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In order to help our students better understand the conceptual connections between spin states and wavefunctions in our junior-level, spins-first quantum course, we have recently begun to emphasize the use of completeness relations to write a state in a particular basis. We describe completeness relations as “special forms of 1” that can be used to project a quantum state vector onto a basis. Using a discrete spin bases, the expansion coefficients are interpreted as probabilities amplitudes. Using a continuous position basis, these probability amplitudes get “smeared out” to become a “probability amplitude density” (a.k.a., wavefunction), similar to how discrete point masses go to mass densities. In this talk, we will share the in-class activities and homework tasks we’ve developed to help students understand make these connections.


AAPT SM 21

Using Completeness Relations to Help Students Understand Wavefunctions

upper division

mathematics

quantum mechanics

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technical paper

Whitewashing science goes back hundreds of years. An early victim was the medieval Arab scientist Ibn al-Haythem. His work combining experimental investigations with mathematical modeling anticipated by many centuries the rise of European science, and his breakthrough theory of light and perception was the first to be both physically grounded and mathematically rigorous. In the late medieval period his theory of light and perception became widely accepted throughout Europe, influencing the development of Renaissance art and providing the foundations for modern geometric optics. In the process, however, his name and ethnic background were obscured—likely due to racial or religious prejudice—and even to this day he is not given due credit for the significant contributions he made to physics. In this talk I will briefly review his work and how it entered into the European cannon while denying him due credit.

The Uncelebrated Arab Genius

The Islamic Golden Age was a time of great cultural and intellectual exchange lasting from approximately the 8th to the 14th century, located in the Middle East, Northern Africa, and parts of Europe. This time period was also characterized by major developments in science and medicine. Even words like “algebra” and “algorithm” can trace their origins back to this period. However, the importance of this era to the development of modern science ought to be better known. This talk will cover some of the background history of the Islamic Golden Age as well as some of the major contributions in mechanics, optics, and astronomy.

Importance of the Islamic Golden Age for Physics and Astronomy

Diversity in physics remains low despite a social shift towards STEM inclusion. According to AIP’s Statistical Research Center, women have seen some increases in representation at the post-secondary level, however that has stagnated even with consistent, equitable participation in high school. Moreover, among underrepresented minority groups, less than 15% of all physical science bachelor degrees awarded in the US are to African-American or Latino/a individuals. After decades of work promoting STEM inclusion, why does this disparity remain? The answer may lie in our marketing, and how we as teachers not only sell the course, but the truth that anyone and everyone can belong. In this talk, we will use the following questions to guide our discussion. What is the current state of diversity in physics? What is identity and how do we define the self? How can teachers better prime their students to see themselves in science careers?

Priming Young Scientists: Identity and Physics

In recent years, addressing diversity, equity, and inclusion in STEM fields, including physics and astronomy, has received growing attention. To effectively promote equity and social justice, we need to acknowledge the role of oppression in disciplinary cultures and thus transform these cultures to be created by and for people of all identities and lived experiences. These actions will improve the teaching and learning of academic content. Here, we propose a culturally responsive framework for astronomy education that incorporates asset-building, reflection, and connectedness, and we present a sample curriculum created using this framework. Students learn astronomy content, such as about Solar System dynamics through physical modeling and simulations, as well as participate in discussions that problematize cultural norms and develop potential solutions. Our approach guides students to critically engage with the content and empowers them to become change agents for making astronomy equitable.

Promoting Equity and Social Justice through Culturally Responsive Astronomy Education

I show some uses of Projective Geometric Algebra for solving problems in geometric optics.

The Equations Match the Drawings: Geometric Algebra for Geometric Optics

Quantum mechanics instruction has remained unchanged for at least 75 years, following a coordinate-space-based formalism that requires significant class time for instruction on the mathematical background for the Frobenius method, delta functions, Fourier transforms, etc. This mathematics instruction, greatly limits the amount of physics that can be included. In this talk, I will tell you how to reverse this trend. In Fall 2020, I taught a one-semester quantum mechanics course at Georgetown that worked within a representation-independent formalism (emphasizing operators, not wavefunctions). It is mathematically simpler and frees up time for discussing conceptual and physical ideas. I discuss important experiments such as Stern-Gerlach, delayed choice, EPR, Bell inequality tests, Hong-Ou-Mandel, Pickering-Fowler lines, discovery of deuterium, proton radius, electron momentum spectroscopy, time of flight, hyperfine interactions and radio astronomy, cyclotron resonance and MRI, single-photon detection, homodyne detection, and LIGO. Come see how you can adopt this for your class too.

Bringing the physics back into the undergraduate quantum classroom

I will report on three practices I implemented in upper division electromagnetic theory courses to improve the impact of homework on student learning. The principal innovation was to adapt some of the principles of calibrated peer review, which has been demonstrated to be beneficial in writing assignments to common physics homework assignments. The other two innovations involved having the students complete the assignments using templates in Jupyter notebooks. This gave them an opportunity to present their homework in a professional manner and to check their symbolic answers against answer verification software running on a server. These innovations improved student persistence until mastery, encouraged students to study alternate approaches to problems, and increased the time students spent on analyzing what they did incorrectly on problems. I will report formal and informal feedback I received from students on the effectiveness of these innovations.

Improving Upper Division Homework Effectiveness

Research-based tutorials---worksheets that guide students in constructing physics knowledge for themselves---have repeatedly proven to be a highly effective complement to lectures in physics courses. Our collaboration has created a collection of tutorials for middle-division quantum mechanics (QM), which has been successfully adopted by QM instructors nationwide. However, many instructors lack the resources and/or institutional support to run tutorials; it can be difficult to find time, space, and personnel to facilitate these activities. As an alternative, we are developing online versions of our QM tutorials that students can work through without instructor facilitation. We are designing the online tutorials to guide students through the worksheets in ways that encourage sense-making and allow students to progress without revealing the answers. For example, they include adaptive guidance pathways based on students’ responses. Early beta versions are available, for free, at acephysics.net.

Online Tutorials for Middle-Division Quantum with Adaptive Guidance

Research-based assessments can provide instructors insights into the efficacy of their teaching and course transformations. As education shifts to focusing more on scientific practices and crosscutting concepts in addition to conceptual knowledge, there is a need for an assessment that can assess these 3-dimensions of learning and inform instructional approaches that foster this learning. In this talk, we present a first-look at an upper-division thermal physics assessment that explicitly targets scientific practices, crosscutting concepts, and conceptual knowledge using coupled, multiple-response items. We will present an example assessment item and discuss ways in which instructors can help with the development process of this assessment while incorporating it into their classrooms. Faculty administering the assessment would receive detailed feedback about their class’s overall performance with recommendations for how to adjust instruction to improve student outcomes.

A Knowledge-in-Use Assessment for Upper-Division Thermal Physics

We present preliminary mixed-methods results for an experiment upon a first-semester senior undergraduate electromagnetism course for physics majors, in which 25 students were given explicit incentive to rework mistakes from unit exams in a first-semester senior-level electromagnetism course. Three unit exam problems were revisited on the final exam. Initial quantitative results for one of the three problems (P1) showed a strong effect from unit exam to final, between students who reworked problems and students who chose not to; however, the other two problems (P2 and P3) presented a ceiling effect on unit exam performance that prevented pre-post analysis. Follow-up interviews with a small sample of students suggest a strong recall of their attempts on problems P1 and P2 despite being over a year removed from said exams; however, some did not recognize mistakes. We conclude by discussing plans for analysis of current/future data and adjustments to this ongoing study.

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Using Completeness Relations to Help Students Understand Wavefunctions

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