As part of ongoing research on student thinking about quantum mechanical concepts and formalism, we explored how students defined and made sense of expectation values.Previous research has focused on student difficulties when defining the expectation value for a generic operator and found that students conflate other quantum mechanical ideas with expectation values.In this study, we analyzed survey data collected from two universities over a number of years and interviews carried out at two points during the semester.With a focus on underlying student thinking, we used a concept image perspective to categorize students' concept definitions of the expectation values in the context of measuring spin in a spin-1=2 system and in the contexts of measuring energy and position for a particle in an infinite square well potential.Analysis of interview data showed that students invoke many different ideas when explaining their reasoning.The two most common definitions for the expectation value were weighted average and most probable value.In interviews, students' definitions were influenced by whether the problem context involves continuous or discrete observables.
As part of ongoing research on student thinking about quantum mechanical concepts and formalism, we explored how students defined and made sense of expectation values.Previous research has focused on student difficulties when defining the expectation value for a generic operator and found that students conflate other quantum mechanical ideas with expectation values.In this study, we analyzed survey data collected from two universities over a number of years and interviews carried out at two points during the semester.With a focus on underlying student thinking, we used a concept image perspective to categorize students' concept definitions of the expectation values in the context of measuring spin in a spin-1=2 system and in the contexts of measuring energy and position for a particle in an infinite square well potential.Analysis of interview data showed that students invoke many different ideas when explaining their reasoning.The two most common definitions for the expectation value were weighted average and most probable value.In interviews, students' definitions were influenced by whether the problem context involves continuous or discrete observables.
Quantum uncertainty is a fundamental concept in quantum mechanics, but challenging for students to master.In this article, we describe student difficulties with visual and conceptual understanding of quantum uncertainty in the context of discrete probability distributions such as those for a spin 1/2 particle.We collected written responses from students at two institutions to a homework activity focusing on uncertainty of spin measurement outcomes, as well as written responses to a test question from one of the institutions.We also conducted interviews with six students to gain further insight into difficulties found.Common incorrect ideas found included a depiction of uncertainty as the error around each of the individual measurement outcomes, not depicting the uncertainty region from the expectation value outwards, and the idea that quantum uncertainty of an observable can never be zero.These ideas may indicate a confusion between quantum uncertainty and errors due to instrumental imperfections of the measurement apparatus, a lack of conceptual understanding of quantum uncertainty as the standard deviation of the probability distribution with respect to its mean, and an incorrect interpretation of the uncertainty relation between two incompatible observables to deduce that quantum uncertainty can never be zero.The results of this study show the importance of supporting students in visual and conceptual understanding of quantum uncertainty.
As part of ongoing research to better understand how students relate Quantum Mechanics concepts to formalisms, we are studying student understanding of expectation values as well as the variety of mathematical representations they use for formalizing them in two different teaching paradigms.Analyzing students' written responses to a series of open-ended research questions has given us some insight into students' common interpretations of this concept, as well as the frequency of the various mathematical formalisms they use to represent expectation values.The results show some common patterns between students in Spin First (SF) and Position First (PF) curricular, as well as some differences.
Not only is Quantum Mechanics (QM) conceptually rich, it is also a theory that physics students have found abstract and technically formidable. Nevertheless, compared to other classical topics of physics, university students’ understanding of QM has received minimal attention in the physics education literature. The principal purpose of this study was to characterize the variation in the ways that undergraduate physics students depict the basic concepts of QM and to extrapolate the results to scaffold possible changes to instructional practices at the university that provided the context for the study. In so doing, an adaptation of a developmental phenomenographic perspective was chosen. Empirically, the study was approached through in-depth interviews with 35 physics students from two Ethiopian governmental universities after they had been exposed to the traditional QM course for one-third of a semester. Interview responses were analyzed using phenomenographic approach where a picture of students’ depictions was established for each quantum concept by expounding the given responses. For each basic quantum concept addressed, the structure of the description categories was separately constructed, and overall, it was found that naive, quasi-classical ontology and/or variants of classical ways of visualization are dominant in students’ responses. For example, it was found that students’ depictions of the photon concept could be described with three distinct categories of description, which are (a) classical intuitive description, (b) mixed model description and (c) quasi-quantum model description. Similarly, the findings revealed that it is possible to establish three qualitatively different categories of description to picture students’ depictions of matter waves, namely, (a) classical and trajectory-based description, (b) an intricate blend of classical and quantum description and (c) incipient quantum model description. Likewise, it was found that students’ depictions of uncertainty principle can be described as: (a) uncertainty as classical ignorance, (b) uncertainty as measurement disturbance and (c) uncertainty as a quasi-quantum principle. \nWith regard to learning QM, the categories of description made clear several issues: most students did not have enough knowledge to depict the basic concepts of QM properly; they were influenced by the perspective of classical physics and their perceptions in making explanations about QM; and they also applied mixed ideas, one based on their classical model and the other from newly introduced QM. These results are also supported by the findings of previous studies in similar domains. Findings from the study were used to guide the design of multiple representations-based instructions and interactive learning tutorials on the conceptual aspects of QM that has been shown to address specific difficulties identified in the study. Theoretical and practical implications of the study, as well as potential future considerations are drawn.
Measurements in quantum mechanics are often taught in an abstract, theoretical context.Compared to what is known about student understanding of experimental data in classical mechanics, it is unclear how students think about measurement and uncertainty in the context of experimental data from quantum mechanical systems.In this paper, we tested how students interpret the variability in data from hypothetical experiments in classical and quantum mechanics.We conducted semi-structured interviews with 20 students who had taken quantum mechanics courses and analyzed to which sources they attribute variability in the data.We found that in the quantum mechanics context, most students interpret any variability in the data as irreducible and inherent to the theory.While acknowledging the influence of experimenter error, limited resolution of measurement equipment, and confounding variables (like air resistance) in classical mechanics, many students did not recognize the influence of such effects in quantum mechanics.Some students expressed the view that there are inherently fewer confounding variables in Quantum Mechanics and the equipment used is more precise.We derive tentative implications for instruction and propose further research to test the influence of framing on the responses to our interview protocol.
This paper is part of the Focused Collection on Upper Division Physics Courses.]Energy measurements play a fundamental role in the theory of quantum mechanics, yet there is evidence that the underlying concepts are difficult for many students, even after all undergraduate instruction.We present results from an investigation into student ability to determine the possible energies that can be measured for a given wave function and Hamiltonian, to determine the probabilities of each energy measurement and how they depend on time, and to recognize how a measurement of energy affects the state.By analyzing student responses to open-ended questions, we identify five broad, interrelated sets of conceptual and reasoning difficulties related to energy measurements.Data are drawn from sophomore-, junior-, and graduate-level quantum mechanics courses.Particular attention is paid to incorrect ideas that persist across all levels.
A common task when problem solving in quantum mechanics, including in a spins-first curriculum, involves changing the basis of a given state.Our research in undergraduate quantum mechanics courses at three institutions explores student thinking about basis, basis expansion coefficients, and change of basis in the context of spin-½ systems.Our investigation is based on conceptual and computational written questions as well as student reasoning interviews.We identify student ideas about whether and how changing basis affects the state, examine how students perceive notation as indicative of choice of basis, explore students' interpretations of the structure and meaning of a basis expansion, and identify the range of methods students employ when changing basis.For instance, we find a recurring idea that changing basis alters the physical system, and observe that some students chose to relabel the ket representing a quantum state vector after changing basis.Together, these results paint a broad, qualitative picture of a variety of ways that students grapple with basis and change of basis, with potential implications for instruction.