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As part of an effort to examine students’ mathematical sensemaking in a spins-first quantum mechanics course, students were asked to construct an eigenvalue equation for an operator that represents the position of a particle constrained to exist along a line. A subset of student responses took the general form of an eigenvalue equation written in Dirac notation. These responses were indicative of a few different ways of thinking about eigenvalue equations: a standard geometric interpretation, operating as measuring, and the eigenvalue equation as it relates to potential outcomes of measurement. In the context of Sherin’s symbolic forms framework, these data suggest three different symbolic forms for an eigenvalue equation that all share a single symbol template but have different conceptual schemata.

AAPT SM 22

Student-Constructed Eigenvalue Equations in Quantum Mechanics: A Symbolic Forms Analysis

AAPT meetings are a great way to reflect on your teaching, recommit to best practices and re-energize your career with new research, products, and techniques. Listen to internationally known plenary speakers, attend sessions, workshops, and special activities. This meeting is an excellent opportunity to share your expertise and learn from and network with leaders in the field.



technical paper

One important 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 not encountered in previous coursework. To investigate student understanding of the relationships between expressions used in these various notations and generic vector expressions, a survey was distributed to students in upper-division QM courses at multiple institutions. The courses surveyed included those structured both as “spins-first” and “wave functions-first.” Network analysis techniques were used to compare student understanding of common expressions used in these courses, in both Dirac notation and wave function notation. Preliminary analysis suggests that, in addition to more distinct morphological grouping, students in “spins-first” courses also appear to conceptualize Dirac expressions as more vector-like than those in “wave-functions first” courses, consistent with the instructional emphasis.

Network Analysis of Student Interpretations of Dirac Expressions Across Curricula

We designed a rubric to assess free-response exam problems in order to compare thinking skills evidenced in exams in classes taught by different pedagogies. The rubric was designed based on Bloom’s taxonomy (revised version) [1-3]. It was then used to code exam problems. We are now using this rubric to analyze the thinking skills of students taught online during covid vs. students taught in class after covid. We analyzed exams from different sections of the algebra-based physics course taught online during covid and in class after covid. We discuss the instrument, present results and future research plans.

Assessing Thinking Skills in Free-response Exam Problems: Covid vs. Non-covid

Quantum mechanics is increasingly being taught “spins-first”: using systems with spin as examples before systems with wave functions. It is important for spins-first instructors to understand how students apply knowledge from spin systems to understand the continuous quantum systems that they study later. We are studying the ideas that students have about discrete bases, continuous bases, and the connections between them. We interviewed six students who were enrolled in a spins-first quantum mechanics course and asked them to participate in a card-sorting task where they organized twenty cards with a variety of quantum mechanics content and representations including braket notation, matrix notation, function notation, and code snippets. In this talk, we will discuss the reasoning participants used to organize cards into “discrete” and “continuous” categories as well as how students described the different types of representations as communicating “discreteness”.

How Students Compare and Contrast the “Discreteness” of Quantum Representations

Previous research has shown that students who demonstrate sufficient skills and conceptual understanding to reason productively may perform inconsistently on analogous questions. Such inconsistencies can be explained via dual process theories of reasoning (DPToR). To gain insight into students’ reasoning trajectories, we developed an exploratory sequence of DPToR-aligned metacognitive prompts and administered the sequence immediately after students answered a physics question containing salient distracting features. The metacognitive prompts asked students to reflect on their initial ideas, the extent to which they had doubts about those ideas, the extent to which their final answer differed from those ideas, the nature of their reasoning approach, and their confidence in their own understanding. In this talk, we describe how we use student responses to these prompts, along with timing data and a measure of cognitive reflection skills, to investigate students’ reasoning trajectories.

Using Metacognitive Prompts to Explore Student Reasoning Trajectories*

Over the past several years, the Partnership for Integrating Computation into Undergraduate Physics (PICUP) has curated a collection of materials to help instructors add in computational modeling to their courses. I’ll discuss the various resources and how you can contribute to the community.

Exercise Sets, Faculty Commons, and Resources of the PICUP Website

The Partnership for the Integration of Computation into Undergraduate Physics (PICUP) has had numerous successes since its beginning. However, challenges do remain as change is a slow process. This talk, by the external evaluator for PICUP, provides an overview of the summative evaluation featured in the PICUP Capstone Conference Report. Potential future avenues are also discussed.

Success and Remaining Challenges for PICUP and Integrating Computation

The North American Nanohertz Observatory for Gravitational Waves (NANOGrav) collaboration is working towards the detection and study of low-frequency gravitational waves using an array of rapidly rotating, highly stable radio pulsars distributed across the galaxy as accurate and precise clocks. A significant amount of research is driven by students from the high school to the graduate level. Students new to the field find themselves engaging with a large cross-section of many fields of astronomy, learning about stellar evolution, compact objects, galaxy evolution, gravitational waves, the interstellar medium, and more, and we strive to make our science accessible to them. I will describe our research program and specifically highlight the efforts within the collaboration to make our science more accessible to new students at many levels, getting them involved in cutting-edge science as quickly and easily as possible.

Simplifying Pulsar Timing Array Science for Students

Pulsars were discovered in 1967 and quickly identified as highly magnetized, fast rotating neutron stars. They are remnants of the explosion of a star known as a supernova. A neutron star has a diameter of about 7 miles and about twice as much mass as our Sun. Pulsars spin several times per second and have a magnetic field that can be a trillion times stronger than the one on Earth. As they age they emit electromagnetic radiation ranging from radio waves, visible light possibly up to gamma-rays. Their rotation causes their emission to be received in the form of pulses on Earth. The regularity of these pulses is the reason why they are often referred to as clocks in the sky. Some of them are more regular than others.

Pulsars -The Story of the Stellar Clocks in the Sky

Most science and engineering degrees at Penn State require that our students transfer to a different campus to complete their degree, so we have targeted first and second-year students and students with non-science majors. We try to provide a meaningful research experience without overwhelming them. The Green Bank Observatory (GBO) has dedicated two smaller telescopes for educational use. Our students have recently studied giant pulses from the Crab pulsar (PSR J0534-3300) and the Vela pulsar (PSR J0835-4510) using the GBO 20-meter telescope remotely. Giant pulses are rare, exceptionally large radio pulses and have only been detected in a small number of pulsars. Interactive Jupyter notebooks were developed to process the telescope data. Text files, produced by the notebooks, listing the detected giant pulses, are downloaded to students' laptops and analyzed using spreadsheet programs.

Radio Astronomy Research Experiences for Undergraduates

The radio sky is accessible at low cost with the RadioJOVE Project, a dual-dipole with receiver that is used by educators, students, and amateur radio astronomers to detect solar flares and radio bursts from Jupiter. The RadioJOVE has been used at Saint Anselm College primarily for a high school afterschool program, but recently, more sophisticated work on data analysis has been undertaken by undergraduates in the physics department. We will share results from current projects, such as progress towards interference reduction and measuring the signal from the Milky Way, and plans for future work that are appropriate for physics undergraduates who may be new to astronomy.

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Student-Constructed Eigenvalue Equations in Quantum Mechanics: A Symbolic Forms Analysis

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