Surveying students understanding of quantum mechanics in one spatial dimension. Guangtian Zhu and Chandralekha Singh

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1 Surveyig studets uderstadig of quatum mechaics i oe spatial dimesio Guagtia Zhu ad Chadralekha Sigh Departmet of Physics ad Astroomy, Uiversity of Pittsburgh, Pittsburgh, Philadelphia, 560, USA Abstract We explore the difficulties that advaced udergraduate ad graduate studets have with o-relativistic quatum mechaics of a sigle particle i oe spatial dimesio. To ivestigate these difficulties we developed a coceptual survey ad admiistered it to more tha two hudred studets at eleve istitutios. The issues targeted i the survey iclude the set of possible wavefuctios, boud ad scatterig states, quatum measuremet, expectatio values, the role of the Hamiltoia, ad the time-depedece of the wavefuctio ad expectatio values. We fid that udergraduate ad graduate studets have may commo difficulties with these cocepts ad that research-based tutorials ad peer-istructio tools ca sigificatly reduce these difficulties. The fidigs also suggest that graduate quatum mechaics courses may ot be effective at helpig studets develop a better coceptual uderstadig of these topics, partly because such courses maily focus o quatitative assessmets. I. INTRODUCTION Learig quatum mechaics is challegig. -4 The cocepts are ot ituitive ad is very differet from the oes which studets are used to from their previous courses ad everyday experieces. 5 Moreover, a good uderstadig of the formalism of quatum mechaics requires a solid grasp of liear algebra, differetial equatios, ad special fuctios. Despite the mathematical facility required to master quatum mechaics, the formalism has a coheret coceptual framework. 6-8 For studet learig to be meaigful, the goals of the course, the istructioal desig, ad the assessmet of learig should all be aliged. 9- Because studets will focus o what is assessed, assessmet should iclude a uderstadig of the coceptual framework ad kowledge structure of quatum mechaics. Without a

2 coceptual framework, studets are ulikely to retai what they have leared whe the course is over. Multiple-choice coceptual surveys are useful tools for evaluatig studets uderstadig of various topics. Such surveys are easy to admiister ad grade. The results are objective ad ameable to statistical aalysis so that differet istructioal methods ad differet studet populatios ca be compared. The Force Cocept Ivetory is a coceptual multiple-choice survey which has helped istructors recogise that may itroductory physics studets do ot develop a fuctioal uderstadig of force cocepts eve if they ca solve quatitative problems. Other coceptual surveys have bee desiged for may physics topics, icludig eergy ad mometum, rotatioal ad rollig motio, circuits, electricity ad magetism, ad Gauss s law. These surveys reveal that studets have may coceptual difficulties with classical physics. Research-based istructioal strategies have bee show to sigificatly improve studets coceptual uderstadig of some of these topics. 0- To explore the coceptual difficulties that udergraduate ad graduate studets have with quatum mechaics, we developed the Quatum Mechaics Survey (QMS), a 3-item multiple-choice test. The survey was developed by cosultig with may quatum mechaics istructors about the goals of their udergraduate courses ad the topics their studets should have leared. We the iterated differet versios of the ope-eded ad multiple-choice questios with a subset of these istructors durig the developmet of the survey. To ivestigate studets difficulties with various cocepts, we admiistered free-respose ad multiple-choice questios ad coducted

3 iterviews with idividual studets usig a thik-aloud protocol. I this iterview protocol, studets were asked to talk aloud while they aswered the questios so that the iterviewer could record their thought processes. Idividual iterviews with studets durig the ivestigatio of difficulties ad the developmet of the survey were useful for a i-depth uderstadig of studets thought processes. Udergraduate quatum mechaics is sometimes taught as a oe semester course. Also, some istructors begi with two-state systems before coverig quatum mechaics of a sigle particle i oe dimesio. Although such courses may help studets develop a good grasp of quatum mechaics, all cocepts covered i the survey may ot be discussed i such courses. Our survey is ot appropriate for such courses i which all relevat cocepts are ot covered. II. SURVEY DESIGN The survey focuses o assessig studets uderstadig of the coceptual framework of quatum mechaics of oe particle i oe spatial dimesio rather tha assessig their mathematical skills. Studets ca aswer the survey questios without performig ay complicated mathematics, although studets eed to uderstad the basics of liear algebra. Because the survey focuses o quatum systems i oe dimesio, the cocept of orbital agular mometum is ot icluded i the survey. We also did ot iclude spi agular mometum ad Dirac otatio to esure that it ca be used after most juior/seior-level quatum mechaics courses regardless of textbook, istitutio, or istructor. While desigig the survey, we paid particular attetio to reliability ad 3

4 validity. 3-4 Reliability refers to the degree of cosistecy betwee idividual scores if someoe immediately repeats the test; validity refers to the appropriateess of iterpretig the test scores. To esure that the survey is valid, we took ito accout the opiios of istructors regardig the goals of juior/seior-level quatum mechaics courses ad the cocepts that their studets should have leared. 5 We also surveyed faculty members who had taught a two semester upper-level udergraduate course about these issues at a 00 Gordo Research Coferece o quatum mechaics. We foud may commoalities about what these istructors expected their studets to have leared. I additio to usig pe ad paper (or olie) surveys, we discussed these issues idividually with several istructors at the Uiversity of Pittsburgh who have taught quatum mechaics at the juior-seior ad/or graduate level. The quatum mechaical models i the survey are all cofied to oe spatial dimesio (D), for example, the ifiite/fiite square well, the simple harmoic oscillator, ad the free particle. The survey icludes a wide rage of topics such as the possible wavefuctio, the expectatio value of a physical observable ad its time depedece, the role of the Hamiltoia, statioary ad o-statioary states ad issues related to their time developmet, ad measuremets. Before developig the questios for the survey, we developed a test blueprit to provide a framework for decidig the desired test attributes. The specificity of the test pla helped us to determie the extet of cotet covered ad the complexity of the questios. I developig good alteratives for the multiple-choice questios, we took 4

5 advatage of prior work o studet difficulties with quatum mechaics. 6-0 To ivestigate studet difficulties further, we admiistered a set of free-respose questios i which studets had to provide their reasoig. The aswers to these ope-eded questios were summarized ad categorized, which helped us develop alteratives for the questios i the survey based o commo difficulties. The icorrect choices ofte had distracters which reflect studets commo miscoceptios to icrease the discrimiatig properties of the questios. Havig good distracters i the alterative choices is importat so that the studets do ot select the correct aswer for the wrog reaso. Statistical aalysis was coducted o the prelimiary versios of the multiple-choice questios to help refie the questios further. We iterviewed idividual studets usig a thik-aloud protocol to develop a better uderstadig of studets reasoig processes whe they were aswerig the ope-eded ad multiple-choice questios. Durig the thik-aloud iterviews, some previously uoticed difficulties ad miscoceptios were revealed. These commo difficulties were icorporated ito ew versios of the writte tests ad ultimately ito the multiple-choice questios i the survey. Four professors at the Uiversity of Pittsburgh reviewed differet versios of the survey several times to examie its appropriateess ad relevace for upper-level udergraduate quatum mechaics courses ad to detect ay ambiguities i item wordig. May professors from other uiversities also provided valuable commets ad feedback to fie-tue the survey. Each questio has oe correct choice ad four icorrect choices. 3 5

6 Some of the questios were based o the research-based learig tools for quatum mechaics such as cocept tests ad Quatum Iteractive Learig Tutorials. 7 Most of the upper-level studets erolled i a two semester quatum mechaics course are able to complete the survey i oe class period after all these topics are covered i class. Experiece i itroductory physics suggests that physics professors ofte take a sigificatly loger time to aswer the questios i the Force Cocept Ivetory whe they take it for the first time compared to studets (most of whom fiish it i less tha 30 miutes both before ad after istructio i relevat cocepts). III. SURVEY RESULTS The survey was admiistered to 6 studets from te uiversities. Although te uiversities were ivolved, 4 differet classes were admiistered the survey because both the upper-level udergraduate ad graduate classes took it at oe istitutio for two cosecutive years. Amog the 6 studets, 33 were first year graduate studets erolled i a two semester graduate quatum mechaics course. The survey was admiistered after the first semester. The other studets were udergraduates who had take at least a oe-semester quatum mechaics course at the juior/seior level. All studets completed the survey i oe class period except those i a class where the istructor taught quatum mechaics i two back-to-back class periods. This istructor requested that his studets be allowed to use both back-to-back class periods to complete the survey. Because there is o statistically sigificat differece betwee the scores of these studets ad those from other classes, we do ot distiguish betwee these studets. Two of the juior/seior classes where studets 6

7 were erolled i a two semester course used research-based learig tools such as cocept tests ad Quatum Iteractive Learig Tutorials. The survey was give twice, oce at the ed of the first semester (8 studets) ad the agai at the ed of the secod semester (6 studets). The average score o the survey for all 6 studets regardless of istructio (icludig oly the first score of studets who took it twice) is 45%. The reliability coefficiet (which is a measure of the iteral cosistecy of the test with a high sigifyig that some studets cosistetly perform well across various questios o the test while others perform poorly) is 0.9, which is quite good by the stadards of test desig. 3 The percetage of studets who correctly aswered each questio is show i Fig. ad raged betwee 0. ad 0.8. Most of the percetages were aroud 0.4. This rage is cosistet with our previous ivestigatios of studet difficulties. Figure shows the item discrimiatio, which represets the ability of a questio to distiguish betwee the high ad low performig studets i the overall survey. A measure of item discrimiatio is the poit biserial discrimiatio coefficiet, 3 which is the correlatio betwee the score o a particular questio for each studet ad the total test score mius the score o that questio for each studet. The poit biserial discrimiatio coefficiet raged from approximately 0.3 to 0.6 with about 3/4 of the questios with poit biserial discrimiatio coefficiets higher tha 0.4. The stadards of test desig 3 idicate that the survey questios have reasoably good item discrimiatio. 7

8 Fig. Item difficulty (fractio correct) for each item o the test for 6 studets. Fig. Item discrimiatio for each item o the test. The average score for the upper-level udergraduate classes that used cocept tests ad Quatum Iteractive Learig Tutorials durig the semester was 7.5% at 8

9 the ed of the first semester (8 studets) ad 69.4% at the ed of the secod semester (6 studets). The average performace of studets who used the research-based active learig tools 7, did ot deteriorate after a secod semester i quatum mechaics. I classes that did ot use the learig tools, the average score was 5.6% for the graduate course (33 studets) ad 39.0% for the udergraduate courses. Note that although studets would score 0% o average if they aswered all questios radomly give a five item multiple choices, experiece with the Force Cocept Ivetory i itroductory physics suggests that with good distracters studets performace ca ofte be worse tha radom because they fid the distracters attractive. Our item aalysis (to be discussed) suggests that studets are ot radomly guessig ad are providig resposes they thik are reasoable. (I idividual iterviews studets ofte claim that the alterative choices are the correct choices for those questios.) Although the graduate studet performace is low, discussios with two graduate quatum mechaics course istructors suggest that they expected their studets to kow all the survey cotet ad perform well. After realizig that the graduate studets had ot doe so, the graduate istructors agreed that may of the graduate studets lacked coceptual uderstadig ecessary for performig well o the survey eve though they do well o the quatitative exams typically give i the graduate level courses. The poor performace of the graduate studets suggests that they would develop a more robust kowledge structure if graduate quatum courses focused o both coceptual ad quatitative problem solvig (rather tha oly 9

10 quatitative problem solvig) by icludig coceptual problems i the assessmet of studet learig. IV. ITEM ANALYSIS The survey is icluded i the supplemetary material. 3 Table I shows a particular categorizatio of the questios i the survey based o the cocepts. The table provides oe of the possible ways to classify the questios. Our prior research shows that istructors categorize a give questio i may differet ways 5 so the categorizatio i Table I is oly oe of those which we foud coveiet. I the group Other, Questio is about the ucertaity priciple; Questio 5 ivolves the cocept of degeeracy i the cotext of a free particle; ad Questio 6 ivolves the Ehrefest theorem, which states that the time depedece of the expectatio value of a physical observable of a quatum system obeys the classical laws. I the followig, we describe the commo difficulties foud by the survey i each of the categories. Cocepts Item Number Possible wavefuctios Boud/scatterig states, 6, 4, 6, 30 8, 9, 4, 7, 3 Measuremet 5, 7, 8,, 3, 7, 0,, 8 Expectatio values 9,, 5 Time depedece of expectatio, 0, 3, 6 values Statioary ad o-statioary states 3, 4, 6, 5, 0, 5, 8, 9 Role of the Hamiltoia 6, 7, 9 Time depedece of wavefuctio 3, 4, 6, 5, 7,, 9 Other, 5, 6 Table I. A possible categorizatio of the QMS items ad the questio umbers belogig to each category A. The Possible Wavefuctios Tables II to VIII show the percetages of studets selectig the choices (a) (e) o 0

11 the problems i differet categories, e.g., the possible wavefuctios, statioary states, etc. The correct resposes are i boldface. I some colums the percetages of choices do ot sum to 00% because some studets left a questio blak. A very commo miscoceptio about the possible wavefuctios is thikig that oly certai forms of the wavefuctios are allowed. Studets usually ecouter the eergy eigestates (or positio eigestates) whe they are learig quatum mechaics, so they ofte thik that the possible wavefuctios must be eergy eigestates or eigestates of a operator correspodig to a physical observable. A superpositio of the eergy eigefuctios is a possible wavefuctio as log as it is ormalized (the overall probability for fidig the particle icludig all space sums to ), cotiuous, ad smooth (the first derivative of the wavefuctio is cotiuous except where the potetial eergy is ifiite). Q Q6 Q4 Q6 Q8 Q9 Q0 Q4 Q7 Q30 Q3 (a) % 3% 7% 40% 8% 5% % 0% 7% % 6% (b) 40% 4% 7% 5% % 5% % % 43% % 4% (c) 5% 8% 9% % 4% 0% 45% 30% 9% 7% 4% (d) 50% 5% 9% 8% 38% 6% 7% % 0% 35% 4% (e) 3% 58% 46% 3% 37% 5% 3% 53% 6% 9% % Table II. Distributio of studets resposes to questios related to the possible wavefuctios. Correct resposes are i boldface.

12 The fact that a possible wavefuctio eed ot be symmetric or ati-symmetric eve if the potetial eergy has symmetry is tested i Questio (pictorial represetatio) ad Questio 30 (writte represetatio). We placed these two questios far from each other i the survey to reduce the possibility that the studets would refer to the picture i Questio while aswerig Questio 30. I Questio, 40% of the studets selected the correct choice that all of the wavefuctios, icludig the asymmetric oe, were possible for the give system. The most commo difficulty, experieced by 50% of the studets, was that the system did ot allow for the asymmetric wavefuctio. Questio 30 was very challegig ad oly 9% of the studets chose the correct respose; 35% claimed that the possible wavefuctios for a particle i a eve potetial eergy well must either be eve or odd, ad aother 7% thought that the wavefuctio must be symmetric but ot ecessarily about x = 0. Whe the wavefuctio was explicitly writte as a liear superpositio of the eergy eigestates, for example, x ) A ( x), may studets recogized that ( this wavefuctio is possible. I Questio 6, over 90% of the studets selected the correct choice (a) that x,0) A ( x) is a possible wavefuctio for a particle ( i a D ifiite square well, where ( x ) are the eergy eigefuctios. However, i Questio 4, the wavefuctio Asi ( x / a) is ot expressed explicitly as a liear superpositio of the eergy eigestates ad more tha 50% of the studets mistakely thought that it is ot a possible wavefuctio. Approximately 40% of the studets chose the distracter choices (a) or (b), idicatig that the possible wavefuctio must

13 satisfy the time idepedet Schrödiger equatio. Aother 9% icorrectly oted that Asi ( x / a) is a possible wavefuctio for two particles ad ot a sigle particle. (A two particle wavefuctio depeds o two variables x ad x.) Some studets kew that the possible wavefuctio must be cotiuous ad smooth. However, they were usure that ay sigle valued, cotiuous, smooth, ad ormalized fuctio satisfyig the boudary coditios of the system is possible. I Questio 6, a sketch of a wavefuctio goig to zero iside a fiite square well was give. Studets kew that for a fiite square well, the particle has a ozero probability of beig i the classically forbidde regio i a statioary state. However, they had the miscoceptio that ay possible wavefuctio for this system must have a ozero probability i the classically forbidde regio. Oly 40% of the studets correctly oted that the wavefuctio i Questio 6 is possible. A subgroup of the possible wavefuctios category is related to the boud ad scatterig states of a quatum system. Whe the eergy of the quatum particle is less tha the potetial eergy V (x) at x = ±, the particle is i a boud state. Otherwise, if the particle s eergy is larger tha V (x) at x = ±, it is i a scatterig state. The boud states have a discrete eergy spectrum ad the scatterig states have a cotiuous eergy spectrum. Questios 8 ad 0 examie studets uderstadig of the shape of the boud/scatterig state wavefuctios. The boud state wavefuctios go to zero at ifiity so they ca always be ormalized. The scatterig state wavefuctios are ot ormalizable because the probability of fidig the particle is ozero at ifiity; a 3

14 ormalized wavefuctio ca be costructed usig their liear superpositios. I Questio 8, 0% of the studets did ot select statemet (3), which suggests that they either thought that the scatterig state wavefuctios are ormalizable or they did ot kow that a liear superpositio of the scatterig state wavefuctios ca be ormalized. I idividual iterviews we foud most studets thought that scatterig states could be ormalized. Studets who kew the geeral shape of the scatterig state wavefuctios usually kew how to costruct a ormalized wave packet by takig the liear superpositio of the scatterig states. Also, i Questio 8, 39% did ot kow that the scatterig states have a cotiuous eergy spectrum ad claimed that eergy is always discrete i quatum mechaics. I Questio 0, studets eeded to uderstad that for a simple harmoic oscillator i its groud state, the probability of fidig the particle is a maximum at the ceter, whereas classically the particle is more likely to be foud close to the classical turig poits. We foud that 0% of the studets who chose statemet (3) i Questio 0 thought that the quatum simple harmoic oscillator caot be foud i the regio where E V (x). Discussios with idividual studets suggest that this difficulty ofte has its origi i their experieces with the turig poits of a classical system. (I very few cases durig the idividual discussios did we fid that this difficulty was due to experiece with the quatum ifiite well.) % of the studets who selected choice (a) did ot kow that the first excited state wavefuctio of the simple harmoic oscillator is zero at x = 0 i the middle of the potetial eergy well; 9% of the studets who chose (b) or (d) did ot realize that for a very high eergy statioary state, the probability distributio for 4

15 fidig the particle is cosistet with the classical distributio accordig to Bohr s correspodece priciple; the groud state of a quatum system ca have very differet behavior from the classical behavior. Questios 4 ad 7 ask that studets decide whether a give potetial eergy V (x) allows for boud states or scatterig states. Questio 4 uses a pictorial represetatio showig four differet potetial eergy wells. The distracter that the studets foud challegig was picture (3) i which the potetial eergy of the well bottom was greater tha the potetial eergy at ifiity (which is zero). Therefore, o boud state ca exist i this potetial eergy well. About /3 of the studets failed to otice the differece betwee pictures (3) ad (4). They had the miscoceptio that ay potetial eergy V (x) that has the shape of a well would allow for boud states if there were classical turig poits. I Questio 4, 85% of the studets had selected picture () as the potetial eergy that allows both boud ad scatterig states. Questio 7 asked studets to choose the Hamiltoia operators that have oly a discrete eergy spectrum from three choices. The most commo mistake, by 40% of the studets, was that the fiite square well allows oly discrete eergies. There are at least two possible sources for studets difficulties i Questio 7: they might have difficulty costructig the correct pictorial represetatio from the mathematical represetatio, or fail to recogize the coectio betwee the boud/scatterig states ad the discrete/cotiuous eergy spectrum. Questios 9 ad 3 focus o the miscoceptio that a give particle may be i a boud or a scatterig state depedig o its locatio. This otio ofte has its 5

16 origi i studets classical experiece. I Questio 9, 5% mistakely thought that the particle could have differet eergies i differet regios. I fact, if a quatum particle is i a eergy eigestate, it has a defiite eergy ad does ot have differet eergies i differet regios. If the particle is ot i a eergy eigestate, it does ot have a defiite eergy util a measuremet of its eergy is performed. I Questio 9, 0% of the studets selected icorrect optio (c), ad 6% selected icorrect optio (d). Idividual discussios suggest that studets who selected optio (c) ofte icorrectly thought that the particle is i a boud state whe it is i the classically allowed regio ad is i a scatterig state whe it is i a classically forbidde regio. A similar difficulty was foud i Questio 3. I particular, 4% of the studets selected icorrect optio (b) ad claimed that statemet (3) is correct, which idicates that the studets did ot realize that whether a state is a boud or a scatterig state depeds oly o the eergy of the particle compared to the potetial eergy at ± ifiity. B. Expectatio Values Questios ad 3 ask studets to evaluate the time depedece of the expectatio values of differet physical observables i a statioary or a o-statioary state respectively. I Questio the iitial state is a eergy eigestate, so the expectatio value of ay time-idepedet operator is time-idepedet. The most commo mistake i Questio was the belief that the expectatio values of the positio ad mometum operators deped o time i a statioary state. The iitial state i Questio 3 is a liear superpositio of the eergy eigestates ( ), which is ot a statioary state. The expectatio value of the eergy is time 6

17 idepedet because the probability of obtaiig eergies E or E is always 50%, but the expectatio value of the positio xˆ depeds o time. Studets eed ot evaluate the itegrals to determie the correct respose if they realize that for a o-statioary state, the probability desity chages with time. Also the positio ad mometum operators do ot commute with the Hamiltoia so their expectatio values deped o time i a o-statioary state. 3% of the studets mistakely thought that all the expectatio values of the positio, mometum, ad eergy deped o time whe the wavefuctio is ot a statioary state. 5% chose optio (c) (oly the expectatio value of the eergy depeds o time), which is the opposite of the correct optio (d). I cotrast, oly 5% of the studets i Questio thought that the expectatio value Ĥ depeds o time, but the expectatio values xˆ or pˆ do ot whe the system is i a statioary state. Questio asks studets to compare the expectatio values of differet physical observables at time t for a ifiite square well for the iitial states ( ) ad ( i ), which are differet liear combiatios of the same eergy eigestates. The expectatio values of the eergy for the two iitial states are the same. Because the relative phases of ad are differet for the two states, the shape of the probability desity is differet at time t. Therefore, the expectatio values of the positio (or mometum) of the particles are ot the same i the two states. Oly 9% of the studets chose the correct respose. 8% thought that the relative phases would ot affect the expectatio values of positio ad mometum. Aother 7% icorrectly thought that the expectatio value of eergy would also be affected by the relative 7

18 phase. Similar to Questio 3, 4% thought that the superpositio of eergy eigestates with differet relative phases would give differet expectatio values of the eergy, but the expectatio value of positio or mometum would ot chage. Questio 9 ivestigates whether the studets uderstad differet ways to represet the expectatio value of the eergy. The expectatio value is the average of a large umber of measuremets o idetically prepared systems ad is equal to the sum of the possible values multiplied by their probabilities. It ca also be writte as E a 0 * ( x,0) Hˆ ( x,0) dx. % of the studets icorrectly thought that E E E E (icorrect sig) ad 8% thought that oly the itegral form 3 3 a * ( x,0) Hˆ ( x,0) dx is correct. They did ot coect the defiitio of the 0 expectatio value with its physical meaig, which is the average of a large umber of measuremets o idetically prepared systems. I Questio 0 the iitial state is the same as i Questio 9, but studets eed to evaluate the expectatio value at time t > 0. 74% of the studets selected the correct aswer to Questio 0. However, may might ot uderstad that the expectatio value of eergy is time-idepedet. I particular, studets who aswered Questio 9 icorrectly might aswer Questio 0 correctly because oly oe of the choices (algorithmic method for calculatig the expectatio value) is correct. I the future versios of the survey, we pla to use E E E as oe of the correct choices i Questio Questio 5 ivolves the degeeracy i a D free particle system. The statioary 8

19 state wavefuctios ikx e ad ikx e have mometum i the opposite directios, but have the same eergy, ad their superpositio e ikx ikx e is a eergy eigestate. The expectatio value of mometum is zero, but that of the eergy is ozero. 3% of the studets did ot kow that e ikx ikx e is a statioary state. Also, 7% of the studets icorrectly selected the choice (a). They kew that e ikx ikx e is a statioary state, but did ot realize that ikx e is a mometum eigestate with a defiite value of mometum, ad the expectatio value of mometum is zero i the state e ikx ikx e. Q Q9 Q0 Q Q3 Q5 Q6 (a) 4% 3% 7% % 6% 7% 45% (b) 8% 4% % 3% 0% % % (c) 5% 8% 6% 9% 5% % 5% (d) 4% 8% 74% 8% 50% 38% % (e) 69% 56% % 7% 3% 5% 3% Table III. Distributio of studets resposes for questios related to expectatio values. C. Statioary State Questios 3 ad 4 require studets to decide whether the iitial state, (x,0), is a statioary state before they calculate the probability desity ( x, t) at time t. I respose to Questio 3, 78% of the studets kew that / a si(5 x / a) is a eergy eigestate with eergy E 5, so the probability desity ( x, t) is time-idepedet, but 8% failed to multiply the complex cojugate correctly whe they calculated the probability desity so their resposes had the icorrect phase factor exp( ie5t / ). I Questio 4, oly 35% realized that Asi 5 ( x / a) is ot a statioary state but a liear superpositio of differet statioary states. I particular, 49% mistakely thought that the probability desity i Questio 4 is time idepedet, similar to Questio 3. 9

20 Whe the potetial eergy of a quatum system is chaged suddely, a statioary state of the old system might ot be a statioary state of the ew system. Whe the ifiite square well was expaded suddely at time t = 0 i Questio 5 the groud state at time t < 0 is ot a statioary state at time t > 0. Oly 4% of the studets correctly oted that the probability desity fuctio evolves i time for all t > 0. The most commo miscoceptio was that the old groud state would evetually evolve ito a ew statioary state. 6% of the studets thought that the wavefuctio would evolve ito the ew groud state, ad 9% thought that the system would evolve ito the ew first excited state because the groud state wavefuctio of the old system is similar i form to the first excited state of the ew system for 0 x a. However, because the iitial wavefuctio is zero i the regio a x a, the old groud state is a liear superpositio of the statioary states of the ew system after the well has expaded. The studets did ot realize that if the iitial state is ot a statioary state of the ew system, the time evolutio would ot cause the wavefuctio to evolve ito a statioary state of the ew system. Questio 8 assesses whether studets ca distiguish betwee the statioary states ad the eigestates of other physical observables. The most commo miscoceptio was that a eigestate of a physical observable is a statioary state. I particular, half of the studets icorrectly thought that statemet () i Questio 8, which states that the statioary states refer to the eigestates of ay operator correspodig to ay physical observable, is correct. Aother 0% did ot choose statemet (), but icorrectly claimed that if the particle has a well-defied positio i 0

21 the iitial state, the positio of the particle is well defied for all future times. Q3 Q4 Q6 Q5 Q5 Q8 Q9 (a) % % 3% 6% 7% 3% 7% (b) 8% 3% 4% 6% % 36% 5% (c) 0% 0% 8% 9% % 5% 43% (d) 78% 49% 5% 4% 38% 0% 4% (e) % 35% 58% 6% 5% % 8% Table IV. Distributio of studets resposes for questios related to the statioary states versus o-statioary states. D. The Role of the Hamiltoia The Hamiltoia govers the time evolutio of the system accordig to the time depedet Schrödiger equatio. I Questio 9 studets were asked about the role of the Hamiltoia i a quatum system. The most commo miscoceptio was that the Hamiltoia determies the shape of a positio eigefuctio. 5% of the studets did ot kow that the Hamiltoia govers the time evolutio. Aother 7% did ot relate the Hamiltoia to the shape of the statioary state wavefuctios. Idividual discussios suggest that sometimes this mistake origiates from their misuderstadig of a statioary state as a eigestate of ay operator correspodig to a physical observable. Studets respose to Questio 6 suggests that most kew that the Hamiltoia is the sum of the potetial eergy ad kietic eergy, but their respose to Questio 7 suggests that more tha half of them had difficulty selectig the Hamiltoia operators that have oly a discrete eergy spectrum. Q6 Q7 Q9 (a) 45% 7% 7% (b) % 43% 5% (c) 5% 9% 43% (d) % 0% 4% (e) 3% 6% 8% Table V. Distributio of studets resposes for questios related to the Hamiltoia.

22 E. Time depedece of the wavefuctio The statioary state wavefuctio at time t satisfies both the time idepedet ad time depedet Schrödiger equatios. However, a liear superpositio of the statioary states does ot have a defiite value of eergy eve at t = 0, for example, H ˆ ( ) E E E( ). I Questio 6 about 70% of the studets icorrectly thought that the superpositio state ( x ) (x) is a eergy eigestate which satisfies the time idepedet Schrödiger equatio. Oly 5% A selected the correct aswer that ( x ) A (x) is ot the solutio of the time idepedet Schrödiger equatio, but its time evolutio ( x, t) A (x) exp( ie t / ) satisfies the time depedet Schrödiger equatio. Further iterviews idicate that may udergraduate ad graduate studets hold the miscoceptio that the time idepedet Schrödiger equatio is satisfied for ay possible wavefuctio. Questio 7 tests the uderstadig of the time depedece of a positio eigefuctio. The positio eigefuctio is a delta fuctio, which ca be writte as a liear superpositio of eergy eigefuctios. The positio eigefuctio is ot a statioary state wavefuctio ad chages with time. 44% of the studets selected the correct statemet (3) [i optios (c) ad (e)], but some of them [who chose optio (c)] did ot aswer the questio correctly because they did ot kow that the wavefuctio would become peaked after a positio measuremet. 39% of the studets selectig statemet () held the miscoceptio that a positio eigefuctio would evolve with time after the measuremet, but evetually retur to the state right

23 before the positio measuremet was performed. Q Q3 Q4 Q6 Q5 Q7 Q Q9 (a) 4% % % 3% 6% 5% 5% 7% (b) 8% 8% 3% 4% 6% 7% 3% 5% (c) 5% 0% 0% 8% 9% 5% 70% 43% (d) 4% 78% 49% 5% 4% 3% 4% 4% (e) 69% % 35% 58% 6% 39% 4% 8% Table VI. Distributio of studets resposes for questios related to the time depedece of the wavefuctio. F. Measuremets Whe calculatig the probability of obtaiig a certai value i a measuremet of a physical observable, studets ofte icorrectly thik that the operator correspodig to the observable must be explicitly ivolved i the expressio. For example, i Questio 5, 30% chose the distractor particle i the regio betwee x ad x dx x ) x ( x dx as the probability of fidig the x dx. They did ot realize that (x) dx is the probability desity of fidig the particle betwee x ad x + dx. I Questio, 33% icorrectly thought that a 0 * ( x) Hˆ ( x,0) dx is the probability of measurig the eergy E at time t = 0 istead of the correct expressio a ( x) ( x,0) dx. 0 * Studets ofte did ot realize that the required iformatio about the eergy measuremet is obtaied by projectig the state of the system alog the eergy eigestate (multiplyig the wavefuctio by * ( x ) before itegratig). Further iterviews idicate that studets held a commo miscoceptio that the Hamiltoia actig o a state represets a eergy measuremet. This icorrect otio is a overgeeralizatio of the fact that the system is i a statioary state after the eergy 3

24 measuremet. Questios 7 ad 8 ivestigate studets uderstadig of the eergy measuremet outcomes for the superpositio state / 7 ( x) 5/ 7 ( ). The oly possible x eergies are the groud state eergy E ad the first excited state eergy E. Whe the eergy E is obtaied, the wavefuctio collapses to ( x ). I Questio 7, 3% icorrectly claimed that the wavefuctio would collapse first but evetually retur to the iitial state / 7 ( x) 5/ 7 ( ). Aother 3% did ot ote that the x wavefuctio would collapse ad thought that the system will remai i the iitial state eve after the measuremet. I Questio 8, 0% claimed they could measure ot oly E ad E, but ay possible eergy E ( is a positive iteger), ad 5% claimed that the probabilities for measurig ay eergy E are equal. Questio 3 examies studets uderstadig of cosecutive quatum measuremets, for example, measurig the eergy of a system immediately after a positio measuremet. For a D ifiite square well with the iitial state ( ), a positio measuremet will collapse the wavefuctio to a delta fuctio which is a superpositio of may eergy eigefuctios. So we ca obtai a higher order eergy E ( > ) for the eergy measuremet of the system after the positio measuremet. Oly 3% of the studets correctly aswered Questio 3 ad realized that the state of the system chaged after the positio measuremet. 40% mistakely thought that the result could be oly eergy E or E, which correspods to the iitial state before the positio measuremet. Questio asks studets to predict a ukow quatum state for a simple 4

25 harmoic oscillator i a liear superpositio of the groud ad third excited states by a measuremet. Whe there is a large esemble of particles i the state a 0 b 3 (a ad b are coefficiets whose absolute square is to be determied), we ca measure the eergy of each particle ad cout the umber of particles collapsig to the states 0 ad 3, ad the calculate the proportios of 0 ad 3 to estimate the absolute squares of a ad b. 70% of the studets kew that the measuremet would chage the state of the particle so they had to prepare the particle i the iitial state agai before makig aother measuremet. 7% of the studets mistakely thought that the wavefuctio would automatically retur to the origial state a log time after the measuremet. The other studets who selected statemet () i Questio did ot realize that the wavefuctio chages after the eergy measuremet. Q5 Q7 Q8 Q Q3 Q7 Q0 Q Q8 (a) 44% 45% 3% 33% 40% 5% % 5% 3% (b) % 8% 74% 43% 3% 7% % 3% 36% (c) 30% 3% 5% % 6% 5% 45% 70% 5% (d) 4% 8% 5% 8% % 3% 7% 4% 0% (e) 9% 4% % 3% 9% 39% 3% 4% % Table VII. Distributio of studets resposes for questios related to quatum measuremet. G. Other The positio-mometum ucertaity priciple is a cetral priciple of quatum mechaics. Writte resposes ad idividual discussios suggest that studets are ofte uclear about the differece betwee the quatum ucertaity priciple ad experimetal ucertaity. Studets ofte have the miscoceptio that the ucertaity 5

26 i positio or mometum is about the expectatio value of the positio or mometum of the particle. I Questio, % of the studets who selected statemet () icorrectly claimed that the ucertaity i positio is smaller whe the expectatio value of the mometum is larger. About 3% of the studets who selected optios (d) or (e) claimed that the expectatio value of the positio is larger whe the expectatio value of the mometum is smaller, that is, x p costat. The studets were uclear that the ucertaity of a physical observable depeds o the stadard deviatio, istead of the expectatio value of that observable for a give wavefuctio. Questio 6 is related to the Ehrefest theorem. I the Schrödiger formalism the expectatio values obey the classical laws of motio. To determie the time-depedece, may studets substituted the classical variables by the quatum operators istead of the expectatio value. For example, 50% of the studets who selected statemet () icorrectly claimed that the mometum operator pˆ is equal to dx m ˆ dt ad about 6% also mistakely claimed that dpˆ V( xˆ). It is importat to dt x help studets build a robust kowledge structure so that they do ot icorrectly over-geeralize their experieces from classical physics. Q Q5 Q6 (a) 59% 7% 45% (b) 5% % % (c) 7% % 5% (d) 6% 38% % (e) 7% 5% 3% Table VIII. Distributio of studets resposes for questios related to other cocepts. 6

27 V. SUMMARY Idetificatio of studets difficulties ca help the desig of better istructioal strategies ad learig tools to improve studets uderstadig. We have developed a research-based multiple choice survey to assess studets coceptual uderstadig of quatum mechaics i oe spatial dimesio. The alterative choices for the multiple-choice questios o the survey ofte deal with the commo difficulties foud i these ivestigatios. We foud that the advaced udergraduate ad graduate studets have may commo difficulties ad miscoceptios about various topics. We also ivestigated the extet to which research-based learig tools 7, ca help studets lear these cocepts ad foud that the difficulties were sigificatly reduced whe studets used cocept tests ad Quatum Iteractive Learig Tutorials. Studets who used research-based learig tools i their quatum mechaics courses ot oly performed better o the survey whe it was admiistered at the ed of the same semester i which the relevat cocepts were covered but performed equally well after a etire semester suggestig good retetio of the cocepts. The survey ca be admiistered to studets i upper-level udergraduate courses after istructio. It ca also be used as a prelimiary test for graduate studets to evaluate their backgroud kowledge i quatum mechaics before they take graduate-level quatum mechaics courses. Those developig istructioal strategies to improve studet uderstadig of quatum mechaics ca take ito accout the difficulties that were brought out by the survey. 7

28 ACKNOWLEDGEMENTS We are very grateful to all the faculty who reviewed the various compoets of the survey at several stages ad provided ivaluable feedback. We are also very thakful to all the faculty who admiistered the test. This work is supported by the Natioal Sciece Foudatio. Refereces D. Styer, Commo miscoceptios regardig quatum mechaics, Am. J. Phys. 64, 3-34 (996). I. D. Johsto, K. Crawford, ad P. R. Fletcher, Studet difficulties i learig quatum mechaics, It. J. Sci. Educ. 0, (998). 3 G. Ireso, The quatum uderstadig of pre-uiversity physics studets, Phys. Educ. 35 (), 5- (000). L. Bao ad E. Redish, Uderstadig probabilistic iterpretatios of physical systems: A prerequisite to learig quatum physics, Am. J. Phys., 70(3), 0-7 (00). 4 M. Wittma, R. Steiberg, ad E. Redish, Ivestigatig studet uderstadig of quatum physics: Spotaeous models of coductivity, Am. J. Phys. 70(3), 8-6, (00). 5 D. J. Griffiths, Itroductio to Quatum Mechaics (Pretice Hall, Upper Saddle River, NJ, 995). 6 D. Zollma, S. Rebello, ad K. Hogg, Quatum physics for everyoe: Hads-o activities itegrated with techology, Am. J. Phys. 70(3), 5-59 (00). 7 L. D. Carr ad S. B. McKaga, Graduate quatum mechaics reform, Am. J. Phys., 77(4), 08-39, (009). 8 Research o Teachig ad Learig of Quatum Mechaics, papers preseted at the Natioal Associatio for Research i Sciece Teachig, Bosto, MA, 999, <perg.phys.ksu.edu/papers/arst/>. Also Am. J. Phys. 70(3), (00) published i cojuctio with Gordo Coferece o physics research ad educatio o quatum mechaics. 9 P. Jolly, D. Zollma, S. Rebello, ad A. Dimitrova, Visualizig potetial eergy diagrams, Am. J. Phys. 66(), (998). J. Hiller, I. Johsto, ad D. Styer, Quatum Mechaics Simulatios (Joh Wiley ad Sos, New York, 995). D. Schroeder ad T. Moore, A computer-simulated Ster-Gerlach laboratory, Am. J. Phys. 6 (9), (003). 0 E. Redish ad R. Steiberg, Teachig physics: Figurig out what works, Phys. Today, 5, 4-30, (999). L. McDermott, P. Shaffer, ad the Physics Educatio Group, Uiversity of Washigto, Tutorials i Itroductory Physics (Pretice Hall, Upper Saddle River, NJ, 998). 8

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