Challenges in addressing student difficulties with measurement uncertainty of two-state quantum systems using a multiple-choice question sequence in online and in-person classes

Research-validated multiple-choice questions comprise an easy-to-implement instructional tool that serves to scaffold student learning and formatively assess students’ knowledge. We present findings from the implementation, in consecutive years, of a research-validated multiple-choice question sequence on measurement uncertainty as it applies to two-state quantum systems. This study was conducted in an advanced undergraduate quantum mechanics course, in online and in-person learning environments for consecutive years. Student learning was assessed after receiving traditional lecture-based instruction in relevant concepts, and their performance was compared with that of a similar assessment given after engaging with the multiple-choice question sequence. We analyze and discuss the similar and differing trends observed in the two modes of instruction.


Introduction
Measurement uncertainty in quantum mechanics (QM) is a foundational concept that has no classical analogue.Quantum measurement uncertainty is illustrated by the fact that when an observable is measured in a quantum state that is not an eigenstate of the corresponding Hermitian operator, measurement outcomes are not certain.In particular, the measurement collapses the state to one of the eigenstates of the operator corresponding to the observable with a certain probability, and if many measurements of this observable are conducted in an ensemble of identically-prepared systems, the standard deviation of those measurement outcomes is the measurement uncertainty.Furthermore, any subsequent measurements of the same observable made in a collapsed state, assuming no time-evolution, will yield the same outcome with 100% certainty.An observable is said to be well-defined when the quantum system is in an eigenstate of the corresponding operator.Also, the uncertainty principle states that if two observables correspond to operators which do not commute, then they cannot both be measured with 100% certainty, i.e., cannot both be welldefined, in the same quantum state.Because the uncertainty principle can be challenging for students, instructional resources have been developed to help students learn this concept in different situations.Measurement uncertainty and the uncertainty principle are fundamental tenets of quantum theory that are relevant in any context involving successive measurements of different observables, including fields of active research, such as the growing field of quantum information science.
While these questions can be successfully implemented without additional technological tools, this research used an electronic response system, generally referred to as "clickers," which automatically tracked student responses in real time.When presented in sequences of validated questions, clicker questions can systematically help students with particular concepts that they may be struggling with.Previously, such multiple-choice question sequences, or Clicker Question Sequences (CQS) related to several key QM concepts have been developed, validated and implemented [39][40][41][42][43][44].Furthermore, previous work has been conducted to investigate student difficulties with the uncertainty principle as it applies to wavefunctions [15] as well as two-state systems [45], but there has not yet been a documented effort to leverage the CQS method to address those difficulties.Here we describe the development, validation, and implementation of a CQS intended to help students learn about measurement uncertainty as it pertains to two-state quantum systems, and we discuss difficulties in identifying well-defined observables in a given state, calculating measurement uncertainty, successive measurements of various spin angular momentum observables, and other difficulties that naturally came up during implementation.

Development and validation
The CQS on quantum measurement uncertainty is intended for use in upper-level undergraduate QM courses.During the development and validation process, we took inspiration from some of the previously-validated learning tools, including determination of learning objectives.In particular, much research involving cognitive task analysis, from both student and expert perspectives, has already been conducted in the development and validation of a QuILT and CQS on measurement uncertainty and the uncertainty principle for wavefunctions (including in the context of the orbital angular momentum), as well as a QuILT on the basics of spin-1/2 systems [35].Ten student interviews had been conducted using a think-aloud protocol in the development of each of these learning tools, and the insights on student difficulties with regards to spin-1/2 systems and uncertainty principle for orbital angular momentum helped to guide the development of this CQS.In addition, we recently conducted four additional think-aloud interviews with physics post-graduate students with this CQS and, among other things, found that they appreciated the flow of CQS questions in terms of how well they build on each other.
We adapted some of those questions while also drafting and iterating new ones for measurement uncertainty related to two-state quantum systems.To ensure that the material could be completed in the allotted class time, while offering maximal value to students, we prioritized the coverage of conceptual knowledge, used common difficulties as a guide, provided checkpoints that could stimulate useful class discussions, and avoided burdensome calculations.We iterated the questions many times amongst ourselves and with other faculty members to minimize unintended interpretations and ensure consistency and simplicity in terminologies and sentence constructions.
We aimed to address common stumbling blocks and emphasize key features that students may have missed in the large information content of a typical lecture.Some questions in the CQS employ a complex multiple-choice question format, in which students are presented a number of options (e.g., options I, II, III) and must select one of several choices (e.g., choices A, B, C, etc.) that consist of a subset of those options.Though this may increase student cognitive load, we and others in physics education research have successfully used such question models for their relative parsimony in addressing multiple relevant facets of a concept at once, with respect to the limited class time.This is also helped by the immediate feedback and scaffolding support students receive during class discussions of the questions.
The seventeen questions in the final version of the CQS focused on the following four learning goals: identifying well-defined observables in a given quantum state (4 questions), calculations related to measurement uncertainty and the generalized uncertainty principle (5 questions), features and commutation relations of the spin operators  ̂2,  ̂,  ̂, and  ̂ (5 questions), and compatible vs. incompatible observables (3 questions).(In the CQS, the words "compatible" and "incompatible" are used to describe observables and their corresponding operators interchangeably.)

Implementation
The data presented here are from administration in a mandatory first-semester junior-/seniorlevel QM course at a large research university in the United States.Given that a physics course was the object of study, we elected to use quantum physics-framed language rather than quantum information-framed language to discuss these concepts.The final version of the CQS was implemented in two consecutive years, one online and one in person.In each instance, the instructor dedicated between two and three consecutive class sessions, each 50 minutes long, to complete the pre-test, CQS, and post-test.
During the online implementation, the CQS was presented as a Zoom poll while the instructor displayed the questions via the "Share Screen" function.For the in-person implementation, the poll was replaced by a functionally similar classroom clicker system.For each question, the instructor displayed the results after all students had voted, before a full class discussion of the validity of the options provided.Some of the questions involved more calculation than is typical of the conceptual questions in Mazur's method for introductory courses, which are intended to take roughly a minute each.Instructors were urged to use their judgment in giving the class more time to answer such questions.
Because of difficulties in adapting to the online environment in a way that remained conducive to small-group student discussion, the Peer Instruction feature was largely forgone in the online administration, but was realized in full for the in-person administration.We note also that the instructors were different for the online and in-person classes.
Table I below summarizes the learning goals, and the CQS questions and pre-test and post-test questions that cover these concepts.

Assessment
To determine the effectiveness of the CQS, we developed and validated a pre-and post-test containing questions on topics covered in the CQS.The post-test was a slightly modified version of the pre-test, containing changes such as a shift from eigenstates of  ̂ to eigenstates of  ̂, but otherwise remaining conceptually similar.In both online and in-person classes, students completed the pre-test immediately following traditional lecture-based instruction on the topic.After administration of the CQS over two to three class sessions, students completed the post-test.Two researchers graded the pre-test and post-test and, after discussion, converged on a rubric, for which the inter-rater reliability was greater than 95%.
During both online and in-person classes, the two to three classes were dedicated exclusively and consecutively to administration of the pre-test, CQS and ensuing discussions, and post-test.Since the CQS questions build on each other, they typically take less time to complete than if the questions were not part of a sequence.The only other content assigned over this duration was a traditional textbook homework set which overlapped with this period; given our past experience, we do not believe this had a significant effect on the more conceptual post-test performance.That said, this study is nonetheless quasi-experimental [46] in design, in light of these and other factors over which we did not have complete control.
The pre-and post-test questions are reproduced in Appendix A. Questions Q1-Q3 on the preand post-test provided students three possible answers from which to choose, and credit was awarded for correctly selecting or omitting each answer, for a total of up to three points per question.For these three questions, correct answers are bolded.For the free-response questions, Q4a was out of one point, while Q4b and all parts of Q5 were scored with two points, one each for answer and reasoning.A more detailed breakdown of the questions is provided in the next section.

Results
The pre-test and post-test results for each question, as well as normalized gains [47] and effect sizes [48], corrected for small sample size through a multiplicative factor equal to , are listed in Tables II (online with  = 27) and III (in-person with  = 23).Overall, the results are encouraging, and students performed well on the post-test, with relatively high normalized gains and generally medium to large effect sizes.The multiple-choice questions (Q1-Q3) had reasonably high pre-test scores across both classes.
Table II.Results of the online administration of the CQS via Zoom.Comparison of pre-and post-test scores, along with normalized gain [47] and effect size as measured by Cohen's d [48], for students who engaged with the CQS ( = 27 Table IV.Summary of notable result trends in CQS responses across both years.For ease of comprehension, correctness rates for questions are referred to as low (0-40%), medium (41-70%), and high (71-100%).Note that a small number of questions administered in the in-person implementation did not appear in the online implementation, and vice versa.

CQS # Overall performance Comments 1.1-1.4 Low to medium
Students were only moderately successful at identifying welldefined observables, especially for less straightforward cases, e.g., in CQS 1.2.
For CQS 1.3, the most popular distractor was the one that asserted that energy could not be well-defined for any state (where the Hamiltonian  ̂= ( ̂ +  ̂)).
For CQS 1.4, many students considered ℏ 2 to be the minimum product of uncertainties for any observables whose operators do not commute; some students also stated that quantum measurement uncertainty comes from apparatus imprecision.2.1-2.5 Generally low to medium CQS 2.1 had medium performance online and high performance inperson CQS 2.1 asked students whether uncertainties of specific observables were zero or nonzero, which most answered with reasonable success.
For CQS 2.2, it was clear that most students knew the formulation   2 = ⟨ 2 ⟩ − ⟨⟩ 2 , but were less comfortable with   2 = ⟨( − ⟨⟩) 2 ⟩ .CQS 2.5 showed that students may have been unfamiliar with applying the generalized uncertainty principle to the spin operators, e.g.,  ̂ and  ̂.

3.1-3.5 Generally high
Low for CQS 3.3 Students had an easy time with commutation relations of simple operators (e.g.,  ̂), but tended to struggle somewhat more as the operators became more complicated (e.g.,  ̂ ̂).
For CQS 3.3 online, the instructor gave a hint reminding students of how to simplify commutation relations when operators are multiplied together.During the second vote, the correctness rate improved to medium.CQS 3.4-3.5 (in-person only) were meant to help students with the results of a measurement of  2 .4.1-4.3Medium to high CQS 4.1, dealing with properties of observables corresponding to compatible operators, was more challenging for the students than the other questions in this cluster.
4.2 (incompatible operators) and 4.3 (generalization of incompatible operators) had high correctness rates; these questions may have been more intuitive or less complicated.
Below, we discuss some difficulties that were successfully addressed during the administration of the CQS for both years, as well as some that remained for smaller percentages of students.

Identifying observables that are well-defined in a state
Questions Q1-Q3 on the pre-and post-test asked students to identify observables that are welldefined in a given state.Most students correctly selected   or   , identifying that the state is an eigenstate of the corresponding operator.In Q1, some students did not select  2 , which is also well-defined in the given state because its corresponding operator is proportional to the identity operator and commutes with  ̂.Some students incorrectly selected   , which may be due to the use of   in class as a frequent example where the Hamiltonian  ̂∝  ̂.Questions such as CQS 1.1 and CQS 2.1 address these issues.On the post-test, the correctness rate on questions Q1-Q3 increased, with the exception of Q3 in the in-person administration, which had an especially high pre-test score (see Tables II-III).Since pre-test scores on all three questions were quite high across both years, the normalized gains and effect sizes are not as informative, but this indicates that students have a relatively strong grasp of the concepts involved in these questions.This being said, Q1 had a large effect size for both implementations, indicating noticeable improvement.

Calculating the measurement uncertainty
On the pre-and post-test, Q4 presented students with two states (Q4a and Q4b).The question asked them to determine whether the uncertainty in measuring a particular component of spin was zero in the given state (similar states involving or -components of spin were provided in the pre-and post-test, respectively), and to calculate the uncertainty if it was not zero in the given state.Q4a provided a state in which the observable could be measured with 100% certainty, so students who indicated this received full credit.Across both implementations for Q4b, we decided to give full credit to students who were able to provide the formulas for calculating uncertainty (e.g., an observable  has uncertainty   = √⟨ 2 ⟩ − ⟨⟩ 2 and the symbols under square root are the expectation values of  2 or ).As the approach for solving the problem, and not the answer of the numerical calculation, was the primary learning goal assessed by this question, our rubric thus avoided penalizing students who made a mathematical error in subsequent steps.Also, some students correctly identified whether the measurement uncertainty is zero or non-zero, but justified their answer only by invoking the probabilities of measuring each possible outcome.These students received half credit based upon our rubric.Questions CQS 2.3 and 2.4 were added for the in-person implementation to address issues related to the calculation of measurement uncertainty, and while Q4a had a high effect size in only the in-person implementation in part due to lower pretest scores, there were reasonably impressive normalized gains and effect sizes for Q4b across both years (see Tables II-III).

Results of successive measurements of identical or non-commuting observables
On the post-test, Q5 asked students for the final outcome of consecutive measurements of some permutation of the observables   ,   , and  2 , specifically testing whether students could recognize what happens when the measurements involved in the question corresponded to operators that did or did not commute with each other.Question Q5a asked about two consecutive measurements of   , Q5b about consecutive measurements of   and   , and Q5c about consecutive measurements of   ,   and   again.For Q5a, while most students correctly answered that the measurement in the collapsed state will yield − ℏ 2 as the outcome, some students did not recognize that  ̂ and  ̂ do not commute.Thus, for question Q5b, they answered that the outcome of the   measurement would still be − , corresponding to |−⟩ , neither of which is correct.Questions such as CQS 2.5 addressed these issues, and in general, the post-test scores show reasonable improvement for all parts of Q5 during the online and in-person administrations (see Tables II-III).

Results of successive measurements of 𝑆 𝑥 and 𝑆 2
Question Q5d asked students for the final outcome of consecutive measurements of   ,  2 and   again.On the pre-test, some students stated that for the final measurement of   , either eigenvalue ± ℏ 2 and eigenstate |±⟩ could be obtained, and some explicitly cited  ̂ and  ̂2 not commuting with each other for their reasoning.CQS 3.5 addresses measurement of  2 immediately after   , and CQS 4.3 and other questions in that sequence helped students generalize from spin-1/2 systems to more generic observables that correspond to operators that do or do not commute, and how such relationships may affect the measurements of those observables in a given quantum state.Student post-test performance showed better understanding of these concepts.

Conflation of eigenstates and eigenvalues
In the in-person implementation, on some questions only on the pre-test (not the post-test), some students wrote, e.g., that "The state will collapse into − ."While the CQS did not explicitly focus on distinguishing between the collapsed state and measured value of the observable, it is likely that the precise language used throughout the CQS helped students distinguish between eigenvalues and eigenstates on the post-test.

Incorrect answers for the results of a measurement
Across both years' implementations, there were some students who on both the preand posttest answered that the result of a measurement was the eigenvalue multiplied by the eigenstate.For example, for question Q5a-e, they stated that making a measurement of   in the state |−⟩ would yield an outcome of − ℏ 2 |−⟩.They also claimed, e.g., that the outcome for Q5a would be This type of reasoning may be closely related to the student difficulty that an operator's action on a quantum state represents a measurement of the corresponding observable in the state and should be investigated further [15].It is interesting that these students' answers remained the same on the pre-test and the post-test.This difficulty did not fall within the scope of this CQS, but it was addressed in another CQS which will be reported upon in a future publication.This may explain why the difficulty was observed to be rare but resistant for the duration of this CQS.
In the in-person implementation, another mistake was observed somewhat frequently even on the post-test.Question Q5d asked for the final outcome when   ,  2 and then   were measured in that order in immediate succession, with the first measurement of   yielding − ℏ 2 . Some students correctly stated that the intervening measurement of  2 did not collapse the state since the state was already an eigenstate of  ̂2, but chose the wrong eigenvalue and eigenstate (+ ℏ 2 , corresponding to |+⟩).These students were given full credit in recognition of their correct reasoning.If this is reflective of a deeper difficulty rather than a careless mistake, we have no compelling speculation with regard to what that difficulty may be.

Conceptual difficulties and ambiguities regarding uncertainty relations
In the in-person implementation, we added an additional free-response question (not shown in Table III) asking students to explain in their own words what it means for two observables, A and B, to have an uncertainty relation between them.Most students gave an answer involving the inability to know with full certainty the values of observables whose corresponding operators do not commute, but some framed their answers in terms of the position-momentum uncertainty relation (commonly cited as      ≥ ℏ 2 ), which is not related to two-state systems, rather than using the generalized uncertainty principle.As a result, these students noted that the measurement uncertainties of two observables must multiply to be greater than or equal to ℏ 2 .These results support the findings of a previous study [45].Though many questions in the CQS implicitly required knowledge of an uncertainty relation, only CQS 2.5 explicitly discussed the uncertainty principle and its applications to spin-1/2 systems, so this concept is worth emphasizing more in the future.
Another related response to this broad question about the uncertainty principle was "You can measure A or B, but not both."This type of response was observed only in the pre-test, and not the post-test, indicating that these students may have realized the difference between the ability to physically measure an observable versus being able to predict with 100% certainty what outcome would be obtained when the measurement of an observable is made.Other students were not strictly incorrect, but were somewhat unclear in their qualitative responses, which included statements such as "If we know the value of A, then B is a point of complete uncertainty," or "if we know A with 100% certainty, we will have 0% certainty for B." One interpretation of these responses is that "complete uncertainty" or "0% certainty" would refer to, in the case of a two-state system, an equal 50% chance of measuring either outcome.However, it is also possible that these students had the position-momentum uncertainty relation in mind, in which it is intuitive that a (continuous) decrease in uncertainty   must be compensated for by an increase in uncertainty    in order to maintain the product to be greater than or equal to ℏ 2 .In the case of a two-state system, rather than refer to this as "complete" uncertainty, a more accurate description may be something like "maximum" uncertainty.(It is worth noting that, in a spin-1/2 system, the maximum uncertainty for any component of spin is exactly ℏ 2 .)Furthermore, in response to this question about the uncertainty principle between observables A and B, many students stated something more general to the effect of "we can never know both quantities exactly at the same time" if their corresponding operators did not commute.Additionally, many students noted that "measurement of one [observable] will possibly affect the other," referencing the collapse to an eigenstate of the observable measured, in which the other observable would not be well-defined.Students were given full credit for all of these responses, since they have all articulated that the more is known about the value of one observable, the less is known about the value of another observable when their corresponding operators do not commute.While on the pre-test, some students were confused about uncertainty relations or left the question blank, nearly every student answered on the post-test in one of the ways discussed and thus earned full credit.

Comparisons between online and in-person implementations
Before comparing the online and in-person implementations, we note that some revisions were made to improve the CQS.These improvements affected the presentation of concepts that are covered in questions Q3, Q4b, and Q5d-e, mostly by providing more scaffolding to help students with solving these problems.The pre-and post-test questions remained nearly unchanged, with the exception of a clarification for questions Q5a-e as described in the next section.A one-to-one comparison can thus be drawn between the online and in-person implementations for pre-and post-test questions Q1, Q2, Q4a, and Q5a-c, as the CQS questions that covered the relevant concepts did not undergo any changes between years.For these questions, students' post-test performance does not differ appreciably between years.However, for the free-response questions Q4a and Q5a-c, the gap between pre-test and post-test performance is larger for the in-person class, as indicated by the larger normalized gains and effect sizes (see Tables II-III).
For all questions on the pre-and post-test, the trend appears to remain similar: The average pretest performance in the in-person implementation was, in general, somewhat lower, but post-test performance for both groups was comparable for all questions aside from Q5e, which is discussed in the next section.It is interesting that students performed about equally well on the post-test for both administrations, given that the online learning environment had greatly reduced opportunity for peers to discuss their responses with each other.We acknowledge that one possibility is students' ability to consult resources, despite being instructed not to do so, during the onlineadministered pre-and post-tests.Even though students were told that the quizzes were closed-book and closed-notes, such a rule could not be enforced when, as was the case, most students had their cameras off.Even so, those students would have had access to the same resources during both the pre-and post-test, so the sizable improvements in the post-test scores of the online class are still a good sign of the benefits of the CQS.We also observed that many more students left some pretest questions completely blank in the in-person implementation compared to the online implementation, despite both classes having received the same amount of time to complete the pre-and post-tests.Since the students were given sufficient time to complete the pre-and posttests, the cause of this higher occurrence of leaving some questions unanswered during the inperson class is unclear.This may be due to students not feeling confident enough to answer, or dealing with additional test anxiety or apathy not experienced by the students in the online implementation.In particular, while online classes have their disadvantages, there were also some benefits and conveniences that would have been lost in the transition back to in-person classes, which could have contributed to this phenomenon.Finally, as we have noted before, the different instructors between years could have also been a factor; e.g., during the online administration, it is possible that more emphasis was placed on content related to two-state spin systems.
The administration of the CQS in an online learning context may have affected student performance differently as compared to the in-person administration.However, regardless of whether the performance across years can be compared one-to-one, it is clear that the CQS has had a beneficial impact on student learning for both the online and in-person implementations.

Result of measurement of 𝑆 2
For the in-person implementation, CQS 3.4-3.5 were added to provide additional scaffolding on concepts relating to the observable  2 .Although the two implementations therefore cannot directly be compared with regard to student learning of these concepts, there are some differences worth examining.During the online implementation of the CQS, students demonstrated difficulties with the outcomes obtained from a measurement of  2 .This is relevant in post-test question Q5e, which asked students to provide the outcomes of successive measurements of   ,   , and  2 in a state, in that order.During online administration, this question had a large variety of responses that were difficult to score, but were still useful in shedding light on student difficulties.In particular, some students did not recognize that the eigenvalues obtainable from a measurement depend on the observable being measured.As an example,  ̂ and  ̂2 share eigenstates, but a measurement of   made in the state |−⟩ would yield − ℏ 2 , whereas a measurement of  2 in that same state would yield 3 4 ℏ 2 .However, multiple students answered that, should the measurements of   and  2 be made in succession in this state, they would both yield eigenvalues of − ℏ 2 .One possible reason for this type of response is that these students had associated the label |−⟩ primarily with the eigenstate of operator  ̂ with eigenvalue − ℏ 2 , so although some students realized that this state is a simultaneous eigenstate of  ̂2, they had difficulty with the corresponding eigenvalue (the state |−⟩ also carries an implicit label for the quantum number , which is dropped for ease of notation, which may contribute to this difficulty).
Question Q5e also revealed that some students may not have realized when answering the question that the only possible measured eigenvalue of  2 in any two-state spin system is In the online administration, the CQS did not explicitly address measurements of the observable  2 .Therefore, it is not surprising that students were left with some alternative conceptions.As a result of this, we refined the CQS to explicitly address the eigenvalues and eigenstates obtained from a measurement of  2 in the following in-person implementation.On Q5a-e of the pre-and post-test for the in-person implementation, we made the small addition of explicitly asking students to provide both the eigenvalue and eigenstate resulting from the measurements posed, instead of simply asking "What is the result of the measurement?"as in the online administration in the preceding year (see Appendix A for details).Previously, for the online implementation, students were given credit for providing either the obtained eigenvalue or the eigenstate after the measurement.During the in-person implementation, as a result of the change, students provided less ambiguous answers to these questions.Though the differences in post-test performances for online and in-person administrations could be attributable to more than one factor, the additional scaffolding provided by CQS 3.4-3.5,which specifically addressed these concepts, appears to have been effective.Across the two years, students' performance rose from 48% percent in the online implementation to 80% on the in-person implementation (see Tables II-III).
A summary of student difficulties observed in the pre-and post-tests is presented in Table V. Giving an incorrect eigenvalue for a measurement of  2 3.4, 3.5 Q5e | Some improvement for online; major improvement inperson

Summary
Validated CQS can be effective tools when implemented alongside classroom lectures.We developed, validated, and found encouraging results from implementation of a CQS on the topic of measurement uncertainty in two-state quantum systems, in both online and in-person settings.Post-test scores improved for every question following the administration of the CQS, with the exception of Q3 in the online implementation; this question tested students on whether energy can be well-defined for various Hamiltonians, and had an exceptionally high pre-test score.While the performance on the multiple-choice questions was high to begin with on the pre-test, there was significant improvement in the free-response questions.Effect sizes varied for the online implementation, but notably were large for nearly every free-response question in the in-person implementation, since the pre-test scores were lower than in the online implementation.This difference in pre-test performance could be reflective of differences in student preparation or behavior, or other factors such as instructor or environment.An examination of the post-test scores in the online and in-person implementations shows comparable performance in both years on most questions, demonstrating the effectiveness of the CQS in both administrations.

Ethical statement
This research was carried out in accordance with the principles outlined in the University of Pittsburgh Institutional Review Board (IRB) ethical policy.Informed consent was obtained from all interviewed students who participated in this investigation.|±⟩ where  = , , • Measurement in a state in which an observable is well-defined will yield a particular eigenvalue with 100% certainty (i.e., the state is an eigenstate of the operator corresponding to the observable).

CQS 1.4
Choose all of the following statements that are true regarding the uncertainty principle and uncertainty of measurement in QM.I.
The uncertainty principle refers to the inability of a measuring apparatus to be infinitely precise.II.
The uncertainty in the measurement of an observable can be determined by making a large number of measurements in identically prepared quantum systems in state |⟩, and calculating the standard deviation of those measurements.III.
The uncertainty in the measurement of an observable can never be zero, because the product of the uncertainties of two observables whose operators do not commute must always be ≥ • The uncertainty principle describes the observation that two observables whose operators do not commute (and thus whose operators do not have a complete set of simultaneous eigenstates) can never be measured with 100% certainty in the same state, i.e., the system cannot be in an eigenstate of both operators at the same time.• The uncertainty in the measurement of an observable must be determined by performing a large number of measurements on identically prepared systems, instead of making repeated measurements on the same system.This is because the first measurement will collapse the state of a system into an eigenstate of the operator corresponding to the observable being measured, which in general will not be the initial state (unless the initial state was an eigenstate of the operator corresponding to the observable being measured).
• Emphasize to students that the generalized uncertainty principle is   2   2 ≥ (  ).

II.
A measurement of   made in immediate succession will have uncertainty    = ℏ 2 .

III.
A measurement of   made in immediate succession will have uncertainty    = 0.

ℏ 2 "
or "It is in the state − ℏ 2

3 4ℏ 2 1 2ℏ 2 , 2 4 4 ℏ 2
, as the only eigenvalue of  ̂2 is ℏ 2 ( + 1) , with  = for spin-1/2 systems.When listing the possible eigenvalues for the measurement of  2 , in addition to the previously-mentioned − which are not unreasonable responses.The response ± ℏ may come from literally squaring the eigenvalues of, e.g.,  ̂ while matching the sign of the eigenstate label |±⟩, in analogy with the eigenvalues of  ̂ ± ℏ 2 being associated with the respective eigenstates |±⟩.Finally, the response − 3 may be analogous to the notion that the state |−⟩ is associated with a negative eigenvalue − ℏ 2 for a measurement of   .

are
|+⟩ and |−⟩, respectively (similar notation is used for the x-and y-components).All other notations are standard.Additionally, students were given the following notes:• The spin operators  ̂,  ̂,   ̂ correspond to the observables Sz, Sx, and Sy, respectively, which in turn correspond to the z-, x-, and y-components of a spin-1/2 particle's spin.All notations are conventional.
of the above E. None of the above Class discussion for CQS 1.4

2 ⟨𝑆 2 .
. You make a measurement of   in this state and obtain ℏ Choose all of the following statements that are true.Note:   made in immediate succession will have uncertainty    =

Table I .
Details of learning goals and their correspondence to questions on the CQS and the pretest and post-test.

Table III .
). Results of the in-person administration of the CQS.Comparison of pre-and post-test scores, along with normalized gain and effect size as measured by Cohen's d, for students who engaged with the CQS ( = 23).

Table V .
Student difficulties addressed by the CQS questions, which are found in Appendix B.

. Energy, if the Hamiltonian is 𝑯 ̂= 𝑪𝑺 ̂𝒙 (where 𝑪 is an appropriate constant) III. Any observable whose corresponding operator commutes with 𝑺
̂ 3. Consider the Hermitian operators  ̂ and  ̂, which correspond to observables.They are incompatible operators.The Hamiltonian is given by  ̂=  ̂+  ̂.Suppose you measure energy and obtain  0 .Choose all of the following statements that are correct immediately after the measurement of energy.Consider the Hermitian operators  ̂,  ̂, and  ̂2.Suppose you made a measurement of the observable   for a system in some state, and obtained the value − Suppose you immediately made another measurement of the observable   .What are the possible values that you can measure?What is the state immediately after the measurement?Explain.b.Suppose you instead immediately measured the observable   after the first measurement of   .What are the possible values that you can measure?What is the state immediately after the measurement?Explain.c.Suppose after the first measurement of   , you measured   in immediate succession, and   once again.What are the possible values that you can measure?What is the state immediately after the measurement?Explain.d.Suppose after the first measurement of   , you measured  2 in immediate succession, and   once again.What are the possible values that you can measure?What is the state immediately after the final measurement?Explain.e.Suppose after the first measurement of   , you measured   in immediate succession, and then  2 immediately after that.What are the possible values that you can measure?What is the state immediately after the final measurement?When the state |⟩ appears in the clicker questions, it refers to a generic state.Consider a system with a Hamiltonian  ̂=  ̂, where  is an appropriate constant.Choose all of the following statements that are correct for a system in an eigenstate of  ̂, i.e., |±z⟩.Consider a system with a Hamiltonian  ̂=  ̂, where  is an appropriate constant.Choose all of the following statements that are correct for the state Consider a system with a Hamiltonian  ̂= ( ̂+  ̂), where  is an appropriate constant.Choose all of the following statements that are correct for the state ⟩ is an eigenstate of  ̂, but   would not be well-defined for arbitrary coefficients of |+⟩ + |−⟩.•Discuss that the Hamiltonian  ̂= ( ̂+  ̂) is perfectly acceptable: not only is it diagonalizable and has its own eigenstates (which are not the same as those of either  ̂ or  ̂), but it can also be realized in an experiment, e.g., by applying a magnetic field of the form  ⃗⃗ = 4. For the following states of a system, does a measurement of the observable   yield a value with 100% probability?If the uncertainty is non-zero, calculate it.a. CQS 1.1 is only the special case for position and momentum in one dimension.Choose all of the following statements that are correct about uncertainties in the measurement of different components of spin, for an ensemble of identical systems in an eigenstate of  ̂.Choose all of the following statements that are correct about the uncertainty in the measurement of an observable  in state |⟩ (which is not an eigenstate of  ̂).A.IIonly B. I and II only C. I and III only D. II and III only E. All of the above CQS 2.3 *Consider the uncertainty    in the observable   in a given quantum state |⟩.Choose all the following statements that are correct.[Please give students sufficient time for this question to perform the necessary calculations.]