Student understanding of quantum mechanics at the beginning of graduate instruction

Student understanding of quantum mechanics at the beginning of graduateinstructionChandralekha Singh Citation: Am. J. Phys. 76, 277 (2008); doi: 10.1119/1.2825387 View online: http://dx.doi.org/10.1119/1.2825387 View Table of Contents: http://ajp.aapt.org/resource/1/AJPIAS/v76/i3 Published by the American Association of Physics Teachers Related ArticlesOntario section (OAPT) newsletter www.oapt.ca/newsletter/ Phys. Teach. 51, 382 (2013) Incorporating Sustainability and 21st-Century Problem Solving into Physics Courses Phys. Teach. 51, 372 (2013) Response times to conceptual questions Am. J. Phys. 81, 703 (2013) Can free-response questions be approximated by multiple-choice equivalents? Am. J. Phys. 81, 624 (2013) Combining two reform curricula: An example from a course with well-prepared students Am. J. Phys. 81, 545 (2013) Additional information on Am. J. Phys.Journal Homepage: http://ajp.aapt.org/ Journal Information: http://ajp.aapt.org/about/about_the_journal Top downloads: http://ajp.aapt.org/most_downloaded Information for Authors: http://ajp.dickinson.edu/Contributors/contGenInfo.html

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PHYSICS EDUCATION RESEARCH SECTION

All submissions to PERS should be sent �preferably electronically� to the Editorial Office of AJP, andthen they will be forwarded to the PERS editor for consideration.

Student understanding of quantum mechanics at the beginning of graduateinstruction

Chandralekha SinghDepartment of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260

�Received 2 March 2005; accepted 22 November 2007�

A survey was developed to probe student understanding of quantum mechanics at the beginning ofgraduate instruction. The survey was administered to 202 physics graduate students enrolled infirst-year quantum mechanics courses from seven universities at the beginning of the first semester.We also conducted one-on-one interviews with fifteen graduate or advanced undergraduate studentswho had just completed a course in which all the content on the survey was covered. Althoughstudents from some universities performed better on average than others, we found that studentsshare universal difficulties understanding the concepts of quantum mechanics. The difficulties wereoften due to overgeneralizations of concepts learned in one context to other contexts where they arenot directly applicable. Difficulties in distinguishing between closely related concepts and makingsense of the formalism of quantum mechanics were common. The results of this study can sensitizeinstructors of first-year graduate quantum physics to some of the difficulties students are likely toface. © 2008 American Association of Physics Teachers.

�DOI: 10.1119/1.2825387�

I. INTRODUCTION

A solid understanding of quantum mechanics is essentialfor most scientists and engineers. Single-electron transistors,superconducting quantum interference devices, quantumwell lasers, and other devices are made possible by the un-derlying quantum processes.1 However, quantum physics is adifficult and abstract subject.2 Unlike classical physics,where position and momentum are deterministic variables, inquantum mechanics they are operators that act on a wave-function which lies in an abstract Hilbert space. In addition,according to the Copenhagen interpretation, which is usuallytaught in quantum courses, an electron in a hydrogen atomdoes not have a definite distance from the nucleus; it is theact of measurement that collapses the wavefunction andmakes it localized at a certain distance. If the wavefunctionis known right before the measurement, quantum theory onlyprovides the probability of measuring the distance in a nar-row range.

Students taking quantum mechanics often develop sur-vival strategies for performing reasonably well in theircourse work. For example, they become proficient at solvingthe time-independent Schroedinger equation with a compli-cated potential energy and boundary conditions. However,students often struggle to make sense of the material andbuild a robust knowledge structure. They have difficultymastering concepts and applying the formalism to answerqualitative questions related to the general formalism, mea-surement of physical observables, time development of thewavefunction, the meaning of expectation values, stationarystates, and properties of wavefunctions, for example.

Several visualization tools have been developed to help
students gain intuition about quantum mechanics
277 Am. J. Phys. 76 �3�, March 2008 http://aapt.org/ajp

Downloaded 03 Sep 2013 to 150.108.161.71. Redistribution subject to AAPT l

concepts.3–8 Recently, low-cost laboratory experiments havebeen developed to introduce students to more contemporaryquantum ideas.9,10 Relating these activities to research onstudent difficulties can lead to the development of research-based tools that can greatly enhance their effectiveness.

Student difficulties in learning physics concepts can bebroadly classified in two categories: gaps in students’ knowl-edge and misconceptions. Knowledge gaps can be due to amismatch between the level at which the material is pre-sented and student’s prior knowledge.11 Misconceptions canalso impede the learning process at all levels ofinstruction.12,13 Without curricula and pedagogies that appro-priately account for common difficulties, instruction is likelyto be ineffective.

Several investigations have striven to improve the teach-ing and learning of quantum mechanics.14–27 Styer14 hasdocumented several common misconceptions in quantummechanics. Zollman and co-workers15,18,28 have proposedthat quantum concepts be introduced much earlier in physicscourse sequences and have designed tutorials and visualiza-tion tools which illustrate concepts that can be used at avariety of levels. Bao and co-workers et al.16,18,28 have con-ducted investigations of student difficulties and developedresearch-based material to teach quantum physics conceptsto a wide range of science and engineering students. Catalo-glu and Robinett17 designed a test related to quantum physicsconcepts that can be administered to students in coursesranging from introductory quantum physics to graduatequantum mechanics. Several other investigations have fo-cused on students’ conceptions about modern physics earlyin college or at the pre-college level.18–23

One of our earlier investigations explored student under-
standing of quantum measurement and time dependence of
277© 2008 American Association of Physics Teachers

icense or copyright; see http://ajp.aapt.org/authors/copyright_permission

the expectation value.24–27 Our analysis of the data obtainedfrom the written test and interviews showed that advancedstudents have common difficulties and misconceptions inde-pendent of their background, teaching style, textbook, andinstitution, analogous to the patterns of misconceptions ob-served in introductory physics courses.

II. METHODOLOGY

Here we describe the findings from a graduate quantummechanics survey which was developed and administered to202 graduate students from seven universities in the UnitedStates. The 50 minute written survey administered at the be-ginning of a first-year, first-semester/quarter graduate quan-tum mechanics course covers a range of concepts. To under-stand the reasoning difficulties in depth, we also interviewedfifteen graduate or advanced undergraduate students enrolledin a quantum mechanics course at the university of Pitts-burgh in which all of the concepts on the survey were cov-ered. Although students from some universities performedbetter on average than others, we find that students havecommon conceptual difficulties regardless of where they areenrolled. By conceptual difficulties, we refer to difficulties inusing one’s knowledge to interpret, explain, and draw infer-ences while answering qualitative questions in different con-texts.

During the design of the survey we consulted three Pitts-burgh faculty members who had taught quantum mechanics.Previously, we discussed with them the concepts that theyexpected the students in a first-semester graduate quantummechanics to know. Because the first year graduate studentsat Pittsburgh are given a placement test in each of the corecourses to determine whether they are better suited for thegraduate or the corresponding undergraduate course, we dis-cussed the kinds of questions instructors would put on theplacement test. The initial longer version of the survey wasdesigned based on the concepts the instructors consideredimportant prior knowledge for the graduate course. In addi-tion to commenting on the wording of the questions to elimi-nate ambiguity, we asked the faculty members to rate thequestions and comment on what should be included in thesurvey. Based upon their feedback, we iterated the surveyquestions several times before using a version that was indi-vidually administered to a few graduate students. We dis-cussed the survey with the students after they had taken itand fine-tuned it based on their responses. As shown in theAppendix, the graduate survey covers topics related to thetime-dependent Schroedinger equation and time-independentSchroedinger equation, the time dependence of the wave-function, the probability of measuring energy and position,the expectation values of the energy, the identification ofallowed wavefunctions for an infinite square well �not thestationary states�, graphical representations of the bound andscattering states for a finite square well, and the formalismassociated with the Stern-Gerlach experiment.

The universities that participated in this study and thenumber of students from each university are given in Table I.For four of the seven universities, the survey was adminis-tered in the first class period in the first week, while in theother three universities, it was administered in the second orthird week. At some universities, students were given thesurvey as a placement test, given a small number of bonuspoints for taking the survey regardless of their actual perfor-
mance on the survey, or were asked to take the survey seri-
278 Am. J. Phys., Vol. 76, No. 3, March 2008


ously because the results could help tailor the graduate in-struction and help in making the undergraduate quantummechanics courses more effective. At universities where thetest was not given as a placement test, the students were toldahead of time that they would be taking a survey on thematerial covered in the undergraduate quantum mechanics incase they wanted to review the material.

To investigate the difficulties with these concepts in moredepth, fifteen students �graduate students or physics seniorsfrom the University of Pittsburgh in an undergraduate quan-tum mechanics course in which all of the survey content wascovered� were interviewed using a think-aloud protocol.29

These interviews were semi-structured in the sense that wehad a list of issues related to each question that we definitelywanted to probe. These issues were not brought up initiallybecause we wanted to give students an opportunity to formu-late their own responses and reasoning. The students wereasked to verbalize their thoughts while they were working onthe survey questions. They were not interrupted unless theyremained quiet for a while, in which case they were re-minded to “keep talking.” After students had finished articu-lating their ideas, they were asked further questions to clarifyissues. Some of this later probing was from the list of issuesthat we had planned to probe initially �and asked students atthe end if they did not bring it up themselves� and otherswere questions designed on-the-spot to get a better under-standing of a particular student’s reasoning.

III. DISCUSSION

The Appendix shows the final version of the graduate sur-vey. Table II shows the responses in percentage of studentsfor each question. These difficulties are categorized and dis-cussed in detail in the following.

A. Time-independent Schroedinger equation is mostfundamental

One difficulty that was pervasive across several questionswas the overemphasis on the time-independent Schroedingerequation. For example, in Question 1, students were asked towrite down the most fundamental equation of quantum me-chanics. We were expecting that students would write downsome form of the time-dependent Schroedinger equation:

i��t��

�t= H��t�� , �1a�

Table I. The number of graduate students from each university that partici-pated in this study.

University Number of students

The Ohio State University 49State University of New York �SUNY�, Buffalo 32University of California, Davis 39University of Iowa 6University of California, Irvine 29University of Pittsburgh 21University of California, Santa Barbara 26

or

278Chandralekha Singh


i��x,t�

�t= H��x,t� . �1b�

Equation �1b� is the time-dependent Schroedinger equation

in one spatial dimension for which the Hamiltonian H= p2 / �2m�+V�x�. Responses that cited the position-momentum uncertainty principle �3 students� or the commu-tation relation between position and momentum �1 student�as the most fundamental equation of quantum mechanicswere also considered correct. Even if students did not explic-itly write down the Hamiltonian in terms of the potential andkinetic energy operators, their responses were consideredcorrect. We also considered the response correct if the stu-dents made mistakes such as forgetting �, the relative signsof various terms in the equation, and the mass of the particlem in the Hamiltonian. Only 32% of the students provided acorrect response with this scoring criterion.

Table II shows that 48% of the students believed that thetime-independent Schroedinger equation H�n=En�n is themost fundamental equation of quantum mechanics �not all ofthe equations given by students had the Hamiltonian writtencorrectly�. It is correct that if the potential energy is timeindependent, we can use separation of variables to obtain thetime-independent Schroedinger equation, which is an eigen-

value equation for the Hamiltonian. The eigenstates of Hobtained by solving the time-independent Schroedingerequation are the stationary states which form a complete setof states. Most advanced undergraduate quantum mechanics

Table II. Percentage of correct responses on each queon each question. In the last column, �i�–�iv� catalog

Quest# % Correct Perce

1 32 �i� 482�a� 43 �i� 31

�ii� 92�b� 67 �i� 7%

value2�c� 39 �i� 172�d� 62+17=79 �i� 172�e� 56 �i� 14

�ii� 82�f� 38 �i� 11

�x��,�ii� 7

3 34 �i� 45allow�ii� 5

4 29 �i� 39�ii� 1of en�iii� 1

5�a� 41 �i� 13direc

5�b� 23 Ques6�a� 57 �i� 20

�ii� 8�iii� 8�iv� 8

6�b� 17 �i� 8%well�

courses de-emphasize the time-dependent Schroedinger



equation, and students recall the quantum mechanics courseas an exercise in solving time-independent Schroedingerequation. As we will discuss, an overemphasis on time-independent Schroedinger equation also leads to the maindifficulty with quantum dynamics.

In Question 4, students were asked to explain why theyagree or disagree with the following statement: “By defini-tion, the Hamiltonian acting on any state of the system � will

give the same state back, that is, H�=E�.” We wanted stu-dents to disagree with the statement and note that it is onlytrue if � is a stationary state. In general, �=�n=1

� Cn�n, where

�n are the stationary states and Cn= ��n ��. Then, H�

=�n=1� CnEn�n�E�. Just writing down “disagree” was not

enough for the response to be counted correct. Students hadto provide the correct reasoning. Only 29% of the studentsprovided the correct response. Thirty-nine percent of stu-dents wrote incorrectly that the statement is unconditionallycorrect. This percentage is slightly lower than the percentage

of students who claimed in Question 1 that H�=E� is themost fundamental equation of quantum mechanics. Typi-cally, these students were confident of their responses, as canbe seen from these examples: �a� “Agree. This is what80 years of experiment has proven. If future experimentsprove this statement wrong, then I’ll update my opinion onthis subject.” �b� “Agree, this is a fundamental postulate ofquantum mechanics which is proved to be highly exact untilpresent.” �c� “Agree. This is what Schroedinger equation im-

and percentage of students with common difficultiesmon difficulties for a given question.

of students with common difficulties

�=E��ommon phase factor�,

time-dependence�ividual measurement versus expectation

rote �� E �� or �� H �� but nothing else�nnormalized wavefunction ��x�=5 /7�2�x��o many wiggles in wavefunction�,correct boundary condition�robability versus expectation value of position

correct application of uncertainty principle�rst wavefunction not allowed, second

ird wavefunction allowed�gree with statement unconditionally�,

amiltonian acting on a state is measurement,agree with statement if energy is conserved�plitting into two spots along wrongncluding5�a� and �b��rst excited or higher excited bound states�,ound state of infinite square well�,correct boundary condition�,

scillatory wavefunction in all three regions�vefunction with exponential decay inside the

stioncom

ntage

% �H% �c% �no

�ind�% �w% �u% �to% �in% �p

% �in% �fied�,% �th% �a

1% �Hergy�0% �% �s

tion itions% �fi% �gr% �in% �o

�wa

plies and it is what quantum mechanics is founded on.” �d�



“Agree. H commutes with all operators which measure ob-servable quantities. Hence, any state � is an eigenstate of thesystem.”

During the interview, one student said, “It seems like youare asking this question because you want me to disagreewith this statement. Unfortunately, it seems correct to me.”After reading Question 4, another interviewed student whohad earlier said that the Schroedinger equation is the mostfundamental equation of quantum mechanics, but could notremember the equation said, “Oh, here is the answer to Ques-tion 1.” Our earlier investigation24–27 found that the difficul-ties related to the time dependence of expectation values arealso often related to applying the properties of stationarystates to non-stationary states �such as the eigenstates of po-sition or momentum operators�.

B. Hamiltonian acting on a state represents energymeasurement

Eleven percent of the students answering Question 4 be-lieved incorrectly that any statement involving a Hamiltonianoperator acting on a state is a statement about the measure-ment of energy. Some of these students who incorrectly

claimed that H�=E� is a statement about energy measure-ment agreed with the statement, while others disagreed.

Those who disagreed often claimed that H�=En�n, because

as soon as H acts on �, the wavefunction will collapse intoone of the stationary states �n and the corresponding energyEn will be obtained. The following examples are typical of

students with this misconception: �a� “Agree. H is the opera-tor for an energy measurement. Once this measurement takesplace, the specific value E of the energy will be known.” �b�“Agree. If you make a measurement of energy by applying Hto a state of an electron in hydrogen atom you will get theenergy.” �c� “Agree except when the system is in a linearsuperposition. In that case, Hamiltonian acting on it willmake it settle into only one of the term corresponding to themeasured energy.” �d� “Hamiltonian acting on a system willcollapse the system into one of the possible energy states.This does not give the same original state unless the previousstate was same as resulting state.” �e� “Disagree. The Hamil-tonian acting on a mixed state will single out one component.The wavefunction will collapse to a different state once theenergy has been determined to be that of one component.”

The interviews and written answers suggest that these stu-dents believed that the measurement of a physical observablein a particular state is achieved by acting with the corre-sponding operator on the state. The incorrect notions areovergeneralizations of the fact that after the measurement of

energy, the system is in a stationary state so H�n=En�n. Thisexample illustrates the difficulty students have in relating theformalism of quantum mechanics to the measurement of aphysical observable.

C. Q�=�� for any physical observable Q

Individual interviews related to Question 4 suggest that

some students believed that if an operator Q correspondingto a physical observable Q acts on any state �, it will yieldthe corresponding eigenvalue � and the same state back, that

ˆ
is, Q�=��. Some of these students were overgeneralizing


their “H�=E�” reasoning and attributing Q�=�� to themeasurement of an observable Q. Before overgeneralizing toany physical observables, these students often agreed with

the H�=E� statement with arguments such as “the Hamil-tonian is the quantum mechanical operator which corre-sponds to the physical observable energy” or “if H did notgive back the same state it would not be a Hermitian operatorand therefore would not correspond to an observable.” Of

course, Q�� unless � is an eigenstate of Q and in gen-

eral �=�n=1� Dn�n, where �n are the eigenstates of Q and

Dn= ��n ��. Then, Q�=�n=1� Dn�n�n �for an observable with

a discrete eigenvalue spectrum�.

D. H�=E� if H does not depend on time

In response to Question 4, 10% of the students agreed withthe statement as long as the Hamiltonian is not time depen-

dent. They often claimed incorrectly that if H is not time

dependent, the energy for the system is conserved, so H�=E� must be correct. The following are typical examples:�a� “Agree, if the potential energy does not depend on time.”�b� “Agree but only if the energy is conserved for this sys-tem.” �c� “Agree because energy is a constant of motion.” �d�“Agree if it is a closed system because H is a linear operatorand gives the same state back multiplied by the energy.”

Although the energy is conserved if the Hamiltonian is

time independent, H�=E� need not be true. For example, ifthe system is in a linear superposition of stationary states,

H��E�, although the energy is conserved.

E. Difficulties related to the time developmentof wavefunction

The most common difficulties with quantum dynamics arecoupled with an overemphasis on the time-independentSchroedinger equation. Equation �1a� and �1b� shows that theevolution of the wavefunction ��x , t� is governed by the

Hamiltonian H of the system via the time-dependent Schro-edinger equation and there is no dynamics in the time-independent Schroedinger equation. Question 2 concerned anelectron in a one-dimensional infinite square well, initially�t=0� in a linear superposition of the ground state �1�x� andthe first excited state �2�x�. In Question 2�a�, students wereasked to write down the wavefunction ��x , t� at time t. Wewere expecting the following response: ��x , t�=2 /7�1�x�e−iE1t/�+5 /7�2�x�e−iE2t/�. Responses were con-sidered correct if students wrote the phase factor for the firstterm as e−iAE1t, where A is any real constant �for example, �in the numerator, incorrect sign, or some other constant, forexample, mass m in the phase were considered minor prob-lems and ignored even though they can make the phase aquantity with dimension�. Some students wrote incorrect in-termediate steps; for example, ��x , t�=��x ,0�e−iEt/�

=2 /7�1�x�e−iE1t/�+5 /7�2�x�e−iE2t/�. Such responses wereconsidered correct. During the individual interviews, a stu-dent proceeded from an intermediate incorrect step to thecorrect time dependence in the second step similar to theabove expression. Further probing showed that the studentwas having difficulty distinguishing between the
Hamiltonian operator and its eigenvalue and was probably


thinking of ��x , t�=e−iHt/��x ,0�=2 /7�1�x�e−iE1t/�

+5 /7�2�x�e−iE2t/�, where the Hamiltonian H acting on thestationary states gives the corresponding energies.

As shown in Table I, 31% of students wrote commonphase factors for both terms; for example, ��x , t�=��x ,0�e−iEt/�. Interviews suggest that these students werehaving difficulty distinguishing between the time depen-dence of stationary and non-stationary states. Because theHamiltonian operator governs the time development of thesystem, the time dependence of a stationary state is via asimple phase factor. In general non-stationary states have anon-trivial time dependence because each term in a linearsuperposition of stationary states evolves via a differentphase factor. Apart from using e−iEt/� as the common phasefactor, other common choices include e−i�t, e−i�t, e−it, e−ixt,and e−ikt.

Interestingly, 9% of the students believed that ��x , t�should not have any time dependence; during the interviewssome students justified their claim by pointing to the time-independent Schroedinger equation and adding that theHamiltonian is not time dependent. Several students thoughtthat the time dependence was a decaying exponential; forexample, of the type ��x ,0�e−xt, ��x ,0�e−Et, ��x ,0�e−ct, or��x ,0�e−t. During the interviews some of these students ex-plained their choice by insisting that the wavefunction mustdecay with time because “this is what happens for all physi-cal systems.” Other incorrect responses were due to the par-tial retrieval of related facts from memory, such as �a�2 /7�1�x+�t�+5 /7�2�x+�t�, �b� 2 /7�1�x�e−i�1t

+5 /7�2�x�e−i�2t, �c� 2 /7�1�x�e−ixt+5 /7�2�x�e−i2xt, and�d� 2 /7�1�x�sin�t�+5 /7�2�x�cos�2t�. The interviewssuggest that these students often correctly remembered thatthe time dependence of non-stationary states cannot be rep-resented by a common time-dependent phase factor, but didnot know how to correctly evaluate ��x , t�.

F. Difficulties with measurement and expectation value

1. Difficulty interpreting the meaning of expectationvalue

Although Question 2�b� was the easiest on the survey with67% correct responses, a comparison with the response forQuestion 2�c�, for which 39% provided the correct response,is revealing. It shows that many students who can calculateprobabilities for the possible outcomes of energy measure-ment were unable to use that information to determine theexpectation value of the energy. In Question 2�b�, studentswere asked about the possible values of the energy of theelectron and the probability of measuring each in an initialstate ��x ,0�. We expected students to note that the only pos-sible values of the energy in state ��x ,0� are E1 and E2 andtheir respective probabilities are 2 /7 and 5 /7. In Question2�c�, students had to calculate the expectation value of theenergy in the state ��x , t�. The expectation value of the en-ergy is time independent because the Hamiltonian does notdepend on time. If ��x , t�=C1�t��1�x�+C2�t��2�x�, then theexpectation value of the energy in this state is �E�= P1E1

+ P2E2= �C1�t��2E1+ �C2�t��2E2= �2 /7�E1+ �5 /7�E2, where Pi

= �Ci�t��2 is the probability of measuring the energy Ei at
time t.


Many students who answered Question 2�b� correctly �in-cluding those who also answered Question �c� correctly� cal-

culated �E� by brute-force: first writing �E�=−�+��*H�dx,

expressing ��x , t� in terms of the linear superposition of two

energy eigenstates, then acting H on the eigenstates, and fi-nally using orthogonality to obtain the answer. Some got lostearly in this process, and others did not remember someother mechanical step, for example, taking the complex con-jugate of the wavefunction, using the orthogonality of sta-tionary states, or not realizing the proper limits of the inte-gral.

The interviews reveal that many did not know or recall theinterpretation of expectation value as an ensemble averageand did not realize that expectation values could be calcu-lated more simply in this case by taking advantage of theiranswer to Question 2�b�. Some believed that the expectationvalue of the energy should depend on time �even those whocorrectly evaluated ��x , t� in Question 2�a��.

In the interview, one student who answered Question 2�b�correctly did not know how to apply it to Question 2�c�. Hewrote an explicit expression involving the wavefunction forthe ground and first excited states, but thought that H�n=En with no �n on the right-hand side of this equation.Therefore, he found a final expression for �E� that involvedwavefunctions. When he was told explicitly by the inter-viewer that the final answer should not be in terms of �1 and�2 and he should try to find his mistake, the student couldnot find his mistake. The interviewer then explicitly pointedto the particular step in which he had made the mistake andasked him to find it. The student still had difficulty becausehe believed H�n=En was correct. Finally, the interviewertold the student that H�n=En�n. At this point, the studentwas able to use orthonormality correctly to obtain the correctresult �E�= �2 /7�E1+ �5 /7�E2. Then, the interviewer askedhim to think about whether it is possible to calculate �E�based on his response to Question 2�b�. The student’s eyesbrightened and he responded, “Oh yes … I never thought ofit this way … I can just multiply the probability of measuringa particular energy with that energy and add them up to getthe expectation value because expectation value is the aver-age value.” Then, pointing to his detailed work for Question2�c� he added, “You can see that the time dependence cancelsout ….” Seventeen percent of the students simply wrote��E�� or ��H�� and did not know how to proceed. Afew students wrote the expectation value of energy as �E1

+E2� /2 or ��2 /7�E1+ �5 /7�E2� /2.

2. Difficulty interpreting the measurement postulate

According to the Copenhagen interpretation, the measure-ment of a physical observable instantaneously collapses thestate to an eigenstate of the corresponding operator. In Ques-tion 2�d�, students were asked to write the wavefunctionright after the measurement of energy if the measurementyields 42�2 / �2ma2�. Because the measurement yields theenergy of the first excited state, the measurement collapsedthe system to the first excited state �2�x�. Students did wellon this question with 79% providing the correct response.However, the 79% includes those students �17%� who wrotethe unnormalized wavefunction 5 /7�2�x�. Interviews sug-gest that some students who did not normalize the wavefunc-
tion strongly believed that the coefficient 5 /7 must be in-


cluded explicitly to represent the wavefunction after themeasurement, not realizing that state �2�x� is already nor-malized.

In response to Question 2�d�, some students thought thatthe system should remain in the original state, which is alinear superposition of the ground and first excited states.One of the interviewed students, said, “Well, the answer tothis question depends upon how much time you wait afterthe measurement. If you are talking about what happens atthe instant you measure the energy, the wavefunction will be�2, but if you wait long enough it will go back to the statebefore the measurement.” The notion that the system must goback to the original state before the measurement was deep-rooted in the student’s mind and could not be dislodged evenafter the interviewer asked several further questions about it.When the interviewer said that it was not clear why thatwould be the case, the student said, “The collapse of thewavefunction is temporary … Something has to happen tothe wavefunction for you to be able to measure energy orposition, but after the measurement the wavefunction mustgo back to what it actually �student’s emphasis� is supposedto be.” When probed further, the student continued, “I re-member that if you measure position you will get a deltafunction, but it will stay that way only if you do repeatedmeasurement … if you let it evolve it will go back to theprevious state �before the measurement�.”

Some students confused the measurement of energy withthe measurement of position and drew a delta function inQuestion 2�e�. They claimed that the wavefunction will be-come very peaked about a given position after the energymeasurement. An interviewed student drew a wavefunctionwhich was a delta function in position. He claimed incor-rectly that because the energy will have a definite value afterthe measurement of energy, the wavefunction should be lo-calized in position. Further probing showed that he was con-fused about the vertical axis of his plot and said that he maybe plotting energy along that axis. When asked explicitlyabout what it means for the energy to be localized at a fixedposition, he said he may be doing something wrong but hewas not sure what else to do.

3. Confusion between the probability of measuringposition and the expectation value of position

Born’s probabilistic interpretation of the wavefunction canalso be confusing for students. In Question 2�f�, studentswere told that immediately after the measurement of energy,a measurement of the electron position is performed. Theywere asked to describe qualitatively the possible values ofthe position they could measure and the probability of mea-suring them. We hoped that students would note that it ispossible to measure position values between x=0 and x=a�except at x=0, a /2, and a where the wavefunction is zero�,and according to Born’s interpretation, ��2�x��2dx gives theprobability of finding the particle between x and x+dx. Only38% of the students provided the correct response. Partialresponses were considered correct for tallying purposes ifstudents wrote anything that was correctly related to theabove wavefunction; for example, “The probability of find-ing the electron is highest at a /4 and 3a /4,” or “The prob-ability of finding the electron is non-zero only in the well.”

Eleven percent of the students tried to find the expectationvalue of position �x� instead of the probability of finding the
electron at a given position. They wrote the expectation


value of position in terms of an integral involving the wave-function. Many of them explicitly wrote that Probability= �2 /a�0

ax sin2�2x /a�dx and believed that instead of �x�they were calculating the probability of measuring the posi-tion of electron. During the interview, one student said �andwrote� that the probability is x��2dx. When the interviewerasked why ��2 should be multiplied by x and if there is anysignificance of ��2dx alone, the student said, “��2 gives theprobability of the wavefunction being at a given position andif you multiply it by x you get the probability of measuring�student’s emphasis� the position x.” When the student wasasked questions about the meaning of the “wavefunction be-ing at a given position,” and the purpose of the integral andits limits, the student was unsure. He said that the reason hewrote the integral is because x��2dx without an integrallooked strange to him.

4. Other difficulties with measurement and expectationvalue

Other difficulties with measurements were observed aswell. For example, in response to Question 2�f�, 7% tried touse the �generalized� uncertainty principle between energyand position or between position and momentum, but mostof their arguments led to incorrect inferences. For example,several students noted that because the energy is well definedimmediately after the measurement of energy, the uncertaintyin position must be infinite according to the uncertainty prin-ciple. Some students even went on to argue that the probabil-ity of measuring the particle’s position is the same every-where. Others restricted themselves only to the inside of thewell and noted that the uncertainty principle says that theprobability of finding the particle is the same everywhereinside the well and for each value of position inside the wellthis constant probability is 1 /a. For example, one studentsaid, “Must be between x=0 and x=a … but by knowing theexact energy, we can know nothing about position so prob-able position is spread evenly across in 0xa region.”Some students thought that the most probable values of po-sition were the only possible values of the position that canbe measured. For example, one student said, “According tothe graph above �in Question 2�e��, we can get positions a /4and 3a /4 each with individual probability 1 /2.” The follow-ing statement was made by a student who believed that itmay not be possible to measure the position after measuringthe energy: “Can you even do that? Doesn’t making a mea-surement change the system in a manner that makes anothermeasurement invalid?” The fact that the student believed thatmaking a measurement of one observable can make the im-mediate measurement of another observable invalid, shedslight on student’s epistemology about quantum theory.

Seven percent of students answering Question 2�b� be-came confused between individual measurements of the en-ergy and its expectation value, and almost none of thesestudents calculated the correct expectation value of the en-ergy. Another common mistake was assuming that all al-lowed energies for the infinite square well were possible andthe ground state is the most probable because it is the lowestenergy state. Some students thought that the probabilities formeasuring E1 and E2 are 4 / �7a� and 10 / �7a�, respectively,because they included the normalization factor for the sta-
tionary state wavefunctions 2 /a while squaring the coeffi-


cients. Some thought that the probability amplitudes were theprobabilities of measuring energy and did not square the co-efficients 2 /7 and 5 /7.

G. Difficulty in determining possible wavefunction

Any smooth function that satisfies the boundary conditionfor a system is a possible wavefunction. We note that Ques-tion �3� asks if the three wavefunctions given are allowed foran electron in a one-dimensional infinite square well, inter-views suggest that students correctly interpreted it to meanwhether they were possible wavefunctions. In this question,we hoped that students would note that the wavefunctionAe−��x − a / 2� / a�2

is not possible because it does not satisfy theboundary conditions �does not go to zero at x=0 and x=a�.The first two wavefunctions A sin3�x /a�,A�2 /5 sin�x /a�+3 /5 sin�2x /a�� with suitable normal-ization constants are both smooth functions that satisfy theboundary condition �each of them goes to zero at x=0 andx=a� so each can be written as a linear superposition of thetwo stationary states. Seventy-nine percent of the studentscould identify that the second wavefunction is a possiblewavefunction because it is explicitly written in the form of alinear superposition of stationary states. Only 34% gave thecorrect answer for all three wavefunctions. Within this sub-set, a majority correctly explained their reasoning based onwhether the boundary conditions are satisfied by these wave-functions. For tallying purposes, responses were consideredcorrect even if the reasoning was not completely correct. Forexample, one student wrote incorrectly: “The first two wave-functions are allowed because they satisfy the equation

H�=E� and the boundary condition works.” The first partof the reasoning provided by this student is incorrect whilethe second part that relates to the boundary condition is cor-rect.

Forty-five percent believed that A sin3�x /a� is not pos-sible but that A�2 /5 sin�x /a�+3 /5 sin�2x /a�� is pos-sible. The interviews suggest that a majority of students didnot know that any smooth single-valued wavefunction thatsatisfies the boundary conditions can be written as a linearsuperposition of stationary states. Interviews and written ex-planations suggest that many students incorrectly believedthat the following two constraints must be independently sat-isfied for a wavefunction to be a possible wavefunction: itmust be a smooth single-valued function that satisfies theboundary conditions and it must either be possible to write itas a linear superposition of stationary states, or it must sat-isfy the time-independent Schroedinger equation.

As in the following example, some who correctly realizedthat A sin3�x /a� satisfies the boundary condition, incor-rectly claimed that it is still not a possible wavefunction:“A sin3�x /a� satisfies b.c. but does not satisfy Schroedingerequation �that is, it cannot represent a particle wave�. Thesecond one is a solution to S.E. �it is a particle wave�. Thethird does not satisfy b.c.”

Many claimed that only pure sinusoidal wavefunctions arepossible, and sin2 or sin3 are not possible. The interviews andwritten explanations suggest that students believed thatA sin3�x /a� cannot be written as a linear superposition ofstationary states and hence it is not a possible wavefunction.The following are examples: �a� “A sin3�x /a� is not al-
lowed because it is not an eigenfunction nor a linear combi-


nation.” �b� “A sin3�x /a� is not allowed because it is not alinear function but Schroedinger equation is linear.” �c�“A sin3�x /a� is not allowed. Only simple sines or cosinesare allowed.” �d� “A sin3�x /a� works for 3 electrons but notone.”

The most common incorrect response claimed incorrectlythat A sin3�x /a� is not a possible wavefunction because it

does not satisfy H�=E�. Students asserted thatA sin3�x /a� does not satisfy the time-independent Schro-edinger equation �which they believed was the equation thatall possible wavefunctions should satisfy� butA�2 /5 sin�x /a�+3 /5 sin�2x /a�� does. Many explicitlywrote the Hamiltonian as �−�2 /2m� / ��2 /�x2� and showedthat the second derivative of A sin3�x /a� will not yield thesame wavefunction back multiplied by a constant. Inciden-tally, the same students did not attempt to take the secondderivative of A�2 /5 sin�x /a�+3 /5 sin�2x /a��; other-wise, they would have realized that even this wavefunctiondoes not give back the same wavefunction multiplied by aconstant. For this latter wavefunction, a majority claimedthat it is possible because it is a linear superposition ofsin�nx /a�. Incidentally, A sin3�x /a� can also be written asa linear superposition of only two stationary states. Thus,students used different reasoning to test the validity of thefirst two wavefunctions as in the following example:“�−�2 /2m� / ��2 /�x2�A sin3�x /a� cannot be equal toEA sin3�x /a� so it isn’t acceptable. Second is acceptablebecause it is linear combination of sine.”

Some students incorrectly noted that A�2 /5 sin�x /a�+3 /5 sin�2x /a�� is possible inside the well and

Ae−��x − a / 2� / a�2is possible outside the well. Others incorrectly

claimed that A sin3�x /a� does not satisfy the boundary con-

dition for the system but A�2 /5 sin�x /a�+3 /5 sin�2x /a�� does. Some dismissed A sin3�x /a�claiming it is an odd function that cannot be a possible wave-function for an infinite square well which is an even poten-tial. In the interview, a student who thought that onlyA�2 /5 sin�x /a�+3 /5 sin�2x /a�� is possible said,“these other two are not linear superpositions.” When theinterviewer asked explicitly how one could tell that the othertwo wavefunctions cannot be written as a linear superposi-tion, the student said, “A sin3�x /a� is clearly multiplicativenot additive … you cannot make a cubic function out oflinear superposition … this exponential cannot be a linearsuperposition either.”

Five percent of students claimed that Ae−��x − a / 2� / a�2is a

possible wavefunction for an infinite square well. These stu-dents did not examine the boundary condition. They some-times claimed that an exponential can be represented bysines and cosines, and hence it is possible or focused only onthe normalization of the wavefunction. Not all students who

correctly wrote that Ae−��x − a / 2� / a�2is not a possible wave-

function provided the correct reasoning. Many studentsclaimed that the possible wavefunctions for an infinite squarewell can only be of the form A sin�nx /a� or that

Ae−��x − a / 2� / a�2is possible only for a simple harmonic oscilla-

tor or a free particle.



H. Difficulty distinguishing between three-dimensionalspace and Hilbert space

In quantum theory, it is necessary to interpret the outcomeof real experiments performed in real space by making con-nection with an abstract Hilbert space �state space� in whichthe wavefunction lies. The physical observables that aremeasured in the laboratory correspond to Hermitian opera-tors in the Hilbert space whose eigenstates span the space.Knowing the initial wavefunction and the Hamiltonian of thesystem allows one to determine the time-evolution of thewavefunction unambiguously and the measurement postulatecan be used to determine the possible outcomes of individualmeasurements and ensemble averages �expectation values�.

It is difficult for students to distinguish between vectors inreal space and Hilbert space. For example, Sx, Sy, and Szdenote the orthogonal components of the spin angular mo-mentum vector of an electron in three dimensions, each ofwhich is a physical observable that can be measured in thelaboratory. However, the Hilbert space corresponding to thespin degree of freedom for a spin-1 /2 particle is two-

dimensional �2D�. In this Hilbert space, Sx, Sy, and Sz areoperators whose eigenstates span 2D space. The eigenstates

of Sx are vectors which span the 2D space and are orthogonal

to each other �but not orthogonal to the eigenstate of Sy or

Sz�. In addition, Sx, Sy, and Sz are operators and not orthogo-nal components of a vector in 2D space. If the electron is ina magnetic field with the gradient in the z direction in thelaboratory �real space� as in a Stern-Gerlach experiment, themagnetic field is a vector in three-dimensional �3D� spacebut not in 2D space. It does not make sense to comparevectors in 3D space with vectors in the 2D space as in state-ments such as “the magnetic field gradient is perpendicular

to the eigenstates of Sx.” These distinctions are difficult forstudents to make and such difficulties are common, as dis-cussed in the following.

Question 5 has two parts, both of which are related to theStern-Gerlach experiment. The notation �↑z� and �↓z� repre-

sent the orthonormal eigenstates of Sz �the z component ofthe spin angular momentum� of a spin-1 /2 particle. In oneversion of this question, a beam of neutral silver atoms withspin-1 /2 was sent through the Stern-Gerlach apparatus. Stu-dents had similar difficulties with both versions. In Question5�a�, a beam of electrons propagating along the y direction�into the page� in spin state ��↑z�+ �↓z�� /2 is sent through aStern-Gerlach apparatus with a vertical magnetic field gradi-ent in the −z direction. Students were asked to sketch theelectron cloud pattern that they expected to see on a distantphosphor screen in the x-z plane and to explain their reason-ing. We wanted students to realize that the magnetic fieldgradient in the −z direction would exert a force on the elec-tron due to its spin angular momentum and that two spotswould be observed on the phosphor screen due to the split-ting of the beam along the z direction corresponding to elec-tron spin component in the �↑z� and �↓z� states. All responsesin which students noted that there will be a splitting alongthe z direction were considered correct even if they did notexplain their reasoning. Only 41% of the students providedthe correct response. Many students thought that there wouldonly be a single spot on the phospor screen, as in thesetypical responses: �a� “SGA �the Stern-Gerlach apparatus�
will pick up the electrons with spin down because the gradi-


ent is in the −z direction. The screen will show electroncloud only in −z part”; �b� “all of the electrons that come outof the SGA will be spin down with expectation value −� /2because the field gradient is in −z direction”; �c� “magneticfield is going to align the spin in that direction so most of theelectrons will align along −z direction. We may still have afew in the +z direction but the probability will be verysmall.”

Students were often confused in the interviews about theorigin of the force on the particles and whether there shouldbe a force on the particles at all as they pass through theStern-Gerlag apparatus.

In Question 5�b�, a beam of electrons propagating alongthe y direction �into the page� in spin state �↑z� is sentthrough an apparatus with a horizontal magnetic field gradi-ent in the −x direction. Students were asked to sketch theelectron cloud pattern they expect to see on a distant phos-phor screen in the x-z plane and to explain their reasoning.This question is more challenging than Question 5�a� be-

cause students have to realize that the eigenstate of Sz, �↑z�can be written as a linear superposition of the eigenstates of

Sx; that is, �↑z�= ��↑x�+ �↓x�� /2. Therefore, the magnetic fieldgradient in the −x direction will split the beam along the xdirection corresponding to the electron spin components in�↑x� and �↓x� states and cause two spots on the phosphorscreen. Only 23% of the students provided the correct re-sponse. The most common difficulty was assuming that be-cause the spin state is �↑z�, there should not be any splittingas in the following examples: �a� “Magnetic field gradientcannot affect the electron because it is perpendicular to thewavefunction.” �b� “Electrons are undeflected or rather the

beam is not split because B� is perpendicular to spin state.”�c� “The direction of the spin state of the beam of electrons isy, and the magnetic field gradient is in the −x direction. Thetwo directions have an angle 90°, so the magnetic field gra-dient gives no force to electrons.” �d� “With the electrons inonly one measurable state, they will experience a force only

in one direction upon interaction with B� .”Thus, many students explained their reasoning by claim-

ing that because the magnetic field gradient is in the −x di-rection but the spin state is along the z direction, they areorthogonal to each other, and therefore, there cannot be anysplitting of the beam. It is clear from the responses that stu-dents incorrectly relate the direction of the magnetic field inreal space with the “direction” of the state vectors in Hilbertspace. Several students, in response to Question 5�b�, drew amonotonically increasing function. One interviewed studentdrew a diagram of a molecular orbital with four lobes andsaid “this question asks about the electron cloud pattern dueto spin … I am wondering what the spin part of the wave-function looks like.” Then he added, “I am totally blankingon what the plot of �↑z� looks like; otherwise, I would havedone better on this question.” It is clear from such responsesthat the abstract nature of spin angular momentum posesspecial problems in teaching quantum physics.

In comparison to Question 5�a�, many more student re-sponses to Question 5�b� mentioned that there would be onlyone spot on the screen, but there was no consensus on thedirection of the deflection despite the fact that students wereasked to ignore the Lorentz force. Some students drew thespot at the origin, some showed deflections along the posi-
tive or negative x direction, and some along the positive or


negative z direction. They often provided interesting reasonsfor their choices. Students were confused about the directionin which the magnetic field gradient would cause the split-ting of the beam. The 13% of the students �including Ques-tions 5�a� and 5�b�� drew the splitting of the beam in thewrong direction �along the x axis in 5�a� and along the z axisin 5�b��. One interviewed student who drew it in the wrongdirection said, “I remember doing this recently and I knowthere is some splitting but I don’t remember in which direc-tion it will be.” Another surprising fact is that a large numberof students did not respond to Question 5. Some explicitlywrote that they don’t recall learning about the Stern-Gerlachexperiment. Instructors of the graduate quantum coursesshould take note of the fact that many students may not havedone much related to Stern-Gerlach experiment in the under-graduate quantum mechanics courses.

I. Difficulty in sketching the shape of the wavefunction

Questions related to the shape of the wavefunction showthat students may not draw a qualitatively correct sketcheven if their mathematical form of the wavefunction is cor-rect; may draw wavefunctions with discontinuities or cusps,or may confuse a scattering state wavefunction for a poten-tial barrier problem with the wavefunction for a potentialwell problem. In Question 2�e�, which was related to Ques-tion 2�d�, students were asked to plot the wavefunction inposition space right after the measurement of energy inQuestion 2�d�. We wanted students to draw the wavefunctionfor the first excited state as a function of position x. Thiswavefunction is sinusoidal and goes to zero at x=0, a /2, a.In Question �6�, students were given the potential energydiagram for a finite square well. In part �a� they were askedto sketch the ground state wavefunction, and in part �b� theyhad to sketch any one scattering state wavefunction. In bothcases, students were asked to comment on the shape of thewavefunction in the three regions. We hoped that in part �a�students would draw the ground state wavefunction as asinusoidal curve with no nodes inside the well and with ex-ponentially decaying tails in the classically forbidden re-gions. The wavefunction and its first derivative should becontinuous everywhere and the wavefunction should besingle valued. In part �b� we expected to see oscillatory be-havior in all regions, but because the potential energy islower in the well, the wavelength is shorter in the well. ForQuestion 6�b�, all responses that were oscillatory in bothregions �regardless of the relative wavelengths or amplitudesin different regions� and showed the wavefunction and itsfirst derivative as continuous were considered correct. If thestudents drew the wavefunction correctly, we consideredtheir response correct even if they did not comment on theshape of the wavefunction in the three regions.

In Question 2�e�, the most common incorrect response,provided by 14% of the students, was drawing too many zerocrossings in the wavefunction. Because many of these stu-dents had answered part 2�d� correctly and had written downthe wavefunction as �2�x�, it is interesting that the math-ematical and graphical representations of �2�x� were incon-sistent. In response to Question 6�a�, 8% of students drew theground state wavefunction for the infinite square well thatgoes to zero in the classically forbidden region, and another8% drew an oscillatory wavefunction in all three regions.Including both Questions 6�a� and �b�, 20% of the students
drew either the first excited state or a higher excited bound


state with many oscillations in the well and exponential de-cay outside �a majority of these were in response to Question6�b��. Several students made comments such as “the particleis bound inside the well but free outside the well.” The com-ments displayed confusion about what “bound state” meansand whether the entire wavefunction is associated with theparticle at a given time or the parts of the wavefunctionoutside and inside the well are associated with the particle atdifferent times. In Question 6�b�, approximately 8% of thestudents drew a scattering state wavefunction that had anexponential decay in the well. Although students were ex-plicitly given a diagram of the potential well, they may beconfusing the potential well with a potential barrier. In re-sponse to Question 6�a�, one interviewed student plotted awavefunction �without labeling the axes� which looked like aparabolic well with the entire function drawn below the hori-zontal axis. The interviewer then asked whether the wave-function can have a positive amplitude, that is, whether hiswavefunction multiplied by an overall minus sign is also avalid ground state wavefunction for this potential well. Thestudent responded, “I don’t think so. How can the wavefunc-tion not follow the sign of the potential?” It was apparentfrom further probing that he was not clear about the fact thatthe wavefunction can have an overall complex phase factor.

Including both Questions 6�a� and �b�, approximately 8%of the students drew wavefunctions with incorrect boundaryconditions or that had discontinuities or cusps in some loca-tions. In Question 2�e�, 8% of the students had incorrectboundary conditions, for example, their wavefunction did notgo to zero at x=0 and x=a and abruptly ended at somenon-zero value, or the sinusoidal wavefunction continued be-yond x=0 and x=a as though it were a free particle for all x.

IV. SUMMARY

The analysis of survey data and interviews indicate thatstudents share common difficulties about quantum mechan-ics. All of the questions were qualitative. They required stu-dents to interpret and draw qualitative inferences from quan-titative tools and make a transition from the mathematicalrepresentation to concrete cases. Even if relevant knowledgewas not completely lacking, it was often difficult for studentsto make correct inferences in specific situations.

Shared misconceptions in quantum mechanics can betraced in large part to incorrect overgeneralizations of con-cepts learned earlier, compounding of misconceptions thatwere never cleared up, or failure to distinguish betweenclosely related concepts. Some of the difficulties includingthose with the time evolution of the wavefunction originatefrom the overemphasis on the time-independent Schro-edinger equation and over-generalizing and attributing theproperties of the stationary states to non-stationary states.Students also had difficulty realizing that all smooth wave-functions that satisfy the boundary conditions for a systemare possible. Their responses displayed that they do not havea good grasp of Fourier analysis and have difficulty interpret-ing that linear superposition of sine functions can result infunctions that are not sinusoidal. Students often focusedsolely on the time-independent Schroedinger equation to de-termine if a wavefunction is a possible wavefunction. Manystudents did not know or recall the interpretation of expecta-tion value as an ensemble average, and did not realize thatthe expectation value of an observable can be calculated
from the knowledge of the probability of measuring different


values of that observable. In the context of the Stern-Gerlachexperiment, students had difficulty distinguishing betweenvectors in the real space of the laboratory and the state space.Many students incorrectly believed that a magnetic field gra-dient in the x direction cannot affect a spin-1 /2 particle in

the eigenstate of Sz because the field gradient and eigenstate

of Sz are orthogonal. Students also had difficulty qualita-tively sketching the bound state and scattering state wave-functions given the potential energy diagram.

Our findings can help the design of better curricula andpedagogies for teaching undergraduate quantum mechanicsand can inform the instructors of the first-year graduatequantum courses of what they can assume about the priorknowledge of the incoming graduate students. We are cur-rently developing and assessing quantum interactive learningtutorials which incorporate paper and pencil tasks and com-puter simulations suitable for use in advanced undergraduatequantum mechanics courses as supplements to lectures andstandard homework assignments. The goal of the tutorials isto actively engage students in the learning process and helpthem build links between the formalism and the conceptualaspects of quantum physics without compromising the tech-nical content.

ACKNOWLEDGMENTS

This work is supported in part by the National ScienceFoundation award PHY-0555434. We are very grateful to allthe faculty and J. Gaffney who were consulted during thesurvey design. We are also very grateful to all the facultywho administered the survey.

APPENDIX: SURVEY QUESTIONS

�1� Write down the most fundamental equation of quantummechanics.

In all of the following problems, assume that the measure-ment of all physical observables is ideal.

�2� The wavefunction of an electron in a one-dimensionalinfinite square well of width a at time t=0 is given by��x ,0�=2 /7�1�x�+5 /7�2�x� where �1�x� and �2�x� arethe ground state and first excited stationary state of the sys-tem. ��n�x�=2 /a sin�nx /a�, En=n22�2 / �2ma2� where n=1,2 ,3. . .�

Answer the following questions about this system:�a� Write down the wavefunction ��x , t� at time t in terms

of �1�x� and �2�x�.�b� You measure the energy of an electron at time t=0.

Write down the possible values of the energy and the prob-ability of measuring each.

�c� Calculate the expectation value of the energy in thestate ��x , t� above.

�d� If the energy measurement yields 42�2 / �2ma2�, writean expression for the wavefunction right after the measure-
ment.


�e� Sketch the above wavefunction in position space rightafter the measurement of energy in the previous question.

�f� Immediately after the energy measurement, you mea-sure the position of the electron. Qualitatively describe thepossible values of position you can measure and the prob-ability of measuring them.

�3� Which of the following wavefunctions at time t=0 areallowed for an electron in a one-dimensional infinite squarewell of width a: A sin3�x /a�, A�2 /5 sin�x /a�+3 /5 sin�2x /a�� and Ae−��x − a / 2� / a�2

? In each of the threecases, A is a suitable normalization constant. You must pro-vide a clear reasoning for each case.

�4� Consider the following statement: “By definition, theHamiltonian acting on any allowed state of the system � will

give the same state back, that is, H�=E�” where E is theenergy of the system. Explain why you agree or disagreewith this statement.

�5� Notation: �↑z� and �↓z� represent the orthonormal eigen-

states of Sz �the z component of the spin angular momentum�of the electron. SGA is an abbreviation for a Stern-Gerlachapparatus. Ignore the Lorentz force on the electron.

�a� A beam of electrons propagating along the y direction�into the page� in spin state ��↑z�+ �↓z�� /2 is sent through anSGA with a vertical magnetic field gradient in the −z direc-tion. Sketch the electron cloud pattern that you expect to seeon a distant phosphor screen in the x-z plane. Explain yourreasoning.

�b� A beam of electrons propagating along the y direction�into the page� in spin state �↑z� is sent through an SGA witha horizontal magnetic field gradient in the −x direction.Sketch the electron cloud pattern that you expect to see on adistant phosphor screen in the x-z plane. Explain your rea-soning.



�6� The potential energy diagram for a finite square well of width a and depth −V0 is shown below.

�a� Below, draw a qualitative sketch of the ground state wavefunction and comment on the shape of the wavefunction in allthe three regions shown above.

�b� Draw a qualitative sketch of any one scattering state wavefunction and comment on the shape of the wavefunction in all
the three regions shown above.
1H. W. C. Postma, T. T. Z. Yao, M. Grifoni, and C. Dekker, “Carbonnanotube single-electron transistors at room temperature,” Science293�5527�, 76–79 �2001�.

2D. J. Griffiths, Introduction to Quantum Mechanics �Prentice Hall, UpperSaddle River, NJ, 1995�, Preface.

3A. Goldberg, H. M. Schey, and J. L. Schwartz, “Computer-generatedMotion pictures of one-dimensional quantum-mechanical transmissionand reflection phenomena,” Am. J. Phys. 35, 177–186 �1967�.

4S. Brandt and H. Dahmen, The Picture Book of Quantum Mechanics�Springer-Verlag, New York, 2001�.

5B. Thaller, Visual Quantum Mechanics �Springer-Verlag, New York,2000�.

6J. Hiller, I. Johnston, and D. Styer, Quantum Mechanics Simulations�John Wiley & Sons, New York, 1995�.

7M. Belloni and W. Christian, “Physlets for quantum mechanics,” Comput.Sci. Eng. 5�1�, 90–96 �2003�; M. Belloni, W. Christian, and A. Cox,Physlet Quantum Physics �Pearson Prentice Hall, Upper Saddle River,NJ, 2006�.

8See, for example, see �www.opensourcephsyics.org�.9E. J. Galvez, C. H. Holbrow, M. J. Pysher, J. W. Martin, N. Courte-manche, L. Heilig, and J. Spencer, “Interference with correlated photons:Five quantum mechanics experiments for undergraduates,” Am. J. Phys.73, 127–140 �2005�.

10T. F. Havel, D. G. Cory, S. Lloyd, N. Bouland, E. M. Fortunato, M. A.Pravia, G. Teklemariam, Y. S. Weinstein, A. Bhattacharyya, and J. Hou,“Quantum information processing by nuclear magnetic resonance spec-troscopy,” Am. J. Phys. 70�3�, 345–362 �2002�.

11 J. R. Anderson, Learning and Memory: An Integrative Approach �JohnWiley & Sons, New York, 1999�, 2nd ed.

12For example, see A. B. Arons, A Guide to Introductory Physics Teaching�John Wiley & Sons, New York, 1990�.

13L. C. McDermott, “Research on conceptual understanding in mechanics,”Phys. Today 37�7�, 24–32 �1984�.

14D. Styer, “Common misconceptions regarding quantum mechanics,” Am.J. Phys. 64, 31–34 �1996�.

15D. Zollman, S. Rebello, and K. Hogg, “Quantum physics for everyone:Hands-on activities integrated with technology,” Am. J. Phys. 70�3�,252–259 �2002�; P. Jolly, D. Zollman, S. Rebello, and A. Dimitrova,“Visualizing potential energy diagrams,” Am. J. Phys. 66�1�, 57–63�1998�.



16L. Bao and E. Redish, “Understanding probabilistic interpretations ofphysical systems: A prerequisite to learning quantum physics,” Am. J.Phys. 70�3�, 210–217 �2002�; M. Wittmann, R. Steinberg, and E. Redish,“Investigating student understanding of quantum physics: Spontaneousmodels of conductivity,” Am. J. Phys. 70�3�, 218–226 �2002�.

17E. Cataloglu and R. W. Robinett, “Testing the development of studentconceptual and visualization skills in quantum mechanics through theundergraduate career,” Am. J. Phys. 70, 238–251 �2002�.

18Research on Teaching and Learning of Quantum Mechanics, papers pre-sented at the National Association for Research in Science Teaching,Boston, 1999 �perg.phys.ksu.edu/papers/narst/�.

19H. Fischler and M. Lichtfeldt, “Modern physics and students’ concep-tions,” Int. J. Sci. Educ. 14�2�, 181–190 �1992�.

20G. Ireson, “The quantum understanding of pre-university physics stu-dents,” Phys. Educ. 35�1�, 15–21 �2000�.

21J. Petri and H. Niedderer, “A learning pathway in high-school level quan-tum atomic physics,” Int. J. Sci. Educ. 20�9�, 1075–1088 �1998�.

22R. Muller and H. Wiesner, “Teaching quantum mechanics on an intro-ductory level,” Am. J. Phys. 70, 200–209 �2002�.

23C. Lawless, “Investigating the cognitive structure of students studyingquantum theory in an open university history of science course: A pilotstudy,” British J. Ed. Tech. 25�3�, 198–216 �1994�.

24C. Singh and M. Belloni, W. Christian, “Improving student’s understand-ing of quantum mechanics,” Phys. Today 59�8�, 43–49 �2006�.

25C. Singh, “Student understanding of quantum mechanics,” Am. J. Phys.69�8�, 885–896 �2001�; C. Singh, “Transfer of learning in quantum me-chanics,” AIP Conf. Proc. 790, 23–26 �2005�; C. Singh, “Improvingstudent understanding of quantum mechanics,” AIP Conf. Proc. 818,69–72 �2006�.

26C. Singh, “Student difficulties with quantum mechanics formalism,” AIPConf. Proc. 883, 185–188 �2007�.

27C. Singh, “Helping students learn quantum mechanics for quantum com-puting,” AIP Conf. Proc. 883, 42–45 �2007�.

28For example, see �http://web.phys.ksu.edu/vqm/� or simulations availableat �www.physics.umd.edu/perg/qm/qmcourse/NewModel�.

29M. T. H. Chi, in The Thinking Aloud Method, edited by M. W. vanSomeren, Y. F. Barnard, and J. A. C. Sandberg �Academic Press, London,1994�, Chap. 1.

30H. Batelaan, T. J. Gay, and J. J. Schwendingman, “Stern-Gerlach effectfor electron beams,” Phys. Rev. Lett. 79, 4517–4521 �1997�.



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Student understanding of quantum mechanics at the beginning of graduate instruction

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