. Advanced topics in the interpretation of quantummechanics

With the double slit experiment, some other importantpects of quantum mechanics can be easily demonstrasuch as state reduction and complementarity. State reduis the phenomenological way of describing the influence omeasurement on the subsequent state of the measured oIn our example, state reduction takes place after the detecof the electron behind one of the slits. As a result of streduction, instead of the usual double-slit pattern, a patsuch as the one shown in Fig. 6~b! builds up behind thescreen.

Complementarity has been a central element in Bohphilosophy of quantum mechanics. A simple form of it is tcomplementarity between interference pattern and path inmation. It can be demonstrated by a series of experimenteach one the intensity of the detection light is decreasedthat more and more electrons escape undetected. Accingly, the visibility of the interference pattern increaseseach experiment.

The discussion of superposition states provides a basisthe discussion of Schro¨dinger’s cat paradox where a supeposition of macroscopically different states is considerThe paradox is resolved by the mechanism of decohere

205 Am. J. Phys., Vol. 70, No. 3, March 2002

alsislysen,

ishend

dea-

np-usy

d.c-arv-m

re-

ae

al-is

-d,

ionaject.onern

’s

r-Insord-

or

.e.

The interaction with its natural environment~for example,photon or air molecule scattering! renders the cat effectivelyclassical. That photon scattering may inhibit interferencealready been observed in the double slit experiment withlight bulb. This experience can be used to make the decoence mechanism plausible.

F. Heisenberg’s uncertainty relation

It has been mentioned that there are many misconceptof the Heisenberg uncertainty relation. In our view, its cleest formulation is as a statement about the preparationquantum objects. It restricts the possibility of simultaneoupreparing certain pairs of observables on an ensemblequantum objects.23 We start from an arbitrarily prepared ensemble of electrons described by the wave functionc. Wenow take a subensemble of this ensemble and perform mposition measurements on its members. We obtain a stacal distribution of measurement results and call the standdeviation of this distributionDx. Now we take another subensemble and perform a large number of momentum msurements to obtainDp. The uncertainty relation states thatis not possible to prepare a statec such that the product oDx andDp is smaller than\/2. This way of reasoning showthat the uncertainty relation is not a statement about simtaneous measurements or the mutual disturbance of two msurements. Quantitatively, the uncertainty relation can belustrated by the well-known example of electrons incidenta single slit.

VI. AN INTRODUCTION TO THE QUANTUMMECHANICAL FORMALISM

In the above we presented the first part of our courwhich is addressed to students who are not likely to chophysics at the university. The advanced course is aimestudents with a special interest in physics~Leistungskurse!.In this part of the course we give an introduction to tformalism of quantum mechanics and its applications amathematically very basic level. The discussion is basedthe qualitative understanding gained in the basic coursetakes up the ideas introduced there~such as preparation oquantum mechanical properties!.

A. Wave functions and operators

The wave function has already been introduced qualtively to describe the state of electrons. We now discmathematically the wave function of free electronsCEkin

. Itdescribes an ensemble of electrons prepared to have avalue of the property ‘‘kinetic energy’’~or, likewise, momen-tum!.

Next we introduce the concept of an operator. We looka mathematical operation that extracts the value of thenetic energy from the wave function of free electrons.trial and error, the students find that the operation

2\2

2m

d2

dx2

applied toCEkin, leads to the desired result. It leaves t

wave function unchanged and pulls a factorEkin out of it.They have found the operator of kinetic energy.

205R. Mu¨ller and H. Wiesner

iowtho-

e

nis

so

lu

t

th‘to

nTheeeityth

tuatedWizm

n

tua

ellefin

or

ihe

tfivTw

tu

ereook

ro-oom

f the

had

ionni-

icalartairere,r to

oxi-op-

thee-ter-in-

sics

ber,-of

ep-gly

a

nts-

naire.

he

tye

re

n

ly

B. Eigenvalue equation

Now we ask the opposite question. Given a wave functC, how can we decide whether it describes an ensemblethe property kinetic energy? The answer is to apply againoperator of kinetic energy. If the wave function is reprduced,

2\2

2m

d2

dx2c5Ekinc,

the corresponding ensemble possesses the kinetic enproperty, and the constantEkin gives the value of the kineticenergy. The wave function is an eigenfunction of kinetic eergy andEkin is the eigenvalue. If the eigenvalue equationnot fulfilled, the ensemble described byC does not possesthe kinetic energy property. If a series of measurementsthe corresponding electrons is made, the measured vawill have a distribution.

C. Schrodinger equation

With the concept of an eigenvalue equation at hand, ionly a small step to the stationary Schro¨dinger equation,which is the eigenvalue equation of total energy. SolvingSchrodinger equation means to search for states with the ‘tal energy’’ property. These are called stationary states.

The Schro¨dinger equation is the basic equation of quatum mechanics, and we can solve it for some cases.simplest example is the infinite potential well where tquantization of energy appears for the first time. The thrdimensional potential well is used to illustrate probabildistributions in three-dimensional space and to introduceidea of orbitals.

D. Atoms

The final part of the course is devoted to atoms. The sdents observe line spectra of atoms and see the quantizof energy in the Franck–Hertz experiment. As mentionthe Bohr model is rooted deeply in the students’ minds.therefore discuss the Bohr model critically and emphasthat it conflicts with some of the basic results of quantumechanics covered so far~for example, Heisenberg relatioand the impossibility of trajectories!. As an alternative, wediscuss the orbital model of the atom.

The mathematics necessary to solve the Schro¨dinger equa-tion for the hydrogen atom is beyond the scope of our sdents. We therefore use an approach proposed in Ref. 29model the Coulomb potential by an infinite potential wwith appropriately chosen parameters. We obtain a sconsistent equation for the possible energy values andthe 21/n2 behavior of the energy eigenvalues. This imptant result completes our course.

VII. EVALUATION OF THE COURSE

An important part in the development of a new courseits evaluation. In a pilot study, preliminary versions of tcourses were taught in five classes by P. Engelhardt andauthors. In the actual evaluation, we tested our course inother Gymnasium school classes with about 60 students.of the classes were nonspecialized physics classes~Grund-kurse! with 3 hours a week, and three classes were for sdents with a special interest in physics~Leistungskurse! with5 hours per week.

206 Am. J. Phys., Vol. 70, No. 3, March 2002

nithe

rgy

-

nes

is

e-

-he

-

e

-ion,ee

-nd

lf-d

-

s

heeo

-

The design of the study was as follows: The students winstructed by their regular teachers. Instead of a textbthey were provided with a text~approximately 100 pages!containing the contents of the course. The simulation pgrams developed for the course were used in the classrand, in part, also at home.

We used several instruments to measure the success ocourse.

~1! Questionnaire on students’ conceptions. The question-naire consisted of two parts. In the first part the studentsto rate statements on a five-point scale from 1~‘‘stronglyagree’’! to 5 ~‘‘strongly disagree’’!. They had to judge 44items from four different subfields of the course: Conceptof the atom, determinism/indeterminism, quantum mechacal properties, and the uncertainty relation. Some typquestions are given in Table I. The questions were in padapted from Ref. 14. The second part of the questionncontained questions with the possibility of answers. Hestudents were asked to explain the uncertainty relation odraw their visual image of an atom~including a commentarywhether there are features that cannot be drawn!.

~2! Student interviews. In two of the classes~N522!, semi-structured interviews were conducted. Each lasted apprmately 1 h. They were taped, transcribed, and analyzed. Tics again included the conception of the atom anduncertainty relation. In addition, we investigated in more dtail the understanding of more complex issues such as deminism, probabilistic laws, state superposition, and thesights gained in the double slit experiment.

~3! Questionnaire on physics interest. We also used thequestionnaire to compare the students’ interest in the phyof waves and in quantum mechanics.

To express the success of the course by a single numwe calculated a statistical indexC from 29 items of the questionnaire on students conceptions. An index valueC51100 corresponds to fully quantum mechanical conctions, C52100 means conceptions that contradict stronto the quantum mechanical ones, andC50 corresponds to anindifferent attitude~for example, rating each statement with3!.

Figure 7~a! shows the distribution of the index valueC forthe students that were taught our course. All of the studehave a positive value ofC which means that quantum mechanical conceptions dominate. The average is155.8 with a

Table I. Some of the statements the students had to rate in the question

An atom has a similar structure as the solar system~planets that orbit thesun!Electrons in an atom are like a smeared charge cloud that surrounds tnucleusElectrons move around the nucleus in definite orbits with a high velociNobody accurately knows the position of an electron in orbit around thnucleus because it is very small and moves very fastAn electron that goes from the source to the screen in the double-slitexperiment takes a definite path, even if it cannot be determinedIf we knew the initial conditions precisely enough we could predict whethe next electron is found on the screenThe wave function determines the distribution of electrons on the screeIn principle, quantum objects can simultaneously possess position andmomentumThe uncertainty relation sets a limit on how good the momentum of anelectron can be determinedIf a precise position measurement is carried out on an electron, it is onpossible to make an imprecise momentum measurement afterwards

206R. Mu¨ller and H. Wiesner

im

n-chm

tr

wc

T

tena

edor.omeol-lar

chel

d-t ofsn is

the

cer-men-

hes

ig-

ilityanre-

the

standard deviation of 19.5. We consider this result as andication that the students successfully learned quantumchanical conceptions.

To compare the results with a group of traditionally istructed students, we gave the questionnaire on studentceptions to a group of 35 first-year university students whad been taught quantum physics during their time in Gynasium. We took care to include in the definition ofC onlythose questions that we considered to be fair to the congroup. The distribution of the indexC in this control group isshown in Fig. 7~b!. The average is135.2 with a standarddeviation of 23.7. A comparison of the two diagrams shothat the experimental group has developed more pronounquantum mechanical conceptions than the control group.difference is highly significant~significance level 0.1%!. Toappreciate this result one has to take into account thatcontrol group is positively selected because these studdecided to major in physics at the university. We have c

Fig. 7. ~a! Distribution of the conception indexC in the experimental group;~b! distribution ofC in the control group.

207 Am. J. Phys., Vol. 70, No. 3, March 2002

n-e-

on-o-

ol

sedhe

hets

l-

culated similar indices for the four subfields mentionabove. In all of them, the experimental group was superi

Figures 8–10 show the responses of both groups to sindividual statements in the questionnaire. In Fig. 8, the flowing statement had to be rated: ‘‘An atom has a simistructure to the solar system~planets that orbit the sun!.’’ Thedistribution of the experimental group@Fig. 8~a!# is peakedtoward complete rejection~average 4.38! in accordance withthe quantum mechanical model. The control group is mumore indifferent and not so critical of the planetary mod~average 3.52!.

In Fig. 9 the question of determinism/indeterminism is adressed. The following statement was given in the contexthe double slit experiment: ‘‘If we knew the initial conditionprecisely enough, we could predict where the next electrofound on the screen.’’ The experimental group@Fig. 9~a!#clearly denies the determinism of classical mechanics inquantum domain~average 4.75!. The control group is muchless certain~average 3.88!.

Figure 10 shows the response to a statement on the untainty relation. The statement was, ‘‘In principle, quantuobjects can possess simultaneously position and momtum.’’ Again, the experimental group definitely rejected tstatement~average 4.89!, whereas there are mixed opinionin the control group~average 2.92!. In all of the examplesmentioned, the difference between the groups is highly snificant.

The response to a question with an open reply possibis shown in Fig. 11. The students were asked to drawatom according to their conceptions. We categorized thesults according to the scheme of Ref. 30:~i! Bohr ~nucleuswith electrons on orbits!, 13% in the experimental group

Fig. 8. Response to the statement: ‘‘An atom has a structure similar tosolar system~planets that orbit the sun!;’’ ~a! experimental group,~b! controlgroup ~15‘‘strongly agree,’’ 55‘‘strongly disagree’’!.

207R. Mu¨ller and H. Wiesner

is-the

ar

bell

aveithinthewncon-lar

andotent

d of

hatugh?’’

then-

y

if itonre

reth

ca

Fig. 9. Response to the statement: ‘‘If we knew the initial conditions pcisely enough, we could predict where the next electron is found onscreen.’’~a! Experimental group,~b! control group.

Fig. 10. Response to the statement: ‘‘In principle, quantum objectssimultaneously possess position and momentum.’’~a! Experimental group,~b! control group.

208 Am. J. Phys., Vol. 70, No. 3, March 2002

versus 32% in the control group;~ii ! cloud ~distributed cloudaround the nucleus!: 61% vs 29%;~iii ! dumb-bell~reminis-cent of ap- or d-orbital!: 16% vs 27%, and~iv! ‘‘no imagepossible’’: 5% vs 11%. In accordance with the results dcussed earlier, the Bohr model played a minor role inexperimental group. Nearly all of the students drew‘‘cloud’’ image. In the control group the classical Bohmodel, the quantum mechanical cloud, and the dumb-image were approximately evenly distributed.

Another interesting question is whether the students hdeveloped the competence to argue freely and unaided wthe new conceptual framework. This ability was tested ininterviews. Students were asked to comment in their owords on several questions and statements related to thetents of the course. The results are instructive in particufor the physically more complex topics. For example,important aim of the course was to realize that electronsnot necessarily possess the position property. To what exwere the students able to justify their statements insteamerely memorizing the correct answer?

One of the interview questions was, ‘‘Someone claims tan electron in the double-slit experiment goes either throthe left or through the right slit. How can you disprove thisTo evaluate the student answers quantitatively, we gradedreplies from 1~physically correct, clear and careful reasoing! to 5 ~insufficient or totally confused answer!. The resultsare shown in Fig. 12. Most of the students~55%! were ableto argue adequately~mark 1 or 2! whereas 32% apparentldid not develop a deeper understanding~mark 4 or 5!. Atypical example of an~oral! student reply marked with 1 isthe following: ‘‘Well, it’s actually this point in physics that ismost fascinating to me, in quantum physics. Becausewould go through the left or the right slit, then the electrwould have to end up in the same region if both slits we

-e

n

Fig. 11. Student drawings of an atom~classification as in Ref. 28!; ~a!experimental group,~b! control group.

208R. Mu¨ller and H. Wiesner

isnceslhu

sasuenthd

heion

mavn

tiath

ati

thsngwal

de

gn

n-

l: A

ec-

ic

ies

p-

cs,’’

in

s of

tu-

be-

s,’’archdu/

v.

m

tik/re

ho-

tik/

D

open or if only one slit was open. But this is not true. Thatif I open only the left slit, then I get a single-slit distributiobehind the left slit. If I open them both, I get the interferenpattern; and on the location of the maximum of the singlepattern there is a minimum of the interference pattern. Tmeans: it cannot go through just one of the slits, there mbe something else~S16!.’’

To summarize the results of the evaluation, we canthat the correct quantum mechanical conceptions werecessfully imparted to most of the students of the experimtal group. This understanding was achieved in spite oflarge conceptual difficulties inherent in the subject. In adtion, many of the common misconceptions encounteredtraditional instruction, e.g., in the uncertainty relation, tdeterminism/indeterminism problem, or the atom concepthave been avoided.

VIII. TEACHER’S TRAINING ON THE WEB

Finally, we briefly mention our Web-project milq~MunichInternet Project for teacher’s training on quantumechanics!.31 In this project we want to use the Internet asnew medium for teacher’s training. Traditional courses haa capacity of 20-40 persons. With the Internet, there isupper limit to the number of participants so that the potenimpact is much larger. In addition, we can investigatechanges and limitations offered by new possibilities suchthe use of multimedia, simulations, and hypertext. Upnow, milq has been available in German only. However,collaboration with the Visual Quantum Mechanics Project5 atKansas State University, an English version is planned.

There are two main content areas in milq. We presentideas of the quantum mechanics course described in thiticle, together with background information and teachimaterial. Interested teachers can acquaint themselvesthe new approach of teaching quantum mechanics. Weprovide additional information relevant~not only! to quan-tum mechanics classes in school. This information inclucurrent results of research in quantum mechanics~e.g., quan-tum information! and also material on the ever-recurrinquestions of quantum mechanics such as the discussiothe EPR paradox and Bell’s inequalitites.

a!Electronic mail: [email protected]!Electronic mail: [email protected]

Fig. 12. Why do electrons not necessarily possess a position property?tribution of marks on the free replies in the interviews.

209 Am. J. Phys., Vol. 70, No. 3, March 2002

,

itisst

yc--ei-in

,

eol

es

on

ear-

ithso

s

of

1R. P. Feynman, R. B. Leighton, and M. Sands,The Feynman Lectures onPhysics, Vol. III~Addison–Wesley, Reading, 1966!.

2A. Rae, Quantum Physics: Illusion or Reality~Cambridge U.P., Cam-bridge, 1986!.

3M. P. Silverman,More Than One Mystery. Explorations in Quantum Iterference~Springer, New York, 1995!.

4D. Z. Albert, Quantum Mechanics and Experience~Harvard U.P., Cam-bridge, 1992!.

5Available online at http://www.phys.ksu.edu/perg/vqm.6H. Niedderer, T. Bethge, and H. Cassens, ‘‘A simplified quantum modeteaching approach and evaluation of understanding,’’ inRelating Macro-scopic Phenomena to Microscopic Particles: A Central Problem in Sondary Science Education, edited by P. L. Lijnseet al. ~CD-b Press,Utrecht, 1990!.

7J. Petri and H. Niedderer, ‘‘A learning pathway in high-school level atomphysics,’’ Int. J. Sci. Educ.20, 1075–1088~1998!.

8H. Fischler and M. Lichtfeldt, ‘‘Learning quantum mechanics,’’ inRe-search in Physics Learning: Theoretical Issues and Empirical Stud,edited by R. Duit, F. Goldberg, and H. Niedderer~IPN, Kiel, 1991!.

9H. Fischler and M. Lichtfeldt, ‘‘Modern physics and students’ concetions,’’ Int. J. Sci. Educ.14, 181–190~1992!.

10A. Mashhadi, ‘‘Students’ conceptions of quantum physics,’’ inResearch inScience Education in Europe: Current Issues and Themes, edited by G.Welford, J. Osborne, and P. Scott~Falmere, London, 1996!, pp. 254–265.

11D. F. Styer, ‘‘Common misconceptions regarding quantum mechaniAm. J. Phys.64, 31–34~1996!.

12I. D. Johnston, K. Crawford, and P. R. Fletcher, ‘‘Student difficultieslearning quantum mechanics,’’ Int. J. Sci. Educ.20, 427–446~1998!.

13L. Bao, E. F. Redish, and R. N. Steinberg, ‘‘Student misunderstandingthe quantum wavefunction’’~unpublished!.

14G. Ireson, ‘‘The quantum understanding of pre-university physics sdents,’’ Phys. Educ.35, 15–21~2000!.

15H. Wiesner,Beitrage zur Didaktik der Unterrichts u¨ber Quantenphysik inder Oberstufe~Westarp, Essen, 1993!.

16H. Wiesner, ‘‘Versta¨ndnisse von Leistungskursschu¨lern uber Quanten-physik. Ergebnisse mu¨ndlicher Befragungen,’’ Phys. in der Schule34,95–99~1996!; 34, 136–140~1996!.

17R. Muller and H. Wiesner, ‘‘Vorstellungen von Lehramtsstudenten zugrifflichen Problemen der Quantenmechanik,’’Didaktik der Physik, Bei-trage zur Vol. 62~Physikertagung, Regensburg, 1998!, pp. S. 458–463.

18R. Muller and H. Wiesner, ‘‘Students’ conceptions on quantum physicpresented at the annual meeting of the National Association for Resein Science Teaching, 1999; available online at http://www.phys.ksu.eperg/papers/narst/QM_papers.pdf.

19H. Niedderer, ‘‘Physiklernen als kognitive Entwicklung,’’Didaktik derPhysik, Beitrage zur~Physikertagung Ludwigsburg, 1999!, pp. 49–66.

20L. Eisenbud,The Conceptual Foundations of Quantum Mechanics~vanNostrand Reinhold, New York, 1971!.

21L. Ballentine,Quantum Mechanics, A Modern Development~World Sci-entific, Singapore, 1998!.

22D. I. Blokhintsev,Quantum Mechanics~Reidel, Dordrecht, 1964!.23L. Ballentine, ‘‘The statistical interpretation of quantum mechanics,’’ Re

Mod. Phys.42, 358–380~1970!.24W. E. Lamb, ‘‘An operational interpretation of nonrelativistic quantu

mechanics,’’ Phys. Today22, 23–28~1969!.25G. Ludwig, Foundations of Quantum Mechanics~Springer, Berlin, 1983!.26It can be downloaded from www.physik.uni-muenchen.de/didak

Computer/interfer/interfere.html~the German and English languages asupported!.

27P. Grangier, G. Roger, and A. Aspect, ‘‘Experimental evidence for a pton anticorrelation effect on a beamsplitter,’’ Europhys. Lett.1, 173–179~1986!.

28It can be downloaded from www.physik.uni-muenchen.de/didakComputer/Doppelspalt/dslit.html.

29W. Theis, ‘‘Begrundung diskreter Energiewerte fu¨r gebundene Zusta¨ndeohne Losung der Eigenwertgleichung,’’ Phys. Didaktik22, 198–204~1994!.

30M. Lichtfeldt, Schulervorstellungen in der Quantenphysik und ihre Mo¨gli-chen Vera¨nderungen durch Unterricht~Westarp, Essen, 1992!.

31The milq pages can be found at www.cip.physik.uni-muenchen.de/;milq.

is-

209R. Mu¨ller and H. Wiesner