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Physics Education Research Conference (PERC)

Student understanding of quantum mechanical expectation values in two different curricula

Homeyra R. Sadaghiani, Gina Passante, Steven J. Pollock

Published Jan 21, 2019• 2 citations
Publisher

Abstract

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.

DOI
10.1119/perc.2018.pr.sadaghiani

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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.
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.
Research into student understanding of quantum mechanics has primarily focused on the identification of student difficulties a nd t he d evelopment o f c urriculum t hat c an h elp i mprove s tudent l earning.M ost o f this work has been performed in courses using a position-first i nstructional p aradigm.R ecently, t here h as been increase in the number of instructors choosing to present material in an alternate order: spins-first.In this work, we makes steps towards answering the question of whether or not the instructional paradigm has an impact on student thinking about energy measurement.We use a resources lens to analyze the different tools and methods students are using to solve questions.
We have investigated student difficulties in understanding and interpreting probability and its relevant technical terms as it relates to quantum measurement. These terms include expectation value, probability density, and uncertainty. From this research, it is evident that students have difficulties in understanding these terms and often fail to differentiate among similar but different concepts. In addition, students' difficulties with the concepts of probability often interfere with their understanding and application of the Uncertainty Principle.
We investigate the difficulties of advanced undergraduate students toward the end of a full year upper-level quantum mechanics course with concepts related to quantum measurements and time development. Our analysis is based upon a test administered to 89 students from six universities and interviews with 9 students. Strikingly, most students shared the same difficulties despite variations in background, teaching styles, and textbooks. Concepts related to stationary states, eigenstates, and time dependence of expectation values were found to be particularly difficult. An analysis of written tests and interviews suggests that widespread misconceptions originate from an inability to discriminate between related concepts and a tendency to overgeneralize.
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.