Search for: All records

Abstract This work investigates how students interpret various eigenequations in different contexts for$$2 \times 2$$ $2\times 2$matrices:$$A\vec {x}=\lambda \vec {x}$$ $A\overrightarrow{x}=\lambda \overrightarrow{x}$in mathematics and either$$\hat{S}_x| + \rangle _x=\frac{\hbar }{2}| + \rangle _x$$ ${\hat{S}}_x{|+⟩}_x=\frac{ħ}{2}{|+⟩}_x$or$$\hat{S}_z| + \rangle =\frac{\hbar }{2}| + \rangle$$ ${\hat{S}}_z|+⟩=\frac{ħ}{2}|+⟩$in quantum mechanics. Data were collected from two sources in a senior-level quantum mechanics course; one is video, transcript and written work of individual, semi-structured interviews; the second is written work from the same course three years later. We found two principal ways in which students reasoned about the equal sign within the mathematics eigenequation and at times within the quantum mechanical eigenequations: with a functional interpretation and/or a relational interpretation. Second, we found three distinct ways in which students explained how they made sense of the physical meaning conveyed by the quantum mechanical eigenequations: via a measurement interpretation, potential measurement interpretation, or correspondence interpretation of the equation. Finally, we present two themes that emerged in the ways that students compared the different eigenequations: attention to form and attention to conceptual (in)compatibility. These findings are discussed in relation to relevant literature, and their instructional implications are also explored. 

One expected outcome of physics instruction is that students develop quantitative reasoning skills, including strategies for evaluating solutions to problems. Examples of well-known “canonical” evaluation strategies include special case analysis, unit analysis, and checking for reasonable numbers. We report on responses from three tasks in different physics contexts prompting students in an introductory calculus-based physics sequence to evaluate expressions for various quantities: the velocity of a block at the bottom of an incline with friction, the final velocities of two masses involved in an elastic collision, and the electric field due to three point charges. Responses from written ( $\mathit{N}=580$) and interview ( $\mathit{N}=18$) data were analyzed using modified grounded theory and phenomenology. We also employed the analytical framework of epistemic frames. Students’ evaluation strategies were classified into three broad categories: consulting external sources, checking through computation, and comparing to the physical world. Some of the evaluation strategies observed in our data, including canonical as well as noncanonical strategies, have been reported in prior research on evaluation, albeit sometimes with different names and with varying levels of generalizability. We note four major, general observations prompted by our results. First, most students did not evaluate solutions to physics problems using an approach that an expert would consider an evaluation strategy. Second, many students used evaluation strategies that emphasized computation. Third, many students used evaluation strategies that are not canonical but are nonetheless useful. Fourth, the relative prevalence of different strategies was highly dependent on the task context. We conclude with remarks including implications for classroom instruction. 

[This paper is part of the Focused Collection in Investigating and Improving Quantum Education through Research.] Instructors teaching upper-division quantum mechanics have had two primary options when it comes to textbook choice and thus curriculum sequence: starting with wave functions and the Schrödinger equation, referred to as “wave functions-first;” and starting with discrete spin-1/2 systems and Dirac notation, known as “spins-first” courses. Given the very different structures of these courses, particularly as it pertains to the notations and formalisms both emphasized and used, it begs the question as to whether and to what extent students in these different courses conceptualize symbolic expressions in Dirac and wave function notations differently. To investigate this, online surveys were administered to students in spins-first courses at six institutions and in wave functions-first courses at four institutions. As a follow-up to a prior study focused on the results from the spins-first courses, network analysis and community detection techniques were used to compare the level of conceptual similarity between expressions as viewed by the students in both curricula. Conceptual interpretations of individual expressions in both Dirac and wave function notations were also directly compared between the two populations. The primary difference observed between the two populations appears to lie in the way they interpret Dirac bras and kets: spins-first students were found to more strongly connect these expressions to vectorlike interpretations, while wave functions-first students were found to interpret them as more wave functionlike. This suggests that the choice of text and/or curricular style should be informed by the interpretation that best matches the goals of the instructor. 

[This paper is part of the Focused Collection in Investigating and Improving Quantum Education through Research.] The ability to relate physical concepts and phenomena to multiple mathematical representations—and to move fluidly between these representations—is a critical outcome expected of physics instruction. In upper-division quantum mechanics, students must work with multiple symbolic notations, including some that they have not previously encountered. Thus, developing the ability to generate and translate expressions in these notations is of great importance, and the extent to which students can relate these expressions to physical quantities and phenomena is crucial to understand. To investigate student understanding of the expressions used in these notations and the ways they relate, clinical think-aloud interviews were conducted with students enrolled in an upper-division quantum mechanics course. Analysis of these interviews used the symbolic forms framework to determine the ways that participants interpret and reason about these expressions. Multiple symbolic forms—internalized connections between symbolic templates and their conceptual interpretations—were identified in both Dirac and wave function notations, suggesting that students develop an understanding of expressions for probability both in terms of their constituent pieces and as larger composite expressions. 

One expected outcome of physics instruction is for students to be capable of relating physical concepts to multiple mathematical representations. In quantum mechanics (QM), students are asked to work across multiple symbolic notations, including some they have not previously encountered. To investigate student understanding of the relationships between expressions used in these various notations, a survey was developed and distributed to students at six different institutions. All of the courses studied were structured as “spins-first,” in which the course begins with spin-1/2 systems and Dirac notation before transitioning to include continuous systems and wave function notation. Network analysis techniques such as community detection methods were used to investigate conceptual connections between commonly used expressions in upper-division QM courses. Our findings suggest that, for spins-first students, Dirac bras and kets share a stronger identity with vectorlike concepts than are associated with quantum state or wave function concepts. This work represents a novel way of using well-developed network analysis techniques and suggests such techniques could be used for other purposes as well. 

As part of an effort to examine students’ mathematical sensemaking (MSM) in a spins-first quantum mechanics course during the transition from discrete (spin) to continuous (position) systems, students were asked to construct an eigenvalue equation for a one-dimensional position operator. A subset of responses took the general form of an eigenvalue equation written in Dirac notation. Symbolic blending, a combination of symbolic forms and conceptual blending, as well as a categorical framework for MSM, were used in the analysis. The data suggest two different symbolic forms for an eigenvalue equation that share a symbol template but have distinct conceptual schemata: A transformation that reproduces the original and to operate is to act. These symbolic forms, when blended with two sets of contextual knowledge, form the basis of three different interpretations of eigenvalue equations modeled here as conceptual blends. The analysis in this study serves as a novel example of, and preliminary evidence for, student engagement in sensemaking activities in the transition from discrete to continuous systems in a spins-first quantum mechanics course. 

Physics students are introduced to vector fields in introductory courses, typically in the contexts of electric and magnetic fields. Vector calculus provides several ways to describe how vector fields vary in space including the gradient, divergence, and curl. Physics majors use vector calculus extensively in a junior-level electricity and magnetism (E&M) course. Our focus here is exploring student reasoning with the partial derivatives that constitute divergence and curl in vector field representations, adding to the current understanding of how students reason with derivatives. 

Much of physics involves the construction and interpretation of equations. Research on physics students’ understanding and application of mathematics has employed Sherin’s symbolic forms or Fauconnier and Turner’s conceptual blending as analytical frameworks. However, previous symbolic forms analyses have commonly treated students’ in-context understanding as their conceptual schema, which was designed to represent the acontextual, mathematical justification of the symbol template (structure of the expression). Furthermore, most conceptual blending analyses in this area have not included a generic space to specify the underlying structure of a math-physics blend. We describe a conceptual blending model for equation construction and interpretation, which we call symbolic blending, that incorporates the components of symbolic forms with the conceptual schema as the generic space that structures the blend of a symbol template space with a contextual input space. This combination complements symbolic forms analysis with contextual meaning and provides an underlying structure for the analysis of student understanding of equations as a conceptual blend. We present this model in the context of student construction of non-Cartesian differential length vectors. We illustrate the affordances of such a model within this context and expand this approach to other contexts within our research. The model further allows us to reinterpret and extend literature that has used either symbolic forms or conceptual blending. 

As part of an effort to examine student understanding of expressions for probability in an upper-division spins-first quantum mechanics (QM) context, clinical think-aloud interviews were conducted with students following relevant instruction. Students were given various tasks to showcase their conceptual understanding of the mathematics and physics underpinning these expressions. The symbolic forms framework was used as an analytical lens. Various symbol templates and conceptual schemata were identified, in Dirac and function notations, with multiple schemata paired with different templates. The overlapping linking suggests that defining strict template-schema pairs may not be feasible or productive for studying student interpretations of expressions for probability in upper-division QM courses.