Singh_QM_test

Student understanding of quantum mechanics h, du s st sh ts be co to The students who participated in this study were advanced undergraduates nearing the end of a full year upper-level quantum mechanics course.4 There have been earlier studies investigating student difficulties with waves5 and quantum mechanics.6 Those quantum mechanics investigations con- centrated on difficulties related to material covered in a mod- ern physics sequence, courses taken as a prerequisite to quantum mechanics or in place of it.6 Common misconcep- tions regarding quantum mechanics have also been documented.7 The present study focuses on quantum mea- surement and time development, advanced topics covered only in upper-level quantum mechanics courses. Quantum measurement theory is particularly difficult because of the statistical nature of the measurement outcome. Although questions about the foundations of the theory of quantum measurement are still being debated and investigated, at present the Copenhagen interpretation8 is widely accepted and universally taught to students. eigenstates of an operator, the calculation of expectation val- ues, and the conditions under which expectation values will be time independent. The test also probes student under- standing of how prior measurements affect future measure- ments, and how the time dependence of spin angular mo- mentum operators compares with operators such as position and linear momentum. III. ANALYSIS OF WRITTEN TEST RESULTS AND INTERVIEWS An analysis of students’ written tests and student inter- views shows that most students share a number of common difficulties and misconceptions, despite variations in their backgrounds and the abstract nature of the subject matter. Table I lists the names of the participating universities, the number of students from each university who took the test, and the textbooks used. Since the number of students from 885 885Am. J. Phys. 69 ~8!, August 2001 http://ojps.aip.org/ajp/ © 2001 American Association of Physics Teachers Chandralekha Singha) Department of Physics, University of Pittsburgh, Pittsburg ~Received 27 July 2000; accepted 18 January 2001! We investigate the difficulties of advanced undergra upper-level quantum mechanics course with concept development. Our analysis is based upon a test admini interviews with 9 students. Strikingly, most students in background, teaching styles, and textbooks. Concep time dependence of expectation values were found to tests and interviews suggests that widespread mis discriminate between related concepts and a tendency of Physics Teachers. @DOI: 10.1119/1.1365404# I. INTRODUCTION Quantum mechanics is a technically difficult and abstract subject. The subject matter makes instruction quite challeng- ing, and able students constantly struggle to master the basic concepts. In this study, we investigate the difficulties stu- dents have with concepts related to quantum measurements and time development. Our analysis is based upon a test that was designed and administered to 89 students from six universities1 and interviews with 9 students. The goal of this study is to identify common difficulties that students have about quantum measurements and time development, and determine whether they are correlated with teaching style, place of study, or textbook. We are also in- terested in comparing difficulties and misconceptions in the upper-level courses with those in the lower-level courses. There is a vast literature2,3 detailing misconceptions in intro- ductory courses, showing that misconceptions are pervasive and often arise from an incorrect ‘‘world-view.’’ The realm of quantum physics deals with phenomena not directly ob- servable in everyday experience; we want to explore whether the ways in which misconceptions arise in upper-level courses are similar to those in introductory courses, or whether fundamentally different processes are involved. II. TEST DESIGN Pennsylvania 15260 ate students toward the end of a full year related to quantum measurements and time ered to 89 students from six universities and ared the same difficulties despite variations related to stationary states, eigenstates, and particularly difficult. An analysis of written nceptions originate from an inability to overgeneralize. © 2001 American Association To aid in the design of test questions, three University of Pittsburgh ~Pitt! faculty members were consulted, each of whom had recently taught a full year quantum mechanics course. Each faculty member was asked about what he or she considered to be the fundamental concepts in quantum mea- surements and time development that advanced undergradu- ate students should know. Many test questions were selected and modified from those used as homework and exam ques- tions that had helped diagnose difficulties. During the design phase, we went through several itera- tions of the test with the three Pitt faculty members and two physics postdocs. A preliminary version was administered to students enrolled in quantum mechanics at Duquesne Uni- versity. After administering the test, there was an extensive discussion in the class, followed by individual discussions with student volunteers. Based upon these discussions, the test was modified before being administered to the students in this study. Appendix A shows the final version of the test,9 which is slightly improved and revised from the test actually presented to students. Half of the test questions deal with measurements and the other half deal with time develop- ment. The test is designed to be administered in one class period ~50 min!. The test explores student understanding of a number of important concepts related to quantum measurements and time development: the basic formalism, the special role of energy eigenstates or ‘‘stationary states,’’ the significance of unsure about their responses, had difficulty in discriminating between concepts, and provided conflicting justifications. were based upon ver, when asked s pursued incor- rect answer, and sponses to that un to refer to all ated in Appendix bservable Q and ator Qˆ . It is also the Hamiltonian e explicitly. Table I. The names of participating universities, the number of participating students ~most students enrolled in quantum mechanics took the test since it counted toward the c quantum mechanics mented the text with Name of un University of Pittsb Carnegie Mellon Un Univ. of Illinois, Ur Boston University Univ. of California, University of Color ei i n niv ng or each university is different, we calculate a weighted average of scores ~in percent! and standard deviations for each ques- tion on the test ~see Table II!. The concepts that are probed in the test were covered in all of the classes that participated in the study. Students were told in advance that they would take a test on quantum measurements on a set date but were not told about the exact nature of the test. In all of the par- ticipating universities, students were given 50 min to take the test and were informed that it counted for one homework grade. We also conducted audiotaped interviews with nine paid student volunteers from Pitt and analyzed the transcripts for a better understanding of the reasoning involved in answer- ing the questions. We believe their verbal responses echo those of students from the other universities; the written re- sponses clearly reflect the universal nature of the difficulties. Each interview lasted approximately 1 h. The students inter- viewed were not given the written test earlier because we wanted them to discuss the test without having seen it before. During the interviews, we provided students with a pen and paper and asked them to ‘‘think aloud’’ 10 while answer- ing the questions. Students first read the questions on their own and answered them without interruptions ~except that we prompted them to think aloud if they were quiet for a long time!. After students had finished answering a particular question to the best of their ability, we asked them to further clarify and elaborate issues which they had not clearly ad- dressed earlier. This process was repeated for every question on the test. After the interviews, we carefully re-analyzed the written responses and the reasoning provided by the 89 stu- dents. Many of the written responses were more easily inter- preted after the interviews. The interviews also helped us to gauge the general confidence level of students while re- sponding to a particular question. Often, students seemed Table II. The percent of correct responses, S¯ , and a w test. These are defined as S¯5S i niSi /N , and s5AS universities, ni is the number of students from the ith u ith university, N589 is the total number of participati theses refer to students who wrote ‘‘yes’’ or ‘‘no’’ c wrong justification. Question 1 2 3 4a 4b 5a S¯% 43 76 83 11 17 95 s% 14 8 7 8 10 4 886 Am. J. Phys., Vol. 69, No. 8, August 2001 Table III lists several common misconceptions evident in student responses to questions ~2!, ~4!, and ~5!. For ease in referring to them, we label the misconceptions ~M1!–~M7!. A. Basic formalism of quantum mechanics Question (1): The eigenvalue equation for an operator Qˆ is given by Qˆ uc i&5l iuc i&, i51,...,N . Using this information, write a mathematical expression for ^fuQˆ uf&, where uf& is a general state.11 Answer (1): ^fuQˆ uf&5( iu^fuc i&u2l i , or simply ( iuCiu2l i , where Ci5^fuc i& . Only 43% of students provided the correct response. Some had difficulty with the principle of linear superposition and could not expand a general state in terms of the complete set of eigenstates of an operator. The common mistakes include the following types of answers: Let uf&5( i uc i&, then ^fuQˆ uf&5( i l i , ~1! ^fuQˆ U( i Cic iL 5( i l iCi^fuc i&5( i l iCi , ~2! ^fuQˆ uf&5^fuQˆ uc&5^fuluc&5l^fuc&. ~3! Let uf&5uc& , ghted standard deviation, s, for each question on the i(S¯2Si)2/N , where the sum i runs over all the six ersity, Si is the average percent score of students from students. For questions 5e–5h, the number in paren- rectly but either did not justify their answer or gave 5b 5c 5d 5e 5f 5g 5h 75 73 73 22 22 13 25 ~29! ~32! ~32! ~55! 12 12 11 12 16 7 12 ~18! ~13! ~13! ~9! 886Chandralekha Singh Some admitted that some of their answers ‘‘gut feeling’’ or ‘‘educated guess.’’ Howe to justify their responses, students sometime rect justifications quite far. Below, we list each test question, the cor then students’ written and interviewed re question. We will use the masculine prono students regardless of their gender. As indic A, for all questions, we refer to a generic o its corresponding quantum mechanical oper noted in Appendix A that for all questions, Hˆ and operators Qˆ do not depend upon tim ourse grade in all universities!, and authors of the textbooks used ~in some cases, instructors supple- additional notes!. iversity Number of students Author of quantum textbook urgh 11 Liboff iversity 7 Shankar bana Champaign 17 Goswami 13 Griffiths Santa Barbara 34 Griffiths ado, Boulder 7 Griffiths ˆ answer. Table III. Common misconceptions of students, the symbols used for ease in referring to them, and the questions to which they relate. co o ep of na op n po em an Qˆ . e te an ble Q and a state uf& , many students assume that the state is an eigenstate of the operator, i.e., Qˆ uf&5luf&, whether it is justified or not. Some students also made mistakes with sum- mation indices. The above responses suggest that many ad- vanced students are uncomfortable with the Dirac formalism and notation, even though it was used in all of the classes in this study. In the interview, in response to question ~1!, one student said that ‘‘the eigenvalue gives the probability of getting a particular eigenstate’’ and expanded the state as ‘‘uf& 5( i l iuc i& .’’ Then, he made another mistake by writing the expectation value as ‘‘^fuQˆ ( i l iuc i&5^fu( i l i2uc i& 5( l i 2 .’’ When asked to explain the final step, he said ‘‘( i l i 2 gets pulled out and this bra and ket states (pointing to the bra and ket explicitly) will give 1.’’ Another student made the same mistake and contracted different bra and ket vectors to obtain 1. He wrote ‘‘^fuQˆ u(n Cncn& 5 (n Cn ^f uQˆ u cn& 5 (n Cn ^f uln ucn& 5 (n Cnln ^f ucn& 5(n Cnln .’’ When asked to explain the final step, he said ‘‘cn will pick out the nth state from f and give 1 assuming that the states are normalized.’’ The fact that many students in the written test and interview could retrieve from memory Answer (2): Yes. The first measurement collapses the wave function into an eigenstate of the operator correspond- ing to the observable being measured. If successive measure- ments are rapid so that the state of the system does not have the time to evolve, the outcomes will be the same every time. Question (3): If you make measurements of a physical observable Q on an ensemble of identically prepared systems which are not in an eigenstate of Qˆ , do you expect the out- come to be the same every time? Justify your answer. Answer (3): No, a measurement on a system in a definite state could yield a multitude of results. Therefore, an en- semble of particles prepared in identical states uf& may col- lapse into different eigenstates uc i& of Qˆ , yielding different eigenvalues l i with probability u^c iuf&u2. In questions ~2! and ~3!, students might have misunder- stood the technical terms ‘‘rapid succession’’ and ‘‘identi- cally prepared.’’ Therefore, regardless of their answer, we considered their response correct if they justified it and showed correct understanding. Students performed relatively well on both questions ~2! and ~3! with weighted average scores of 76% and 83%, respectively. Therefore, it appears that most advanced students have some idea that the mea- surement of an observable collapses the wave function into 887 887Am. J. Phys., Vol. 69, No. 8, August 2001 Chandralekha Singh Then ^cuQˆ uc&5l , ~4! ^fuQˆ uf&5l i^fuf&5l i . ~5! Six percent of students based their answers on Eq. ~1!. Nine percent initially expanded the wave function correctly but ended up with an incorrect answer. Six percent of the stu- dents did not realize that ^fuc i& is not unity and Ci 5^c iuf& and provided a response similar to Eq. ~2!. Four- teen percent of students wrote l without any subscript in their final answer similar in spirit to Eqs. ~3! and ~4!. Eq. ~3!–Eq. ~5! show that when presented with an operator Symbol Mis M1 If the system is initially in an eigenstate of another operator Qˆ 8 will be time ind M2 If the system is initially in an eigenstate that operator is time independent. M3.1 An eigenstate of any operator is a statio M3.2 If the system is in an eigenstate of any Qˆ forever unless an external perturbatio M3.3 The statement ‘‘the time dependent ex value’’ is synonymous with the stat eigenstate.’’ M4 The expectation value of an operator in M5 If the expectation value of an operator value cannot have any time dependence M6 Individual terms (Hˆ 0 ,Hˆ 1 , . . . ) in a tim can cause transitions from one eigensta M7 Time evolution of an arbitrary state c particular outcome when any observa operator is of the form exp(2iHˆ t/\). that a general state uf& can be expanded as (n Cnucn& , but did not realize that ^fucn& is not unity, shows that students lack a clear understanding of what the expansion uf& 5(n Cnucn& means and that Cn5^cnuf& ~which implies ^fucn&5Cn*!. B. Effect of prior measurements on future measurement and measurements on identically prepared systems Question (2): If you make measurements of a physical observable Q on a system in rapid succession, do you expect the outcome to be the same every time? Justify your 11 nception Question Nos. f any operator Qˆ , then the expectation value endent if @Qˆ ,Qˆ 8#50. ~4! and ~5! an operator Qˆ , then the expectation value of ~4! and ~5! ry state. ~2!, ~4!, and ~5! erator Qˆ , then it remains in the eigenstate of is applied. nential factors cancel out in the expectation ent ‘‘the system does not evolve in an energy eigenstate may depend upon time. ~4! and ~5! is zero in some initial state, the expectation ~5! -independent Hamiltonian Hˆ 5Hˆ 01Hˆ 11fl of Hˆ to another. ~5! not change the probability of obtaining a is measured because the time evolution ~5! an eigenstate of the corresponding operator, that a measure- ment on a system in a definite state could yield a multitude get the same position.’’ The student had apparently forgotten that identically prepared systems can yield different out- of results, and that prior measurements affect future mea- surements. Asking additional questions similar to ~2! and ~3! but for specific systems would provide further insight into the depth of student understanding. In response to question ~2!, one student wrote ‘‘No, for example, from the uncer- tainty principle in the book there is a 50-50 chance of mea- suring up and down spins everytime you measure Sz .’’ This student seems oblivious of the collapse of the wave function upon measurement of an operator. In the interview, in response to question ~2!, one student began with a correct statement: ‘‘if you measure Q, the sys- tem will collapse into an eigenstate of that operator. Then, if you wait for a while the measurement will be different.’’ But then he added incorrectly: ‘‘if Q has a continuous spectrum then the system would gently evolve and the next measure- ment won’t be very different from the first one. But if the spectrum of eigenvalues is discrete then you will get very different answers even if you did the next measurement after a very short time.’’ When the student was asked to elaborate, he said: ‘‘For example, imagine measuring the position of an electron. It is a continuous function so the time dependence is gentle and after a few seconds you can only go from A to its neighboring point. @Pointing to an x vs t graph that he sketches on the paper#...you cannot go from this place to this without going through this intermediate space.’’ When asked to elaborate on the discrete spectrum case, he said: ‘‘...think of discrete variables like spin...they can give you very differ- ent values in a short time because the system must flip from up to down. I find it a little strange that such [large] changes can happen almost instantaneously. But that’s what quantum mechanics predicts... .’’This student had the misconception that successive measurements of continuous variables, e.g., position, produce ‘‘somewhat’’ deterministic outcomes whereas successive measurements of discrete variables, e.g., spin, can produce very different outcomes. This type of re- sponse may also be due to the difficulty in reconciling clas- sical and quantum mechanical ideas; in classical mechanics the position of a particle is deterministic and can be unam- biguously predicted for all times from the knowledge of the initial conditions and potential. In response to question ~2!, one student who had earlier claimed that the system is stuck in an eigenstate unless you apply an external perturbation said, ‘‘Yes, once the first mea- surement is made...the wave function collapses to an eigen- state where it will stay for all times.’’ Such a response re- flects misconception ~M3.2! ~see Table III! that if the system is in an eigenstate of any operator Qˆ , then it remains in that eigenstate. In response to question ~3!, student S1 ~we call him stu- dent S1 for ease in referring to him later!, who appeared not to remember that the wave function collapses into an eigen- state of the operator that is measured, said ‘‘If Q is not in an eigenstate then Quc&5luc& is not true...so if you measure Q you won’t be able to get l and your results will be different every time.’’ Another student claimed that identically pre- pared systems should give the same measured value of Qˆ . Even when explicitly to