Effectiveness of interactive tutorials in promoting "which-path" information reasoning in advanced quantum mechanics
EEffectiveness of interactive tutorials in promoting “ which-path ” informationreasoning in advanced quantum mechanics Alexandru Maries, Ryan Sayer, and Chandralekha Singh Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221, USA Department of Physics, Bemidji State University, Bemidji, Minnesota 56601, USA Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA (Received 24 May 2016; published 11 September 2017)Research suggests that introductory physics students often have difficulty using a concept in contextsdifferent from the ones in which they learned it without explicit guidance to help them make the connectionbetween the different contexts. We have been investigating advanced students ’ learning of quantummechanics concepts and have developed interactive tutorials which strive to help students learn theseconcepts. Two such tutorials, focused on the Mach-Zehnder interferometer (MZI) and the double-slitexperiment (DSE), help students learn how to use the concept of “ which-path ” information to reason aboutthe presence or absence of interference in these two experiments in different situations. After working on apretest that asked students to predict interference in the MZI with single photons and polarizers of variousorientations placed in one or both paths of the MZI, students worked on the MZI tutorial which, amongother things, guided them to reason in terms of which-path information in order to predict interference insimilar situations. We investigated the extent to which students were able to use reasoning related to which-path information learned in the MZI tutorial to answer analogous questions on the DSE (before working onthe DSE tutorial). After students worked on the DSE pretest they worked on a DSE tutorial in which theylearned to use the concept of which-path information to answer questions about interference in the DSEwith single particles with mass sent through the two slits and a monochromatic lamp placed between theslits and the screen. We investigated if this additional exposure to the concept of which-path informationpromoted improved learning and performance on the DSE questions with single photons and polarizersplaced after one or both slits. We find evidence that both tutorials promoted which-path informationreasoning and helped students use this reasoning appropriately in contexts different from the ones in whichthey had learned it. DOI: 10.1103/PhysRevPhysEducRes.13.020115
I. INTRODUCTION
Prior research suggests that in quantum mechanics,students have many common difficulties due the unintuitiveand abstract nature of the subject [1 – ’ (upper-level under-graduate and graduate students) reasoning difficulties withquantum mechanics concepts and has been developingand evaluating research-based tools to help students learnquantum mechanics effectively [1,3,4,10 – – “ which-path ” information (WPI)reasoning promoted by Wheeler [32] (we provide adefinition of WPI and how it can be used to reason aboutinterference in the MZI and DSE in Sec. II). The MZI andDSE provide great contexts to investigate whether the Published by the American Physical Society under the terms ofthe Creative Commons Attribution 3.0 License. Further distri-bution of this work must maintain attribution to the author(s) andthe published article ’ s title, journal citation, and DOI. PHYSICAL REVIEW PHYSICS EDUCATION RESEARCH = = = II. ISOMORPHISM BETWEEN MZIAND DSE QUESTIONS
We note that a comprehensive discussion of the isomor-phism between questions about interference in the MZIand DSE contexts and how WPI reasoning can be used topredict interference is provided in Appendix A. Below, weprovide only a brief explanation.Before recognizing the isomorphism between questionsabout interference in the MZI and the DSE contexts, onemust first understand how the concept of WPI can be usedto reason about interference in each experiment. Theconcept of WPI at a detector may be useful when the stateof the system is a superposition of two different spatial pathstates (e.g., MZI, DSE with single photons). In general,when a detector can project both components of the pathstate, then WPI is unknown. On the other hand, when a detector can project only one component of the path state,then we have complete which path information, or WPI isknown. For example, when there are no polarizers placed infront of either slit in the DSE or in either path of the MZI,the state of a photon before being detected is a product of alinear superposition of path states with a linear super-position of polarization states (vertical, horizontal). So eachpolarization state component is associated with both pathstate components. Thus, the detector or screen can projectboth path state components for each polarization state,which means that WPI is not known for either polarization.Thus, full interference will be observed on the screen.Suppose that instead, a vertical polarizer is placed in frontof one slit (say, upper slit) in the DSE or is placed in one ofthe paths (say, upper path) of the MZI. Since vertical andhorizontal polarization states are orthogonal, placing thevertical polarizer (in front of the upper slit in the DSE or inthe upper path of the MZI) will cancel the horizontalcomponent associated with only the upper path state. Thus,for the horizontal component of the photon state, thedetector can only project the lower path state component,which means that WPI is known for horizontally polarizedphotons detected at the screen. On the other hand, WPI isnot known for vertically polarized photons because for thevertical component of the photon state, the detector canproject both path states. Thus, the horizontally polarizedphotons detected at the screen will not interfere, whereasthe vertically polarized photons will interfere. Similarreasoning can be applied in other situations (see Q1 throughQ5 described in Sec. III B).
III. PARTICIPANTS, MATERIALS, RESEARCHQUESTIONS AND STUDY DESIGNA. Participants
The participants in this study were 46 undergraduatestudents enrolled in an upper-level quantum mechanicscourse (mostly juniors and seniors in physics) and 59physics graduate students enrolled in a mandatorysemester-long TA professional development course whichmet for two hours each week. With very few exceptions(several students), all of the students had typical ages youwould expect for the level they are at, around 20 –
22 yearsfor the undergraduate students and 23 –
25 for the graduatestudents. For the undergraduate students, the MZI and DSEwere part of the course material and therefore the QuILTsand post-tests (described in detail below) were graded forcorrectness and the post-tests were counted as regularquizzes. In addition, the undergraduate students were awarethat topics discussed in these QuILTs can also appear infuture exams. After completing a pretest on a particularQuILT, students worked on that tutorial during an hour longclass and whatever they did not finish, they completed athome. None of the undergraduate students completed eitherQuILT in class. For the graduate students, one of the topicsMARIES, SAYER, and SINGH PHYS. REV. PHYS. EDUC. RES.13,
25 for the graduatestudents. For the undergraduate students, the MZI and DSEwere part of the course material and therefore the QuILTsand post-tests (described in detail below) were graded forcorrectness and the post-tests were counted as regularquizzes. In addition, the undergraduate students were awarethat topics discussed in these QuILTs can also appear infuture exams. After completing a pretest on a particularQuILT, students worked on that tutorial during an hour longclass and whatever they did not finish, they completed athome. None of the undergraduate students completed eitherQuILT in class. For the graduate students, one of the topicsMARIES, SAYER, and SINGH PHYS. REV. PHYS. EDUC. RES.13,
B. Materials
The materials used in this study are two research-basedQuILTs on the MZI and DSE, each of which includepretests and post-tests. The DSE pretests and post-tests alsoinclude the DSE polarizer questions, a topic which is notdiscussed in the DSE tutorial (the DSE pretest is included inAppendix B). These polarizer questions (described in detailbelow) were designed specifically to investigate the extentto which students are able to use WPI reasoning theylearned in the context of the MZI to answer isomorphicquestions in the context of the DSE. Both the MZI and DSEQuILTs focus on helping students learn about topics suchas the wave-particle duality (in the context of singlephotons in the MZI and in the context of particles withmass in the DSE), interference of single photons (MZI)-particles with mass (DSE), probabilistic nature of quantummeasurements, and collapse of a quantum state uponmeasurement. Both QuILTs make use of interactive sim-ulations in which students can manipulate the MZI andDSE setups to predict and observe what happens at thephoto-detectors (MZI)-screen (DSE) for various setups.The development of both QuILTs included interviews withboth graduate and undergraduate students in which studentsworked on the tutorials while thinking out loud. Whilestudents worked on the tutorials, they were not disturbed.After they were finished, they were asked for clarificationon points they had not made clear earlier while thinking out loud. For the DSE QuILT alone, approximately 85 h ofindividual student interviews were conducted, each inter-view lasting 2 – … PHYS. REV. PHYS. EDUC. RES.13,
The materials used in this study are two research-basedQuILTs on the MZI and DSE, each of which includepretests and post-tests. The DSE pretests and post-tests alsoinclude the DSE polarizer questions, a topic which is notdiscussed in the DSE tutorial (the DSE pretest is included inAppendix B). These polarizer questions (described in detailbelow) were designed specifically to investigate the extentto which students are able to use WPI reasoning theylearned in the context of the MZI to answer isomorphicquestions in the context of the DSE. Both the MZI and DSEQuILTs focus on helping students learn about topics suchas the wave-particle duality (in the context of singlephotons in the MZI and in the context of particles withmass in the DSE), interference of single photons (MZI)-particles with mass (DSE), probabilistic nature of quantummeasurements, and collapse of a quantum state uponmeasurement. Both QuILTs make use of interactive sim-ulations in which students can manipulate the MZI andDSE setups to predict and observe what happens at thephoto-detectors (MZI)-screen (DSE) for various setups.The development of both QuILTs included interviews withboth graduate and undergraduate students in which studentsworked on the tutorials while thinking out loud. Whilestudents worked on the tutorials, they were not disturbed.After they were finished, they were asked for clarificationon points they had not made clear earlier while thinking out loud. For the DSE QuILT alone, approximately 85 h ofindividual student interviews were conducted, each inter-view lasting 2 – … PHYS. REV. PHYS. EDUC. RES.13, “ DSE lamp ques-tions. ” These questions were explicitly discussed inthe DSE QuILT, which helped students make senseof them by using reasoning related to WPI.(ii) DSE polarizer questions related to interference ofsingle photons passing through the slits and theeffect on the interference pattern of placing polar-izers of various orientations after one or bothslits. These topics were not discussed in the DSEQuILT.The DSE polarizer questions are summarized as follows: “ You perform a DSE in which photons that are polarizedat þ ° are sent one at a time towards the double slit. Thewavelength of the photons is comparable to the slit widthand the separation between the slits is more than twice theslit width. In all questions, assume that the same largenumber N of photons reaches the screen. In each situation,describe the pattern you expect to observe on the screen.Explain your reasoning. ” Q1. Situation described above.Q2. Vertical polarizer placed in front of one slit.Q3. Vertical polarizer placed in front of each slit.Q4. Vertical and horizontal polarizer placed in front ofslits 1 and 2, respectively.Q5. Vertical and horizontal polarizer placed in front ofslits 1 and 2, respectively. Additionally, a polarizer whichmakes an angle of þ ° with the horizontal is placed inbetween the slits and the screen.These questions are isomorphic to questions studentsconsidered in the context of the MZI: Q1 — no polarizersplaced in either path of the MZI, Q2 — vertical polarizerplaced in one path of the MZI, etc. The MZI pre- and post-tests were comprised of the MZI polarizer questions andmany other questions in other situations (e.g., effect ofremoving BS2 on interference at the detectors D1 or D2and the percentages of photons of a given polarizationarriving at D1 and D2 in different situations) which are verydifferent from the polarizer questions. C. Research questions and study design
We investigated three research questions. The first twoare related to the MZI QuILT and are referred to as RQ1.aand RQ1.b and the other is related to the DSE QuILT andis referred to as RQ2. We describe the research questionsand the approach used to investigate them below andsummarize them in Figs. 1 and 2 (RQ1.a and RQ1.b aresummarized in Fig. 1 and RQ2 in Fig. 2). We hypothesized that at least some students who learnhow to reason about the MZI polarizer questions in termsof WPI from the MZI QuILT may be able to use thisreasoning appropriately when answering the DSE polarizerquestions which were isomorphic to the MZI polarizerquestions. If students are indeed able to use WPI reasoningwhen answering the DSE polarizer questions, this wouldbe an indication that the MZI QuILT may be effective atpromoting WPI reasoning.Thus, we first wanted to determine the percentage ofstudents who use WPI reasoning on the MZI and DSEpolarizer questions before having the opportunity to learnfrom the MZI QuILT and compare to after they work on theMZI QuILT (this is the focus of RQ1.a).(We note that in all that follows, “ MZI pretest ” refersto MZI questions given before the MZI QuILT, “ DSEpretest ” refers to DSE questions given before the DSEQuILT, and “ DSE post-test ” refers to DSE questionsgiven after the DSE QuILT. We have described studentperformance and use of appropriate reasoning on theMZI post-test elsewhere [37], and thus, this is notdiscussed here.) RQ1.a. What percentage of students use WPI reasoningbefore working on the MZI QuILT and how does thatcompare to after working on the MZI QuILT?
To investigate this question, we gave the MZI pretest,which includes the MZI polarizer questions, to 46 under-graduate and 45 graduate students before these studentsworked on the MZI QuILT. We refer to these students asthe MZI → DSE cohort because they worked on the MZIQuILT (pretest, tutorial, post-test) before working on theDSE QuILT. Thus, these students did not have an oppor-tunity to learn from the MZI QuILT before answering theMZI polarizer questions. After they answered the MZIpolarizer questions, these students worked on the MZIQuILT and then answered the DSE polarizer questions.We compared the percentage of students who used WPIreasoning after working on the MZI QuILT to beforeworking on the MZI QuILT.A second cohort of 14 graduate students (referred toas the DSE → MZI cohort) worked on the DSE QuILTfirst (pretest, tutorial, post-test) before they worked onthe MZI QuILT. Thus, these students did not have theopportunity to learn about WPI from the MZI QuILTbefore answering the DSE polarizer questions. We com-pared the percentage of graduate students from theDSE → MZI cohort who used WPI reasoning to answerthe DSE polarizer questions with the percentage ofgraduate students from the MZI → DSE cohort who usedWPI on these questions.Furthermore, if the MZI QuILT is effective in promotingthe use of WPI reasoning, students who work on the MZIQuILT before answering the DSE polarizer questionsshould perform better on these questions compared tostudents who do not work on the MZI QuILT beforeMARIES, SAYER, and SINGH PHYS. REV. PHYS. EDUC. RES.13,
To investigate this question, we gave the MZI pretest,which includes the MZI polarizer questions, to 46 under-graduate and 45 graduate students before these studentsworked on the MZI QuILT. We refer to these students asthe MZI → DSE cohort because they worked on the MZIQuILT (pretest, tutorial, post-test) before working on theDSE QuILT. Thus, these students did not have an oppor-tunity to learn from the MZI QuILT before answering theMZI polarizer questions. After they answered the MZIpolarizer questions, these students worked on the MZIQuILT and then answered the DSE polarizer questions.We compared the percentage of students who used WPIreasoning after working on the MZI QuILT to beforeworking on the MZI QuILT.A second cohort of 14 graduate students (referred toas the DSE → MZI cohort) worked on the DSE QuILTfirst (pretest, tutorial, post-test) before they worked onthe MZI QuILT. Thus, these students did not have theopportunity to learn about WPI from the MZI QuILTbefore answering the DSE polarizer questions. We com-pared the percentage of graduate students from theDSE → MZI cohort who used WPI reasoning to answerthe DSE polarizer questions with the percentage ofgraduate students from the MZI → DSE cohort who usedWPI on these questions.Furthermore, if the MZI QuILT is effective in promotingthe use of WPI reasoning, students who work on the MZIQuILT before answering the DSE polarizer questionsshould perform better on these questions compared tostudents who do not work on the MZI QuILT beforeMARIES, SAYER, and SINGH PHYS. REV. PHYS. EDUC. RES.13,
RQ1.b. How well do students who have worked on theMZI QuILT perform on the DSE polarizer and lampquestions compared to students who have not worked onthe MZI QuILT?
To investigate this question, for the two graduatestudent cohorts who either worked or did not work onthe MZI QuILT before answering the DSE pretest ques-tions, we used a one way repeated measures ANOVA [38]
FIG. 2. Schematic description of the design used to investigate RQ2. UG, GS, and N have the same meaning as in Fig. 1. The timelineof chronological order is from left to right as depicted by the blue arrow.FIG. 1. Schematic description of the design used to investigate RQ1.a and RQ1.b. Abbreviations in the figure are undergraduatestudents (UG), graduate students (GS), and N refers to the number of students. The timeline of chronological order is from left to right asdepicted by the blue arrows. All of the questions were given as pretests, i.e., the DSE polarizer questions in both RQ1.a and RQ1.b werealways given before students worked on the DSE QuILT, and the MZI polarizer questions in RQ1.a were given before students workedon the MZI QuILT. EFFECTIVENESS OF INTERACTIVE TUTORIALS … PHYS. REV. PHYS. EDUC. RES.13,
FIG. 2. Schematic description of the design used to investigate RQ2. UG, GS, and N have the same meaning as in Fig. 1. The timelineof chronological order is from left to right as depicted by the blue arrow.FIG. 1. Schematic description of the design used to investigate RQ1.a and RQ1.b. Abbreviations in the figure are undergraduatestudents (UG), graduate students (GS), and N refers to the number of students. The timeline of chronological order is from left to right asdepicted by the blue arrows. All of the questions were given as pretests, i.e., the DSE polarizer questions in both RQ1.a and RQ1.b werealways given before students worked on the DSE QuILT, and the MZI polarizer questions in RQ1.a were given before students workedon the MZI QuILT. EFFECTIVENESS OF INTERACTIVE TUTORIALS … PHYS. REV. PHYS. EDUC. RES.13,
RQ2. To what extent is the DSE QuILT effective inpromoting WPI reasoning from one context of the DSE(single particles and a monochromatic lamp placedbetween the slits and the screen) to a different contextof the DSE (single photons and polarizers placed in front ofone or both slits) without an instructional interventiondesigned to help them make the connection between thesedifferent contexts?
Students in all cohorts worked on the DSE pretest, andthen worked on the DSE QuILT after which they worked onthe DSE post-test. Some of the students worked on the MZIQuILT before working on the DSE (pretest, tutorial, post-test), while other students worked on the MZI QuILT afterworking on the DSE. We therefore investigated the extentto which the DSE QuILT, which discussed WPI reasoningin the context of the DSE with single particles with massand a monochromatic lamp placed between the slits and thescreen, promoted WPI reasoning to a different context(DSE polarizer questions) by comparing their performanceand use of WPI reasoning before working on the DSEQuILT to after working on it. The study design for RQ2including chronology and number of participants is sum-marized in Fig. 2.The summary of the rubric used to grade students ’ performance on the DSE polarizer questions is shown inTable I. This rubric is designed to evaluate students ’ conceptual understanding of the effect of placing polarizersof various orientations in front of one or both slits in the DSEby considering responses for multiple questions together.For example, the third conceptual point (recognize that “ which-path ” information can be lost) is based on students ’ answers to the last two questions (orthogonal polarizers andquantum eraser). Similarly, the second conceptual point isbased on students ’ answers to all the questions: studentsshould recognize that the situation in which two polarizersare orthogonal and there is no quantum eraser is the onlycase in which no interference is observed on the screen. Ifstudents claimed that the interference pattern vanishes inmore than one situation, it was considered that they did notunderstand this conceptual point.Another rubric was designed for grading the DSE lampquestions with the same goal: assess student understandingof concepts across questions. However, discussing thisrubric in detail here would require discussing those ques-tions along with the correct answers, which we have doneelsewhere [34].The two rubrics were used to score students ’ perfor-mance on the DSE polarizer questions and DSE lampquestions. A subset of the responses for all questions(20% – (cid:129) The percentage of students who used WPI reasoningamong those who wrote down any reasoning (Tables IIand VI). (cid:129)
The percentage of students who answered each ofthe DSE polarizer questions correctly (Table III).Students who did not answer a particular questionwere excluded from the data analysis for that question.However, students were given more than enough timeto complete the DSE pretest, and nearly all studentshanded in their pretests voluntarily. In Appendix C,we provide statistics for how many students did notprovide a response on each of these questions. (cid:129)
For student performance on a group of questions, inparticular, the DSE polarizer questions and the DSElamp questions (data shown in Table IV and Table V),students ’ performance was graded using rubrics (the TABLE II. Percentages of undergraduate students (US) andgraduate students (GS) who used WPI reasoning out of those whoprovided reasoning on DSE polarizer questions 2-5 (Q2-Q5) inthe DSE pretest. All these students worked on the MZI QuILTprior to taking the DSE pretest. We note that on average, 67% ofthe undergraduate students and 28% of the graduate studentsprovided reasoning for their answers. There data are based on 46undergraduate and 45 graduate students.Q2 Q3 Q4 Q5US 37 37 57 62GS 33 20 60 44TABLE I. Summary of the rubric used to grade students ’ performance on the DSE polarizer questions.Recognize that photons exhibit interference þ , 0Recognize that only a when two polarizers areorthogonal and there is no “ quantum eraser ”— the interference pattern vanishes þ , 0Recognize that “ which-path ” informationcan be lost þ , 0Correctly interpret the effect of one polarizeron the interference pattern þ , 0Correctly interpret the effect of two polarizerson the interference pattern (both questions) þ , 1, 0 a if a student said that the interference pattern vanishes in morethan 1 situation → points. MARIES, SAYER, and SINGH PHYS. REV. PHYS. EDUC. RES.13,
For student performance on a group of questions, inparticular, the DSE polarizer questions and the DSElamp questions (data shown in Table IV and Table V),students ’ performance was graded using rubrics (the TABLE II. Percentages of undergraduate students (US) andgraduate students (GS) who used WPI reasoning out of those whoprovided reasoning on DSE polarizer questions 2-5 (Q2-Q5) inthe DSE pretest. All these students worked on the MZI QuILTprior to taking the DSE pretest. We note that on average, 67% ofthe undergraduate students and 28% of the graduate studentsprovided reasoning for their answers. There data are based on 46undergraduate and 45 graduate students.Q2 Q3 Q4 Q5US 37 37 57 62GS 33 20 60 44TABLE I. Summary of the rubric used to grade students ’ performance on the DSE polarizer questions.Recognize that photons exhibit interference þ , 0Recognize that only a when two polarizers areorthogonal and there is no “ quantum eraser ”— the interference pattern vanishes þ , 0Recognize that “ which-path ” informationcan be lost þ , 0Correctly interpret the effect of one polarizeron the interference pattern þ , 0Correctly interpret the effect of two polarizerson the interference pattern (both questions) þ , 1, 0 a if a student said that the interference pattern vanishes in morethan 1 situation → points. MARIES, SAYER, and SINGH PHYS. REV. PHYS. EDUC. RES.13, (cid:129)
For the data shown in Table V, we used a simple t testfor both populations because we were comparing theperformance of each group (undergraduate and gradu-ate students) from before to after working on the DSEQuILT. We also report effect sizes (Cohen ’ s d [38]) forthe improvement in performance on the DSE polarizerquestions from before to after working on the DSEQuILT. The guidelines for interpreting Cohen ’ s d arethat a value of 0.2 corresponds to a small effect, avalue of 0.5 to a medium effect and a value of 0.8 to alarge effect [38]. (cid:129) For the data in Table p values were obtained byconducting MacNemar ’ s test, and the effect sizes wereport are Cramer ’ s V (equivalent to phi when 2groups are being compared) [38]. The general guide-lines for Cramer ’ s V (when two groups are beingcompared) is that a value of 0.1 corresponds to a smalleffect, 0.3 to a medium effect and 0.5 to a largeeffect [38]. (cid:129) For the data in Table IV, we used a one way repeatedmeasures ANOVA [38] to investigate the interactionsbetween condition (working on the MZI QuILT vsnot working on the MZI QuILT) and two performancemeasures (lamp questions and polarizer questions).Also, in all cases for which undergraduate performanceor percentage correct is shown, the data are based on all46 undergraduate students. For graduate students, there are either 45 or 14 depending on the cohort (45 in theMZI → DSE cohort, 14 in the DSE → MZI cohort).
IV. RESULTS
RQ1.a. What percentage of students use WPI reasoningbefore working on the MZI QuILT and how does thatcompare to after working on the MZI QuILT?
For the MZI polarizer questions, we found that withoutany prior instruction related to WPI, only one graduatestudent (out of 45) used WPI reasoning and only to answerone question. None of the 46 undergraduate students usedthis reasoning. For the DSE polarizer questions, we alsofound that prior to any instruction on WPI (from the MZIQuILT), only one graduate student (out of 14) used WPI
TABLE III. Percentage of graduate students from theDSE → MZI cohort and the MZI → DSE cohort who answeredDSE polarizer questions Q1 through Q5 correctly on the DSEpretest. In the table, N refers to the number of graduate students ineach cohort. N Q1 Q2 Q3 Q4 Q5DSE → MZI cohort 14 77 8 58 80 50MZI → DSE cohort 45 88 46 78 81 71 TABLE V. Performance of undergraduate (UG) and graduatestudents (GS) on the DSE polarizer questions before and afterworking on the DSE QuILT and p values comparing theperformance before to after working on the DSE QuILT foreach group (obtained via a simple t test) as well as effect sizes(Cohen ’ s d ). N refers to the number of undergraduate or graduatestudents. Std. dev. refers to standard deviation.DSE polarizer questions N Average Std. dev.UG-before QuILT 46 57% 35%UG-after QuILT 46 88% 20%UG p value < . UG effect size 1.09GS-before QuILT 59 60% 35%GS-after QuILT 59 75% 28%GS p value 0.018GS effect size 0.47TABLE IV. Performance of two graduate student cohorts(depending on the order in which they worked on the MZIand DSE tutorials) on the DSE transfer questions and on the DSElamp questions in the DSE pretest. The p values were obtained bycarrying out a one-way repeated measures ANOVA to identify thesignificance of the interactions between condition (MZI → DSEcohort or DSE → MZI cohort) and two performance measures(DSE lamp questions or DSE polarizer questions). We also reportCohen ’ s d effect sizes to compare the performance of the twodifferent cohorts. Std. dev. refers to standard deviation.MZI → DSEcohort( N ¼ ) DSE → MZIcohort( N ¼ )Average Std.dev. Average Std.dev. p Cohen ’ s d DSE polarizerquestions 65% 13% 38% 26% 0.011 0.831DSE lampquestions 42% 34% 42% 27% 0.955 0.016 Phi, or the phi coefficient, defines the strength ofthe relationship described in a × contingency table (in otherwords, phi is a measure of the effect of the difference betweentwo groups for nominal data, i.e., students who used WPIreasoning, students who didn ’ t use WPI reasoning before andafter working on the DSE QuILT; in our case, phi is a measure ofthe effect size when comparing the percentage of undergraduate/graduate students who used WPI reasoning before working on theDSE QuILT to after working on the DSE QuILT). Cramer ’ s V isan extension of the phi coefficient for contingency tables withmore than 2 rows and 2 columns (i.e., similar to how ANOVA isan extension to a t test when more than 2 groups are beingcompared). For more information see Ref. [38]. EFFECTIVENESS OF INTERACTIVE TUTORIALS … PHYS. REV. PHYS. EDUC. RES.13,
TABLE III. Percentage of graduate students from theDSE → MZI cohort and the MZI → DSE cohort who answeredDSE polarizer questions Q1 through Q5 correctly on the DSEpretest. In the table, N refers to the number of graduate students ineach cohort. N Q1 Q2 Q3 Q4 Q5DSE → MZI cohort 14 77 8 58 80 50MZI → DSE cohort 45 88 46 78 81 71 TABLE V. Performance of undergraduate (UG) and graduatestudents (GS) on the DSE polarizer questions before and afterworking on the DSE QuILT and p values comparing theperformance before to after working on the DSE QuILT foreach group (obtained via a simple t test) as well as effect sizes(Cohen ’ s d ). N refers to the number of undergraduate or graduatestudents. Std. dev. refers to standard deviation.DSE polarizer questions N Average Std. dev.UG-before QuILT 46 57% 35%UG-after QuILT 46 88% 20%UG p value < . UG effect size 1.09GS-before QuILT 59 60% 35%GS-after QuILT 59 75% 28%GS p value 0.018GS effect size 0.47TABLE IV. Performance of two graduate student cohorts(depending on the order in which they worked on the MZIand DSE tutorials) on the DSE transfer questions and on the DSElamp questions in the DSE pretest. The p values were obtained bycarrying out a one-way repeated measures ANOVA to identify thesignificance of the interactions between condition (MZI → DSEcohort or DSE → MZI cohort) and two performance measures(DSE lamp questions or DSE polarizer questions). We also reportCohen ’ s d effect sizes to compare the performance of the twodifferent cohorts. Std. dev. refers to standard deviation.MZI → DSEcohort( N ¼ ) DSE → MZIcohort( N ¼ )Average Std.dev. Average Std.dev. p Cohen ’ s d DSE polarizerquestions 65% 13% 38% 26% 0.011 0.831DSE lampquestions 42% 34% 42% 27% 0.955 0.016 Phi, or the phi coefficient, defines the strength ofthe relationship described in a × contingency table (in otherwords, phi is a measure of the effect of the difference betweentwo groups for nominal data, i.e., students who used WPIreasoning, students who didn ’ t use WPI reasoning before andafter working on the DSE QuILT; in our case, phi is a measure ofthe effect size when comparing the percentage of undergraduate/graduate students who used WPI reasoning before working on theDSE QuILT to after working on the DSE QuILT). Cramer ’ s V isan extension of the phi coefficient for contingency tables withmore than 2 rows and 2 columns (i.e., similar to how ANOVA isan extension to a t test when more than 2 groups are beingcompared). For more information see Ref. [38]. EFFECTIVENESS OF INTERACTIVE TUTORIALS … PHYS. REV. PHYS. EDUC. RES.13, –
5, thepercentages of both undergraduate and graduate students(from the MZI → DSE cohort) who used WPI reasoningout of the students who provided any reasoning for theiranswers on the DSE pretest. All these students worked onthe MZI QuILT before answering the DSE polarizerquestions, but had not worked on the DSE QuILT. Wenote that despite the fact that all questions explicitly askedfor reasoning, some students did not provide any reasoning.On average, 67% of the undergraduate students and 28%of the graduate students provided reasoning for theiranswers on the DSE polarizer questions in the DSE pretest.However, it is possible that some of the students who didnot explicitly write down their reasoning may haveanswered the questions by reasoning about WPI, and ifthat was the case, that would imply these students learnedhow to use WPI reasoning from the MZI QuILT, just thatthey did not explicitly provide their reasoning. Table IIshows that students who worked on the MZI QuILT beforeanswering the DSE polarizer questions on the DSE pretestoften used WPI reasoning to answer these questions,especially on the last two questions. In addition, out ofall instances in which a graduate or undergraduate studentused WPI reasoning and described their reasoning toanswer a question, he/she used it correctly 79% of thetime, thus indicating appropriate usage of WPI reasoninglearned in the MZI context to answer questions in the DSEcontext. In contrast, as mentioned earlier, students almostnever used such reasoning when answering the MZI orDSE polarizer questions when they did not have theopportunity to learn about the concept of WPI from theMZI QuILT. This suggests that students ’ use of the WPIreasoning on the DSE polarizer questions may be due torecognizing how to use this reasoning which they learned inthe MZI QuILT to answer questions in a different con-text (DSE). RQ1.b. How well do students who have worked on theMZI QuILT perform on the DSE polarizer and lampquestions compared to students who have not worked onthe MZI QuILT?
Table III shows the percentage of graduate students fromthe DSE → MZI and MZI → DSE cohorts who answeredDSE polarizer questions Q1 through Q5 correctly on theDSE pretest. Although the numbers are too small toperform meaningful statistics on each individual question,Table III suggests that students who had the opportunity tolearn from the MZI QuILT were more likely to answer thesequestions correctly (meaningful statistics can, however, beperformed on the aggregate data, i.e., overall performanceon the DSE polarizer questions; see Table IV and thediscussion related to the data shown in Table IV). Table IV shows the average performance of the twograduate student cohorts on the DSE polarizer questions (asgraded using the rubric shown in Table I), as well as theirperformance on the DSE lamp questions, which were quitedifferent from the DSE polarizer questions. Students inthe MZI → DSE cohort worked on the MZI QuILT beforeanswering these questions whereas students in the DSE → MZI cohort did not. A repeated measures ANOVA carriedout on these data showed a statistically significant inter-action between working on the MZI QuILT and perfor-mance on the polarizer questions ( F ð ; Þ ¼ . , p ¼ . ). The other interaction (working on MZIQuILT and lamp questions) was not significant, thussuggesting that the MZI QuILT helped students on theDSE polarizer questions only. Furthermore, the effect size[38] for comparing the performance on the DSE polarizerquestions of graduate students who worked on the MZIQuILT with the performance of those who did not was 0.83,thus suggesting a large effect of working on the MZI QuILTon these questions. RQ2. To what extent is the DSE QuILT effective inpromoting WPI reasoning from one context of the DSE(single particles and a monochromatic lamp placedbetween the slits and the screen) to a different contextof the DSE (single photons and polarizers placed in front ofone or both slits) without an instructional interventiondesigned to help them make the connection between thesedifferent contexts?
Table V shows the overall performance of undergraduateand graduate students (averages and standard deviations)on the DSE polarizer questions as graded by the rubricshown in Table I both before working on the DSE QuILTand after (all graduate students from both cohorts areincluded in the graduate student data). Table V alsolists p values obtained via a simple t test and effect sizes(Cohen ’ s d [38]) comparing students ’ performance frombefore to after working on the DSE QuILT. The p valuesshow that both undergraduate and graduate studentsimproved significantly. The effect is large for undergradu-ate students, but only medium for graduate students. Theimprovement may seem surprising because the DSE QuILTdid not address any of the situations in the DSE polarizerquestions at all, and did not even mention interference of photons in the DSE. We discuss some possible reasons forthis improvement in detail in Sec. V.Table VI shows, for DSE questions 2 –
5, the percentagesof both undergraduate and graduate students (similarly toTable V, all the graduate students are included in these data)who provided reasoning related to WPI among those whoprovided any reasoning for their answers both before andafter working on the DSE QuILT. As mentioned earlier,students did not always provide reasoning for their answerseven though the questions explicitly asked for reasoning.The percentages of both undergraduate and graduatestudents who provided reasoning on the DSE polarizerMARIES, SAYER, and SINGH PHYS. REV. PHYS. EDUC. RES.13,
5, the percentagesof both undergraduate and graduate students (similarly toTable V, all the graduate students are included in these data)who provided reasoning related to WPI among those whoprovided any reasoning for their answers both before andafter working on the DSE QuILT. As mentioned earlier,students did not always provide reasoning for their answerseven though the questions explicitly asked for reasoning.The percentages of both undergraduate and graduatestudents who provided reasoning on the DSE polarizerMARIES, SAYER, and SINGH PHYS. REV. PHYS. EDUC. RES.13,
RQ1.a ); after workingon the DSE QuILT, on average 96% of the undergraduatestudents provided reasoning for their answers while onaverage, 49% of the graduate students provided reasoningfor their answers. The p values listed in Table VI show thatundergraduate students were statistically significantly morelikely to provide reasoning related to WPI after working onthe DSE QuILT on three out of the four questions, and forthe graduate students it was two out of the four questions.The effect sizes (Cramer ’ s V) shown in Table VI suggestthat for most questions the magnitude of the effect ismedium. Given that students who used WPI reasoningused it correctly 79% of the time, it appears that increasedusage of WPI reasoning may play an important role in theimprovement observed in Table V for both graduate andundergraduate students. V. DISCUSSION
As evidenced in Tables V and VI, both the graduate andundergraduate students exhibited improved performanceon the DSE polarizer questions after working on the DSEQuILT, and they were also more likely to make use of WPIreasoning to motivate their answers (and most studentswho used WPI reasoning did so correctly). This improvedperformance on the DSE polarizer questions may seemsurprising. However, the DSE QuILT did guide studentsthrough the concept of WPI and how it can be used todetermine whether interference is observed in the DSE withsingle particles when a monochromatic lamp which emitsphotons that scatter with the particles (with mass) is placedbetween the slits and the screen. In some of these situations,scattering between the particles emitted by the source andthe photons emitted by the lamp can provide WPI for theparticles and destroy the interference pattern. It is possiblethat students who engaged with the DSE QuILT deeply can recognize on their own how this type of WPI reasoning canbe applied to answer the DSE polarizer questions.To test this hypothesis we conducted think-aloud inter-views with students who had completed the study ofModern Physics 1, which (typically) discusses the basicset up of the DSE. In an interview, students answered theDSE pretest questions, worked on the DSE QuILT, and thenanswered the DSE post-test questions while thinking aloud.We note that these students had not worked on the MZIQuILT so there was no possibility of transfer of the WPIconcept and its relation to interference from the MZIcontext to the DSE context. Students were not disturbedduring the interviews except when they became quiet for along time, in which case the interviewer prompted thestudent to keep talking. After working on each part (e.g.,pretest), students were asked for clarification on points theyhad not made clear earlier while talking aloud. We alsoshould stress that this qualitative investigation is not theprimary focus of this investigation and thus the datacollected was not analyzed in great detail, e.g., by tran-scribing and coding it, but rather the interviewer paid closeattention to how students were guided by the DSE QuILTand took careful notes during the interviews.The interviews suggested that the DSE QuILT helpedstudents reason using WPI to determine the patternobserved on the screen for a given DSE setup. In manycases, they were able to transfer this reasoning correctly tothe DSE polarizer questions. For example Andrew, oneinterviewed student, when answering DSE polarizer ques-tion 3 (a vertical polarizer placed in front of each slit) beforeworking on the DSE QuILT noted that a full interferencepattern will form, however, he was not sure why. When theinterviewer probed further (after the student had answeredall pretest questions) it appeared that the student wasprimarily guessing on this question and he did not havea very good reason for his answer. On the other hand, afterworking on the DSE QuILT, when answering the samequestion he said: “ There will be interference. If the photonis vertical (vertically polarized), there is no which pathknowledge, so there is interference. If (the photon is)horizontal, it doesn ’ t go through. ” Thus, Andrew reasoned correctly using the concept ofWPI, which he had learned in the DSE QuILT in com-pletely different situations involving placing a monochro-matic lamp between the slits and the screen for a DSE withsingle particles with mass instead of using single photonsand placing polarizers of various orientations in front ofone or both slits (polarizer questions). After working on theDSE QuILT, Andrew used WPI reasoning to answer theother DSE polarizer questions, and for the most part, usedthis reasoning correctly. For example, on DSE polarizerquestion 4 (two orthogonal polarizers) he recognized thatWPI is known for all photons and therefore no interferenceis observed on the screen.
TABLE VI. Percentage of undergraduate (UG) and graduate(GS) students who used WPI reasoning among those whoprovided reasoning on DSE polarizer questions 2 – p values and effect sizes for comparing these percentages frombefore to after working on the DSE QuILT via MacNemar ’ stests. N Q2 Q3 Q4 Q5UG-before QuILT 46 37 37 57 62UG-after QuILT 46 88 52 87 88 p value < . p value 0.070 0.016 0.073 0.013Effect size 0.23 0.43 0.32 0.42 EFFECTIVENESS OF INTERACTIVE TUTORIALS … PHYS. REV. PHYS. EDUC. RES.13,
TABLE VI. Percentage of undergraduate (UG) and graduate(GS) students who used WPI reasoning among those whoprovided reasoning on DSE polarizer questions 2 – p values and effect sizes for comparing these percentages frombefore to after working on the DSE QuILT via MacNemar ’ stests. N Q2 Q3 Q4 Q5UG-before QuILT 46 37 37 57 62UG-after QuILT 46 88 52 87 88 p value < . p value 0.070 0.016 0.073 0.013Effect size 0.23 0.43 0.32 0.42 EFFECTIVENESS OF INTERACTIVE TUTORIALS … PHYS. REV. PHYS. EDUC. RES.13, “ So thereare two cases to consider: one where there ’ s ( … ) ahorizontal photon coming in and the other is when there ’ sa vertical photon coming in. So if it ’ s a horizontal photoncoming in, it only goes through the right one [slit withhorizontal polarizer] and you get an [interference] pattern,and if the vertical one [photon] comes in, it only goesthrough the left one and you get an [interference] pattern. Idon ’ t know if those patterns are going to overlap ( … ) Ifthey overlap you ’ d just get a normal [interference] pattern,but if they don ’ t overlap, you ’ d get a continuum [randombackground] ” Discussions suggest that initially John thought that boththe horizontally and the vertically polarized photons willcreate an interference pattern, and depending on where thetwo patterns form, they can either overlap perfectly, or areoffset by a half of a wavelength so that the highs of onepattern overlap over the lows of the other pattern to producean overall homogeneous distribution.On the other hand, after working on the DSE QuILT,John correctly reasoned that a horizontally and a verticallypolarized photon each goes through only one slit, andtherefore no interference pattern is observed because WPIis known. In all the questions with polarizers, he reasonedby thinking about WPI, which is a concept he learned inthe DSE QuILT in a different context. Interestingly, whenreading the first DSE polarizer question in the post-test hestated “ Hmm … So I don ’ t think this was in the tutorial, but Iassume something in the tutorial should help me answerthese (questions) ” . It appeared that he was able to use whathe learned about how gaining WPI affects the patternobserved on the screen to reason about the DSE polarizerquestions. It is possible that similar reasoning applies toother students like John who improved on the DSE polarizerquestions after working on the DSE QuILT, which did notdiscuss the setups in the DSE polarizer questions.It is important to keep in mind that these students onlyworked on the DSE QuILT and were not exposed to theMZI QuILT at all. It appears that they were able to makeconnections between what they learned in the DSE QuILT,in particular how to reason in terms of WPI to determinewhether an interference pattern is formed, to answer theDSE polarizer questions. It is possible that if they hadalso worked on the MZI QuILT earlier, they would havebeen able to make connections between the type of WPIreasoning used in the MZI context and similar reasoningused in the DSE context. In that case, working on bothtutorials is likely to consolidate their knowledge of WPIfurther and can lead to even better performance on the post-test, similar to the undergraduate students for whom both tutorials were a part of their course, as shown in the resultssection. We note that the interviews provide a good startingpoint for understanding possible reasons for the QuILTspromoting WPI reasoning, and future investigations willprobe these issues further.Finally, we should note that even though the DSE QuILTdid not discuss interference of single photons, sincestudents answered the DSE polarizer questions as part ofthe DSE pretest, it is possible that this may have primedthem to think about the DSE polarizer questions whileworking on the DSE QuILT, and this may have partlyhelped them make connections between what they werelearning in the DSE QuILT (using WPI reasoning toexplain interference for particles with mass) and theDSE polarizer questions. While this may have aided them,we also note that giving a pretest before instruction in aparticular topics and then giving the identical post-test is acommon practice in introductory physics (for example,giving the Force Concept Inventory [39] before and afterinstruction). However, in the context of introductoryphysics, it has been found that giving the FCI as a pretestdoes not bias post-test results [40]. VI. SUMMARY
In this study, we find evidence that a QuILT on the MZIwas effective in promoting WPI reasoning from the MZIcontext to help upper-level undergraduate and graduatestudents answer isomorphic questions in the context of theDSE. The MZI QuILT introduced students to the concept ofWPI and guided them to use this concept to reason aboutwhether or not interference is observed at the detectors in aparticular MZI setup. Among students who did not work onthe MZI QuILT, almost none of them made use of WPIreasoning when answering either the MZI or DSE polarizerquestions. In contrast, after working on the MZI QuILT,the percentages of students who used WPI reasoning on theDSE polarizer questions (among those who providedreasoning) ranges from 20% to 60% for the graduatestudents and 37% to 62% for the undergraduate students.Additionally, the graduate students who worked on the MZIQuILT before answering the DSE polarizer questions onthe pretest performed significantly better on these questionsthan the graduate students who did not. These two cohortsof graduate students showed identical performance onthe other DSE questions, which did not have analogoussituations discussed in the MZI QuILT, suggesting that theimproved performance on the DSE polarizer questions islikely due to the MZI QuILT helping students discern theunderlying principles required to answer questions aboutinterference by using WPI reasoning. We note however thatthe number of graduate students the DSE → MZI cohortwas small (14), and thus it is possible that the encouragingresults can be at least in part accounted for by the smallnumber of students. We also note that due to lack ofparticipation from faculty members teaching the upper levelMARIES, SAYER, and SINGH PHYS. REV. PHYS. EDUC. RES.13,
In this study, we find evidence that a QuILT on the MZIwas effective in promoting WPI reasoning from the MZIcontext to help upper-level undergraduate and graduatestudents answer isomorphic questions in the context of theDSE. The MZI QuILT introduced students to the concept ofWPI and guided them to use this concept to reason aboutwhether or not interference is observed at the detectors in aparticular MZI setup. Among students who did not work onthe MZI QuILT, almost none of them made use of WPIreasoning when answering either the MZI or DSE polarizerquestions. In contrast, after working on the MZI QuILT,the percentages of students who used WPI reasoning on theDSE polarizer questions (among those who providedreasoning) ranges from 20% to 60% for the graduatestudents and 37% to 62% for the undergraduate students.Additionally, the graduate students who worked on the MZIQuILT before answering the DSE polarizer questions onthe pretest performed significantly better on these questionsthan the graduate students who did not. These two cohortsof graduate students showed identical performance onthe other DSE questions, which did not have analogoussituations discussed in the MZI QuILT, suggesting that theimproved performance on the DSE polarizer questions islikely due to the MZI QuILT helping students discern theunderlying principles required to answer questions aboutinterference by using WPI reasoning. We note however thatthe number of graduate students the DSE → MZI cohortwas small (14), and thus it is possible that the encouragingresults can be at least in part accounted for by the smallnumber of students. We also note that due to lack ofparticipation from faculty members teaching the upper levelMARIES, SAYER, and SINGH PHYS. REV. PHYS. EDUC. RES.13, → MZI condition for undergraduatestudents. However, our research with the DSE QuILT [34]and the MZI QuILT [37] indicated that the undergraduatestudents learned more from these QuILTs compared tograduate students. In both the DSE and MZI, on thepretest, undergraduate students ’ performance was loweron average than graduate students ’ performance, but on thepost-test, undergraduate students outperformed graduatestudents, thus indicating higher normalized gains for theundergraduate students. In this study too, we found thatundergraduate students ’ performance on the DSE polarizerquestions improved more than graduate students ’ performance — data shown in Table V. We hypothesize thatif we had investigated the DSE → MZI condition forundergraduate students, we may have found similar, ifnot more encouraging results.In addition, we found that after working on the DSEQuILT, both undergraduate and graduate students improvedsignificantly on the DSE polarizer questions despite thefact that the DSE QuILT did not mention anything aboutpolarizers thus indicating that the DSE QuILT also pro-moted WPI reasoning from one context to another.Interviews with students who worked only on the DSEQuILT suggest that this improved performance may partlybe due to students correctly recognizing the utility of WPIreasoning when considering situations described in theDSE polarizer questions. It is likely that students who workon the MZI QuILT before working on the DSE QuILT andengage with both tutorials well, e.g., the undergraduateswho worked on both QuILTs as part of their quantummechanics course, consolidate their knowledge of WPIfurther by making connections between the DSE and MZIcontexts.Finally, we note that the results of this investigationcan be interpreted from the lens of transfer of learning[20,41 – ACKNOWLEDGMENTS
We thank the National Science Foundation for GrantNo. PHY-1505460, Albert Huber for creating the simu-lation used in the MZI QuILT, and Klaus Muthsam forcreating the simulation used in the DSE QuILT. We alsothank F. Reif, R. P. Devaty, and E. Marshman for helpfuldiscussions.
APPRENDIX A: ISOMORPHISM BETWEENMZI AND DSE QUESTIONS
Since WPI reasoning can be used to reason aboutinterference in the MZI and DSE, we begin by definingWPI: in general, when a detector can project both compo-nents of the path state, then WPI is unknown. On the otherhand, when a detector can project only one component ofthe path state, then we have complete which path informa-tion, or WPI is known.We first consider the most basic MZI setup shown inFig. 3. BS1 and BS2 are beam splitters; BS1 is orientedsuch that it puts the single photon emitted from the sourceinto an equal superposition of the U and L path statesshown (which we represent as j U i and j L i , respectively).Mirrors are for proper alignment. BS2 ensures that thecomponents of the single photon state from both the U and L paths can be projected into each (photo) detector D1 andD2 after BS2 so that constructive or destructive interference(or anything in between) can be observed (depending on thepath length difference between the U and L paths). If an additional detector is placed anywhere in the lower path L between BS1 and BS2, after encountering the detector,the superposition of the U and L path states of a photoncollapses and if the photon does not get absorbed by thedetector, the state of the photon inside the MZI is the upperpath state j U i . Conversely, if an additional detector isplaced in the upper path U , after encountering the detector,if the photon is not absorbed by that detector, the state ofthe photon inside the MZI collapses to the lower path state j L i . In these situations (additional detector in the U or L path of the MZI), if a photon arrives at the detector D1 orD2 after BS2, we have WPI because either detector canonly project the component of the photon state along theU or L path and no interference is observed at D1 or D2. Ifinstead, no detector is placed in either of the U or L path ofEFFECTIVENESS OF INTERACTIVE TUTORIALS … PHYS. REV. PHYS. EDUC. RES.13,
Since WPI reasoning can be used to reason aboutinterference in the MZI and DSE, we begin by definingWPI: in general, when a detector can project both compo-nents of the path state, then WPI is unknown. On the otherhand, when a detector can project only one component ofthe path state, then we have complete which path informa-tion, or WPI is known.We first consider the most basic MZI setup shown inFig. 3. BS1 and BS2 are beam splitters; BS1 is orientedsuch that it puts the single photon emitted from the sourceinto an equal superposition of the U and L path statesshown (which we represent as j U i and j L i , respectively).Mirrors are for proper alignment. BS2 ensures that thecomponents of the single photon state from both the U and L paths can be projected into each (photo) detector D1 andD2 after BS2 so that constructive or destructive interference(or anything in between) can be observed (depending on thepath length difference between the U and L paths). If an additional detector is placed anywhere in the lower path L between BS1 and BS2, after encountering the detector,the superposition of the U and L path states of a photoncollapses and if the photon does not get absorbed by thedetector, the state of the photon inside the MZI is the upperpath state j U i . Conversely, if an additional detector isplaced in the upper path U , after encountering the detector,if the photon is not absorbed by that detector, the state ofthe photon inside the MZI collapses to the lower path state j L i . In these situations (additional detector in the U or L path of the MZI), if a photon arrives at the detector D1 orD2 after BS2, we have WPI because either detector canonly project the component of the photon state along theU or L path and no interference is observed at D1 or D2. Ifinstead, no detector is placed in either of the U or L path ofEFFECTIVENESS OF INTERACTIVE TUTORIALS … PHYS. REV. PHYS. EDUC. RES.13, U and L path states,WPI is unknown (because the detectors can project both the j U i and j L i components of the photon state) and thereforeinterference is observed at D1 and D2.Now consider the DSE setup shown in Fig. 4. If slit 2 isblocked, the state of a photon inside the DSE (after passingthrough the slits) collapses to, e.g., j Ψ i and if slit 1 isblocked, the state of a photon collapses to, e.g., j Ψ i . If thisphoton arrives at the screen (the screen is the detectiondevice in the DSE, equivalent to detectors D1 and D2 in theMZI), we have WPI because the screen can only projectone component of the photon ’ s path state (either j Ψ i or j Ψ i ) and therefore, no interference is observed. If neitherslit is blocked, the state remains an equal superposition of j Ψ i and j Ψ i . In other words, j U i and j L i in the MZI areanalogous to j Ψ i and j Ψ i in the DSE. In the situations inwhich there is no detector in either path of the MZI andneither slit is blocked for the DSE, we do not have WPI andeach photon interferes with itself.Now consider the situation shown in Fig. 5 in whichwe place a vertical polarizer in the upper path of the MZIand the source emits þ ° polarized single photons. Thissituation is analogous to the situation shown in Fig. 6 in theDSE in which a vertical polarizer is placed after slit 1 (andthe source emits þ ° polarized single photons). We nowhave to use a four dimensional Hilbert space, two dimen-sions for path/slit states, j U i , j L i = j Ψ i , j Ψ i , and twodimensions for polarization states, for which the convenientbasis for the situations described in Figs. 5 and 6 is fj V i ; j H ig (vertical, horizontal polarization states, respec-tively). If a vertical polarizer is placed in the upper path of the MZI, the j U i state will be associated with a verticalpolarization state ( j U ij V i ) and the j L i state is still asso-ciated with both vertical and horizontal polarization states( j L ij V i þ j L ij H i ). In both experiments we assume thatthe detectors are sensitive to polarization (they are coveredwith polarizers with a particular orientation, e.g., verticalor horizontal), which means that the collapse of thephoton state after it is measured by the detectors D1 orD2 provides information about the polarization of thephoton. Therefore, in the situation depicted in Fig. 5, wehave WPI for horizontally polarized photons arriving at D1and D2 because the horizontal polarization is associated withthe lower path state only — each detector can only project the j L i component of the state of a horizontally polarizedphoton. We do not have WPI for the vertically polarizedphotons because the vertical polarization is associated bothwith the upper and the lower path states — each detector canproject both the j L i and j U i components of the state of avertically polarized photon. The fact that we have WPI forhorizontally polarized photons and we do not have WPI forvertically polarized photons implies that the photons thatarrive at the detectors in the j V i state interfere and those inthe j H i state do not. In the DSE, the situation is analogous(Fig. 6): if a vertical polarizer is placed after slit 1,horizontally polarized photons arriving at the screen willnot interfere, while vertically polarized photons arriving atthe screen will show interference.It is important to note that while questions aboutinterference in the DSE and MZI contexts are isomorphic, FIG. 3. Basic MZI setup.FIG. 4. Basic DSE setup with single photons. FIG. 5. MZI setup with a vertical polarizer placed in theupper path.FIG. 6. DSE setup with a vertical polarizer placed after slit 1.
MARIES, SAYER, and SINGH PHYS. REV. PHYS. EDUC. RES.13,
MARIES, SAYER, and SINGH PHYS. REV. PHYS. EDUC. RES.13, “ surface ” features of these two experiments are ratherdifferent. In the MZI, the paths are restricted and thephotons arrive at point detectors D1 and D2, while in theDSE the photons are delocalized in the space betweenthe slits and the screen and can be detected anywhere on theextended screen. In addition, in the DSE, there is no explicitoptical element corresponding to BS2 in the MZI whichmixes the components of the photon state from the twopaths. These differences suggest that the surface featuresof these problems are quite different, which can make itchallenging for novices to recognize the isomorphism. Inorder to recognize the isomorphism between the MZI andDSE questions, students must be able to reason about thedeep features of the contexts and recognize the utility of theconcept of WPI and its relation to whether or not interfer-ence will take place in both contexts. Thus, even if studentsfully understand the underlying physics principles in theMZI context, they may have difficulty recognizing how thesame physics principles apply to the DSE.Also, it is worthwhile to keep in mind that while theupper-level undergraduates and graduate students havesome knowledge of the DSE, almost none of them havebeen introduced to the concept of WPI and learned how toreason using WPI to answer questions similar to the onesdiscussed here. We found that among the graduate studentswho were not introduced to WPI via our tutorials, only oneused WPI reasoning to answer only one question (out offive) on the DSE with polarizers placed in front of one slit.Similarly, in the MZI context before being introduced toWPI reasoning via tutorials, we also found that only onegraduate student used this reasoning. The physics under-graduate students in this study were almost all nearly at theend of the undergraduate curriculum (more than 80% wereseniors) and the physics graduate students were all in theirfirst year. Therefore, for the purposes of this study, the twopopulations (undergraduates and graduate students) are notvery different in terms of background knowledge on theDSE and MZI. APPRENDIX B: DSE PRETEST
Note that the DSE post-test was identical with someminor differences (e.g., electrons in one problem beingreplaced by Na atoms). Questions 4 – “ DSE lamp questions ” and questions 9(i) through9(v) are the DSE polarizer questions Q1 through Q5discussed at length in the article.For all questions, ignore relativistic effects. For allquestions that ask about a double slit setup, assume thatthe screen can detect the particles used and that the distancefrom the slits to the screen is much larger than the distancebetween the slits.For any constant, e.g., the mass of an electron or muon orPlanck ’ s constant, use the following values: (cid:129) eV ¼ . × − J (cid:129) keV ¼ kilo electron volt ¼ eV, meV ¼ millielectron volt ¼ − eV (cid:129) mm ¼ − m, μ m ¼ − m, nm ¼ − m, pm ¼ − m (cid:129) Planck ’ s constant ¼ h ¼ . × − Js (cid:129) magnitude of elementary charge ð on an electron orproton Þ ¼ e ¼ . × − C (cid:129) speed of light ¼ c ¼ . × m = s (cid:129) mass of electron ¼ . × − kg (cid:129) mass of neutron ¼ mass of proton ¼ . × − kg (cid:129) mass of muon ¼ . × − kg (cid:129) mass of helium atom ¼ . × − kg (cid:129) mass of sodium atom ¼ . × − kg
1. Pretest (1) What is the de Broglie relation? In one or twosentences, explain its significance.(2) You are conducting a double-slit experiment inwhich you send a large number of nonrelativisticelectrons of the same kinetic energy one at a timetowards a double-slit plate. The slit width is 50 pm,the slit separation is 1 nm and the distance betweenthe slits and the screen is 3 m.(i) If the wavelength of the electrons is 9 pm,describe the pattern you expect to observe onthe screen after a large number of electrons havepassed through. Explain your reasoning.(ii) Suppose the experiment is modified by usingprotons instead of electrons while all followingparameters are held fixed: kinetic energy, slitwidth, slit separation, distance between slitsand screen. How does the pattern change, ifat all?(iii) Explain your reasoning for your answer in2 (ii).(3) Consider particles of sand, which can be approxi-mated as spheres of a radius of about = of amillimeter.(i) Do you expect that a double slit experimentwith well-chosen parameters would show aninterference pattern?(ii) Explain your reasoning for your answer in 3 (i).In questions 4 –
8, assume that particles aresent one at a time from the particle source.Figure 7 shows a double-slit experiment whichwas modified by adding a lamp (light bulb)between the double slit and the screen (slightlyoff to the side so it is not directly in front ofthe slits). (cid:129)
Assume that when the lamp is turned on, if scatteringoccurs between a particle used in the double-slitexperiment and a photon from the lamp, this scatteringoccurs at the slits only. (cid:129)
Assume that ALL the particles scattered by photonsstill reach the screen.EFFECTIVENESS OF INTERACTIVE TUTORIALS … PHYS. REV. PHYS. EDUC. RES.13,
Assume that ALL the particles scattered by photonsstill reach the screen.EFFECTIVENESS OF INTERACTIVE TUTORIALS … PHYS. REV. PHYS. EDUC. RES.13, – (cid:129) If slit 2 is closed, the wave function of a Na atom thatgoes through slit 1 and arrives at a point x on thescreen is Ψ ð x Þ . If instead, slit 1 is closed, the wavefunction of a Na atom that goes through slit 2 andarrives at a point x on the screen is Ψ ð x Þ . (cid:129) For this example, if slit 2 is closed, and a total number N of particles arrives at the screen, the number densityof the particles at a point x on the screen is N j Ψ ð x Þj . (cid:129) For questions 5 –
8, both slits are open.(5) For (i) and (ii) below, suppose that the wavelength ofthe photons is significantly smaller than the distancebetween the slits and the intensity of the lamp is suchthat each Na atom scatters off a photon. Also,assume that all the scattered atoms still reach thescreen. (i) Write down an expression for the numberdensity of Na atoms at a point x on the screenin terms of Ψ ð x Þ and Ψ ð x Þ after a largenumber N of Na atoms arrive at the screen.(ii) Describe the pattern you expect to observe onthe screen after a large number N of Na atomshave arrived at the screen. Explain yourreasoning.(6) For (i) and (ii) below, suppose that the wavelength ofthe photons is significantly larger than the distancebetween the slits and the intensity of the lamp issuch that each Na atom scatters off a photon. Also,assume that all scattered atoms still reach the screen.(i) Write down an expression for the numberdensity of Na atoms at a point x on the screenin terms of Ψ ð x Þ and Ψ ð x Þ after a largenumber N of Na atoms arrive at the screen.(ii) Describe the pattern you expect to observe onthe screen after a large number N of Na atomshave arrived at the screen. How, if at all, isthis pattern different from the pattern in 5(ii)?Explain your reasoning.(7) For (i) and (ii) below, suppose that the wavelength ofthe photons is significantly smaller than the distancebetween the slits and the intensity of the lamp is suchthat about half of the Na atoms scatter off a photon.Also, both slits are open and all the atoms reach thescreen, including the ones that scatter.(i) Write down an expression for the numberdensity of Na atoms at a point x on the screenin terms of Ψ ð x Þ and Ψ ð x Þ after a largenumber N of Na atoms arrive at the screen.(ii) Describe the pattern you expect to observe onthe screen after a large number N of Na atoms FIG. 7. Double slit setup with the addition of a lamp (image reproduced with permission from a simulation developed by KlausMuthsam, [email protected]).
MARIES, SAYER, and SINGH PHYS. REV. PHYS. EDUC. RES.13,
MARIES, SAYER, and SINGH PHYS. REV. PHYS. EDUC. RES.13, x on the screenin terms of Ψ ð x Þ and Ψ ð x Þ after a largenumber N of Na atoms arrive at the screen.(ii) Describe the pattern you expect to observe onthe screen after a large number N of Na atomshave arrived at the screen. How, if at all, is thispattern different from the pattern in 6(ii)?Explain your reasoning.(9) You perform a double slit-experiment in whichphotons that are polarized at þ ° are sent one ata time towards the double slit. The wavelength of thephotons is comparable to the width of the slits andthe separation between the slits is more than twicethe slit width. In all parts (i) through (vi) below,assume that the same large number N of photonsreach the screen (in other words, you wait longenough in each case to clearly observe the patternthat forms on the screen).(i) Describe the pattern you expect to observe onthe screen after a large number N of photonsreach the screen. Explain your reasoning.(ii) Suppose that a vertical polarizer is placed infront of only one of the slits. Describe thepattern you expect to observe on the screen aftera large number N of photons reach the screen.How does this pattern differ, if at all, fromthe pattern observed in 9(i)? Explain yourreasoning.(iii) Suppose that a vertical polarizer is placed infront of each of the two slits. Describe thepattern you expect to observe on the screen aftera large number N of photons reach the screen.How does this pattern differ, if at all, fromthe pattern observed in 9(i)? Explain yourreasoning.(iv) Suppose that a vertical polarizer is placed infront of one of the slits and a horizontalpolarizer is placed in front of the other slit.Describe the pattern you expect to observe onthe screen after a large number N of photonsreach the screen. How does this pattern differ, ifat all, from the pattern observed in 9(i)? Explainyour reasoning.(v) Suppose that a vertical polarizer is placed infront of one of the slits and a horizontal polarizer is placed in front of the other slit.Furthermore, an additional polarizer whichmakes an angle of þ ° with the horizontalis placed right before the screen. Describe thepattern you expect to observe on the screen aftera large number N of photons reach the screen.How does this pattern differ, if at all, from thepattern observed in 9(i)? Explain. APPRENDIX C: COMMON STUDENTDIFFICULTIES
Here, we discuss common student difficulties on theDSE polarizer questions both before and after studentsworked on the DSE QuILT. Since the data were quali-tatively similar for the graduate students regardless ofwhether they had completed the MZI QuILT beforetaking the DSE pretest, the graduate students from allcohorts are combined. We also carried out think-aloudinterviews with undergraduate and graduate students tofurther understand the common types of incorrect rea-soning they used to answer these questions, which oftenprovided further insight into their difficulties.
Difficulties with interference of single photons — nopolarizers Among the students who answered Q1, the vastmajority of both undergraduate and graduate studentsanswered it correctly (clear interference pattern shown) asshown in Table VII. A small percentage of studentsselected answers which indicated that no interferencepattern is observed, but none provided reasoning for theiranswers. On the pretest, roughly one quarter of theundergraduate students and one sixth of the graduatestudents either did not respond or indicated that they didnot know whether photons will exhibit interference inthis case. These percentages drop to nearly zero in thepost-test.
TABLE VII. Percentages of undergraduate (UG) and graduatestudents (GS) with different answers on Q1 (interference, nointerference, other, and no response or “ I don ’ t know ” ). Bolditalic indicates correct responses.Interference Nointerference Other No response/ “ I don ’ t know ” UG-beforeQuILT
11 2 16GS-afterQuILT EFFECTIVENESS OF INTERACTIVE TUTORIALS … PHYS. REV. PHYS. EDUC. RES.13,
11 2 16GS-afterQuILT EFFECTIVENESS OF INTERACTIVE TUTORIALS … PHYS. REV. PHYS. EDUC. RES.13, ifficulties with the effect of one polarizer on theinterference pattern
Q2 involves a DSE in which a vertical polarizer is placedin front of only one of the slits. In this situation, WPI will beknown for horizontally polarized photons and will not beknown for vertically polarized photons (as explained in thesection discussing the isomorphism between the DSEand MZI questions). Therefore, the pattern observed onthe screen will consist of an interference pattern (neglectingsingle slit diffraction since the slit is sufficiently narrow)provided by the vertically polarized photons (which dointerfere) on top of a uniform background provided by thehorizontally polarized photons which do not interfere. Thiswas the most challenging question for both student pop-ulations. As shown in Table VIII, for both populations, themost common incorrect answer choice is that no interfer-ence is observed in this situation. Students with this answertypically reasoned that WPI is known for all photonsbecause the polarizer “ tags ” the photons that go throughit by polarizing them (this reasoning did not alwaysmention WPI explicitly). For example, one student stated: “ no interference because you are essentially ‘ tagging ’ halfthe photons ” and another stated “ no interference since thepolarizer tells us which slit the photon went through. ” Thisdifficulty is also common in the MZI context when avertical polarizer is placed in one of the paths: manystudents thought that no interference is observed at eitherdetector because the polarizer provides WPI for the photonsthat take that path by ‘ tagging ’ them.Interestingly, more graduate students use this type ofreasoning after working on the DSE QuILT than before.This may be because before working on the DSE QuILT,some students (21%) provided responses that were difficultto categorize, and some (16%) did not provide a response, but after working on the DSE QuILT, the majority of thesestudents provided responses that could be categorized,some of which used the incorrect reasoning that the verticalpolarizer provides WPI for vertically polarized photonsdetected at the screen.For students who attempted to explicitly reason in termsof WPI on the DSE polarizer questions, 67% of them(including both undergraduate and graduate students)reasoned correctly (note that this is the most challengingquestion for both undergraduate and graduate students). Forexample, one student wrote “ The interference pattern willbe fuzzier because we do have which-path data for anyphotons that are not vertically polarized ” (common correctreasoning) and another wrote “ I only see two lines on thescreen because we have which-path information about oneof the slits. ” The second student is using WPI reasoningincorrectly, but at the very least, he is recognizing that thisreasoning may be useful in the DSE context.
Difficulties with the effect of two polarizers on theinterference pattern
Q3 and Q4 evaluate student understanding of theeffect of two polarizers on the interference pattern.Students showed significant improvement after workingon the DSE QuILT on these two questions as shown inTables IX and X. Among the students who answered thesequestions before working on the DSE QuILT, the majorityof them answered them correctly. Also, on these questions,the performance of undergraduate students after workingon the DSE QuILT is close to 100%. It appears that theundergraduate students were able to use the concept ofWPI learned from the MZI QuILT to answer the DSEpolarizer questions, and also, after working on the DSEQuILT, they were able to consolidate their learning todevelop a solid understanding of the effect of two polarizers
TABLE VIII. Percentages of undergraduate (UG) and graduate students (GS) with different answers on Q2 (partialinterference, no interference, full interference, other, and no response/ ” I don ’ t know ” ). Bold italic indicates correctresponses. Partial interference No interference Full interference Other No response/ “ I don ’ t know ” UG-before QuILT
17 5 16 24UG-after QuILT
16 2 12 0GS-before QuILT
19 12 21 16GS-after QuILT
32 10 5 2TABLE IX. Percentages of undergraduate (UG) and graduate students (GS) with different answers on Q3 (fullinterference, partial interference, no interference, other, and no response/ “ I don ’ t know ” ). Bold italic indicatescorrect responses. Full interference Partial interference No interference Other No response/ “ I don ’ t know ” UG-before QuILT MARIES, SAYER, and SINGH PHYS. REV. PHYS. EDUC. RES.13,
32 10 5 2TABLE IX. Percentages of undergraduate (UG) and graduate students (GS) with different answers on Q3 (fullinterference, partial interference, no interference, other, and no response/ “ I don ’ t know ” ). Bold italic indicatescorrect responses. Full interference Partial interference No interference Other No response/ “ I don ’ t know ” UG-before QuILT MARIES, SAYER, and SINGH PHYS. REV. PHYS. EDUC. RES.13, N of photons reach the screen, this interfer-ence pattern is no different from the pattern observed whenno polarizers are placed after either slit. As shown inTable IX, the most common incorrect answer for both theundergraduate and graduate students is that there will be nointerference. A common incorrect reasoning, especiallybefore students worked on the DSE QuILT, is that in thissituation, WPI will be known for all photons.If a vertical polarizer is placed after one slit (say the topslit) and a horizontal polarizer is placed after the other slit(bottom slit) as in Q4, then WPI is known for all photonsbecause a horizontally polarized photon detected at thescreen must have gone through the bottom slit and avertically polarized photon detected at the screen musthave gone through the top slit. On this question, the mostcommon incorrect answer was that a full interferencepattern should form. Students who provided responses ofthis type may have had difficulty recognizing that thepolarizers provide WPI for all photons, or may believe thateven though WPI is known for all photons, an interferencepattern is still observed. For example, one graduate studentrecognized that WPI can be obtained both for a verticallyand a horizontally polarized photon detected at the screen,and concluded that neither horizontally nor verticallypolarized photons interfere with themselves. However,she thought that they can interfere with each other and said: “ I don ’ t know … would they [photons coming fromone slit] be able to interfere with the ones [photons] comingfrom the other slit … ? ” When probed further, she said “ If it [photon] can only gothrough one slit or the other it can ’ t interfere with itself, butonce it goes through it, there would still be wave propa-gation [ … ] would it [a vertically polarized photon] be ableto interfere with the horizontally polarized photons or not … I don ’ t know. ” When the interviewer asked, “ So what you ’ re saying isthat a single photon can only go through one slit or theother but you ’ re not sure if that implies that there ’ s nointerference because that photon might interfere withanother photon that ’ s coming through the other slit, is thatright? ” , she responded, “ Yeah. ” Difficulties with quantum eraser
The last situation (vertical polarizer after one slit,horizontal polarizer after the other, 45° polarizer in frontof the screen) is a quantum eraser because the last polarizererases WPI that could be obtained due to the effect of theother two polarizers. Table XI shows that the most commonincorrect answer for both undergraduate and graduatestudents was that there will be no interference in thissituation. Many students who provided these types ofresponses ignored the third polarizer. For example, onestudent stated “ I don ’ t think interference is possiblebecause you are still identifying the path of one side ofphotons as different from the other. ” Another student stated, “ See no interference since one is horizontally and theother vertically polarized. ” These types of reasoningindicate that students essentially ignored the effect of thethird polarizer, which erases WPI. As further evidence ofstudents recognizing the similarity between the MZI and
TABLE X. Percentages of undergraduate (UG) and graduate students (GS) with different answers on Q4 (partialinterference, no interference, full interference, other, and no response/ “ I don ’ t know ” ). Bold italic indicates correctresponses. Full interference Partial interference No interference Other No response/ “ I don ’ t know ” UG-before QuILT 10 0 “ I don ’ t know ” ), including percentages ofstudents who mention MZI or quantum eraser when responding to Q5. Bold italic indicates correct response.Fullinterference Partialinterference Nointerference Other No response/don ’ t know Mention MZI orquantum eraserUG-before QuILT
10 12 2 0 27
EFFECTIVENESS OF INTERACTIVE TUTORIALS … PHYS. REV. PHYS. EDUC. RES.13,
EFFECTIVENESS OF INTERACTIVE TUTORIALS … PHYS. REV. PHYS. EDUC. RES.13, “ quantum eraser ” or reasoned in a manner which could have been learned only in the context of the MZI(e.g., the third polarizer erases the WPI obtained from theother two polarizers) even though such things were notmentioned in the DSE QuILT. [1] C. Singh, Student understanding of quantum mechanics,Am. J. Phys. , 885 (2001).[2] D. Zollman and S. Rebello, Quantum mechanics foreveryone: Hands-on activities integrated with technology,Am. J. Phys. , 252 (2002).[3] R. Sayer et al. , A case study evaluating Just-in-TimeTeaching and Peer Instruction using clickers in a quantummechanics course, Phys. Rev. Phys. Educ. Res. , 020133(2016); E. Marshman and C. Singh, Investigating andimproving student understanding of the probability dis-tributions for measuring physical observables in quantummechanics, Eur. J. Phys. , 025705 (2017); Investigatingand improving student understanding of the expectationvalues of observables in quantum mechanics, Eur. J. Phys. , 045701 (2017); S. Siddiqui and C. Singh, How diverseare physics instructors ’ attitudes and approaches to teach-ing undergraduate-level quantum mechanics?, Eur. J. Phys. , 035703 (2017); C. Singh, M. Belloni, and W. Christian,Approaches for improving students ’ understanding ofquantum mechanics: Response to letters, Phys. Today , No. 3, 12 (2007).[4] G. Zhu and C. Singh, Improving students ’ understandingof quantum mechanics via the Stern-Gerlach experiment,Am. J. Phys. , 499 (2011).[5] P. Jolly, D. Zollman, S. Rebello, and A. Dimitrova, Visu-alizing motion in potential wells, Am. J. Phys. , 57 (1998).[6] M. C. Wittmann, R. N. Steinberg, and E. F. Redish, Inves-tigating student understanding of quantum physics: Sponta-neous models of conductivity, Am. J. Phys. , 218 (2002).[7] J. Morgan and M. Wittmann, Examining the evolution ofstudent ideas about quantum tunneling, in Proceedingsof the Physics Education Research Conference 2005, SaltLake City, UT , edited by P. Heron, L. McCullough, andJ. Marx (AIP, New York, 2006), pp. 73 – Proceedings of thePhysics Education Research Conference 2011, Omaha,NE , edited by N. Rebello, P. Engelhardt, and C. Singh(AIP, New York, 2012), pp. 55 – , 760 (2010).[10] C. Singh, M. Belloni, and W. Christian, Improving students ’ understanding of quantum mechanics, Phys. Today , 43(2006).[11] C. Singh, Assessing, and improving student understandingof quantum mechanics, AIP Conf. Proc. , 69 (2006).[12] C. Singh, Student difficulties with quantum mechanicsformalism, AIP Conf. Proc. , 185 (2007). [13] C. Singh, Helping students learn quantum mechanics forquantum computing, AIP Conf. Proc. , 42 (2007).[14] E. Marshman and C. Singh, Framework for under-standing the patterns of student difficulties in quantummechanics, Phys. Rev. ST Phys. Educ. Res. , 020119(2015).[15] C. Singh and E. Marshman, Review of student difficultiesin upper-level quantum mechanics, Phys. Rev. ST Phys.Educ. Res. , 020117 (2015).[16] M. Belloni, W. Christian, and D. Brown, Open sourcephysics curricular material for quantum mechanics,Comput. Sci. Eng. , 24 (2007).[17] A. Kohnle, I. Bozhinova, D. Browne, M. Everitt, A.Fomins, P. Kok, G. Kulaitis, M. Prokopas, D. Raine,and E. Swinbank, A new introductory quantum mechanicscurriculum, Eur. J. Phys. , 015001 (2014).[18] G. Passante, P. Emigh, and P. Shaffer, Investigating studentunderstanding of basic quantum mechanics in the contextof time-dependent perturbation theory, in Proceedingsof the Physics Education Research Conference 2013,Portland, OR , edited by A. Churukian, P. Engelhardt,and D. Jones (AIP, New York, 2014), pp. 269 – , 010121(2016).[20] C. Singh, Transfer of learning in quantum mechanics, AIPConf. Proc. , 23 (2005).[21] C. Singh, Student understanding of quantum mechanics atthe beginning of graduate instruction, Am. J. Phys. , 277(2008).[22] C. Singh, Interactive learning tutorials on quantummechanics, Am. J. Phys. , 400 (2008).[23] C. Singh and G. Zhu, Cognitive issues in learningadvanced physics: An Example from Quantum Mechanics,AIP Conf. Proc. , 63 (2009).[24] G. Zhu and C. Singh, Surveying students ’ understanding ofquantum mechanics in one spatial dimension, Am. J. Phys. , 252 (2012).[25] G. Zhu and C. Singh, Improving students ’ understandingof quantum measurement I: Investigation of difficulties,Phys. Rev. ST Phys. Educ. Res. , 010117 (2012).[26] G. Zhu and C. Singh, Improving students ’ understandingof quantum measurement II: Development of Research-based learning tools, Phys. Rev. ST Phys. Educ. Res. ,010118 (2012).[27] G. Zhu and C. Singh, Improving student understandingof addition of angular momentum in quantum mechanics,Phys. Rev. ST Phys. Educ. Res. , 010101 (2013). MARIES, SAYER, and SINGH PHYS. REV. PHYS. EDUC. RES.13,
EFFECTIVENESS OF INTERACTIVE TUTORIALS … PHYS. REV. PHYS. EDUC. RES.13, “ quantum eraser ” or reasoned in a manner which could have been learned only in the context of the MZI(e.g., the third polarizer erases the WPI obtained from theother two polarizers) even though such things were notmentioned in the DSE QuILT. [1] C. Singh, Student understanding of quantum mechanics,Am. J. Phys. , 885 (2001).[2] D. Zollman and S. Rebello, Quantum mechanics foreveryone: Hands-on activities integrated with technology,Am. J. Phys. , 252 (2002).[3] R. Sayer et al. , A case study evaluating Just-in-TimeTeaching and Peer Instruction using clickers in a quantummechanics course, Phys. Rev. Phys. Educ. Res. , 020133(2016); E. Marshman and C. Singh, Investigating andimproving student understanding of the probability dis-tributions for measuring physical observables in quantummechanics, Eur. J. Phys. , 025705 (2017); Investigatingand improving student understanding of the expectationvalues of observables in quantum mechanics, Eur. J. Phys. , 045701 (2017); S. Siddiqui and C. Singh, How diverseare physics instructors ’ attitudes and approaches to teach-ing undergraduate-level quantum mechanics?, Eur. J. Phys. , 035703 (2017); C. Singh, M. Belloni, and W. Christian,Approaches for improving students ’ understanding ofquantum mechanics: Response to letters, Phys. Today , No. 3, 12 (2007).[4] G. Zhu and C. Singh, Improving students ’ understandingof quantum mechanics via the Stern-Gerlach experiment,Am. J. Phys. , 499 (2011).[5] P. Jolly, D. Zollman, S. Rebello, and A. Dimitrova, Visu-alizing motion in potential wells, Am. J. Phys. , 57 (1998).[6] M. C. Wittmann, R. N. Steinberg, and E. F. Redish, Inves-tigating student understanding of quantum physics: Sponta-neous models of conductivity, Am. J. Phys. , 218 (2002).[7] J. Morgan and M. Wittmann, Examining the evolution ofstudent ideas about quantum tunneling, in Proceedingsof the Physics Education Research Conference 2005, SaltLake City, UT , edited by P. Heron, L. McCullough, andJ. Marx (AIP, New York, 2006), pp. 73 – Proceedings of thePhysics Education Research Conference 2011, Omaha,NE , edited by N. Rebello, P. Engelhardt, and C. Singh(AIP, New York, 2012), pp. 55 – , 760 (2010).[10] C. Singh, M. Belloni, and W. Christian, Improving students ’ understanding of quantum mechanics, Phys. Today , 43(2006).[11] C. Singh, Assessing, and improving student understandingof quantum mechanics, AIP Conf. Proc. , 69 (2006).[12] C. Singh, Student difficulties with quantum mechanicsformalism, AIP Conf. Proc. , 185 (2007). [13] C. Singh, Helping students learn quantum mechanics forquantum computing, AIP Conf. Proc. , 42 (2007).[14] E. Marshman and C. Singh, Framework for under-standing the patterns of student difficulties in quantummechanics, Phys. Rev. ST Phys. Educ. Res. , 020119(2015).[15] C. Singh and E. Marshman, Review of student difficultiesin upper-level quantum mechanics, Phys. Rev. ST Phys.Educ. Res. , 020117 (2015).[16] M. Belloni, W. Christian, and D. Brown, Open sourcephysics curricular material for quantum mechanics,Comput. Sci. Eng. , 24 (2007).[17] A. Kohnle, I. Bozhinova, D. Browne, M. Everitt, A.Fomins, P. Kok, G. Kulaitis, M. Prokopas, D. Raine,and E. Swinbank, A new introductory quantum mechanicscurriculum, Eur. J. Phys. , 015001 (2014).[18] G. Passante, P. Emigh, and P. Shaffer, Investigating studentunderstanding of basic quantum mechanics in the contextof time-dependent perturbation theory, in Proceedingsof the Physics Education Research Conference 2013,Portland, OR , edited by A. Churukian, P. Engelhardt,and D. Jones (AIP, New York, 2014), pp. 269 – , 010121(2016).[20] C. Singh, Transfer of learning in quantum mechanics, AIPConf. Proc. , 23 (2005).[21] C. Singh, Student understanding of quantum mechanics atthe beginning of graduate instruction, Am. J. Phys. , 277(2008).[22] C. Singh, Interactive learning tutorials on quantummechanics, Am. J. Phys. , 400 (2008).[23] C. Singh and G. Zhu, Cognitive issues in learningadvanced physics: An Example from Quantum Mechanics,AIP Conf. Proc. , 63 (2009).[24] G. Zhu and C. Singh, Surveying students ’ understanding ofquantum mechanics in one spatial dimension, Am. J. Phys. , 252 (2012).[25] G. Zhu and C. Singh, Improving students ’ understandingof quantum measurement I: Investigation of difficulties,Phys. Rev. ST Phys. Educ. Res. , 010117 (2012).[26] G. Zhu and C. Singh, Improving students ’ understandingof quantum measurement II: Development of Research-based learning tools, Phys. Rev. ST Phys. Educ. Res. ,010118 (2012).[27] G. Zhu and C. Singh, Improving student understandingof addition of angular momentum in quantum mechanics,Phys. Rev. ST Phys. Educ. Res. , 010101 (2013). MARIES, SAYER, and SINGH PHYS. REV. PHYS. EDUC. RES.13,
28] S. Y. Lin and C. Singh, Assessing Expertise in QuantumMechanics using Categorization Task, AIP Conf. Proc. , 185 (2009).[29] S. Y. Lin and C. Singh, Categorization of quantum me-chanics problems by professors and students, Eur. J. Phys. , 57 (2010).[30] R. Feynman et al. , Feynman Lectures on Physics (AddisonWesley, Reading, MA, 1989), Vol III.[31] M. Schneider and I. LaPuma, A simple experiment fordiscussion of quantum interference and which-way meas-urement, Am. J. Phys. , 266 (2002).[32] J. A. Wheeler, The “ Past ” and the “ Delayed-Choice ” Double-Slit Experiment, in
Mathematical Foundationsof Quantum Theory , edited by A. R. Marlow (AcademicPress, New York, 1979).[33] E. Marshman and C. Singh, Interactive tutorial to improvestudent understanding of single-photon experimentsinvolving a Mach-Zehnder interferometer, Eur. J. Phys. , 024001 (2016).[34] R. Sayer, A. Maries, and C. Singh, Quantum interactivelearning tutorial on the double-slit experiment to improvestudent understanding of quantum mechanics, Phys. Rev.Phys. Educ. Res. , 010123 (2017).[35] https://sites.google.com/site/quiltbeta/.[36] K. Ericsson and H. Simon, Verbal reports as data,Psychol. Rev. , 215 (1980).[37] E. Marshman and C. Singh, Investigating and improvingstudent understanding of quantum mechanics in the contextof single photon interference, Phys. Rev. Phys. Educ. Res. , 010117 (2017).[38] G. V. Glass and K. D. Hopkins, Statistical Methods inEducation and Psychology (Allyn & Bacon, Boston, MA,1996); A. Agresti,
Introduction to Categorical DataAnalysis (John Wiley and Sons, New York, 1996).[39] D. Hestenes, M. Wells, and G. Swackhammer, ForceConcept Inventory, Phys. Teach. , 141 (1992).[40] C. Henderson, Common concerns about the Force ConceptInventory, Phys. Teach. , 542 (2002).[41] M. Gick and K. Holyoak, The cognitive basis of knowl-edge transfer, in Transfer of Learning: Contemporary Research and Applications , edited by Cornier and Hagman(Academic Press, New York, NY, 1987), pp. 9 – , 010105 (2008).[43] H. A. Simon and J. R. Hayes, The understanding Process:Problem Isomorphs, Cogn. Psychol. , 165 (1976).[44] L. Cui, N. S. Rebello, and A. G. Bennett, College students ’ transfer from calculus to physics, AIP Conf. Proc. , 37(2006).[45] N. S. Rebello, D. A. Zollman, A. R. Allbaugh, P. V.Engelhardt, K. E. Gray, Z. Hrepic, and S. F. Itza-Ortiz,Dynamic dransfer: A perspective from Physics EducationResearch, in Transfer of Learning from a ModernMultidisciplinary Perspective , edited by J. P. Mester(Information Age Publishing, Greenwich, CT, 2005),pp. 217 – Proceedings of the NARST 2007 AnnualConference (New Orleans, LA 2007).[47] N. S. Rebello, L. Cui, A. G. Bennett, D. A. Zollman, andD. J. Ozimek, Transfer of learning in problem solving inthe context of mathematics and physics, in
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