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Wuli yu gongcheng.

NUMERICAL EXPERIMENTAL DESIGN FOR ANGULAR MOMENTUM VISUALIZATION

Hang Cheng, 邓志姣, Yong SHEN, Pingxing CHEN

Published Feb 1, 2026• 0 citations
Publisher

Abstract

Angular momentum is a difficult knowledge point for undergraduates to understand in quantum mechanics. In this paper, a number of numerical experiments are designed to help students build visual images of angular momentum. The design concept is to break the boundaries between elementary and advanced quantum mechanics, basic theory and cutting-edge research, guide students to calculate and draw by themselves, discuss with each other, and build their own physical pictures. Through this efficient way of learning by doing, students can get an intuitive sense of abstract concepts from some simple calculations. The discussion starts from rotation operation, gradually increases the difficulty to squeezing operation, and introduces the basic concepts of spin squeezed state and Husimi Q representation. This design skillfully links classroom teaching with frontier research, bringing deeper thinking and inspiration to undergraduates.

DOI
10.26599/phys.2026.9320112

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Angular momentum is a difficult knowledge point for undergraduates to understand in quantum mechanics. In this paper, a number of numerical experiments are designed to help students build visual images of angular momentum. The design concept is to break the boundaries between elementary and advanced quantum mechanics, basic theory and cutting-edge research, guide students to calculate and draw by themselves, discuss with each other, and build their own physical pictures. Through this efficient way of learning by doing, students can get an intuitive sense of abstract concepts from some simple calculations. The discussion starts from rotation operation, gradually increases the difficulty to squeezing operation, and introduces the basic concepts of spin squeezed state and Husimi Q representation. This design skillfully links classroom teaching with frontier research, bringing deeper thinking and inspiration to undergraduates.
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Abstract Quantum mechanics (QM) forms the most crucial ingredient of modern-era physical science curricula at undergraduate level. The abstract ideas involved in QM related concepts pose a challenge towards appropriate visualization as a consequence of their counter-intuitive nature and lack of experiment-assisted visualization tools. At the heart of the quantum mechanical formulation lies the concept of ‘wavefunction’, which forms the basis for understanding the behavior of physical systems. At undergraduate level, the concept of ‘wavefunction’ is introduced in an abstract framework using mathematical tools and therefore opens up an enormous scope for alternative conceptions and erroneous visualization. The present work is an attempt towards exploring the visualization models constructed by undergraduate students for appreciating the concept of ‘wavefunction’. We present a qualitative analysis of the data obtained from administering a questionnaire containing four visualization based questions on the topic of ‘wavefunction’ to a group of ten undergraduate-level students at an institute in India which excels in teaching and research of basic sciences. Based on the written responses, all ten students were interviewed in detail to unravel the exact areas of difficulty in visualization of ‘wavefunction’. The outcome of present study not only reveals the gray areas in students’ conceptualization, but also provides a plausible route to address the issues at the pedagogical level within the classroom.
The Stern Gerlach experiment has played a central role in the discovery of spin angular momentum and it has also played a pivotal role in elucidating foundational issues in quantum mechanics. Here, we discuss investigation of students’ difficulties related to the Stern Gerlach experiment by giving written tests and interviewing advanced undergraduate and graduate students in quantum mechanics. We also discuss preliminary data that suggest that the Quantum Interactive Learning Tutorial (QuILT) related to the Stern Gerlach experiment is helpful in improving students’ understanding of these concepts.
Designed for a two-semester advanced undergraduate or graduate level course, this distinctive and modern textbook provides students with the physical intuition and mathematical skills to tackle even complex problems in quantum mechanics with ease and fluency. Beginning with a detailed introduction to quantum states and Dirac notation, the book then develops the overarching theoretical framework of quantum mechanics, before explaining physical quantum mechanical properties such as angular momentum and spin. Symmetries and groups in quantum mechanics, important components of current research, are covered at length. The second part of the text focuses on applications, and includes a detailed chapter on quantum entanglement, one of the most exciting modern applications of quantum mechanics, and of key importance in quantum information and computation. Numerous exercises are interspersed throughout the text, expanding upon key concepts and further developing students' understanding. A fully worked solutions manual and lecture slides are available for instructors.
Based on years of teaching experience, this book provides a clearly structured and accessible introduction to the fundamental principles of quantum mechanics, tailored for undergraduate students. The text builds a strong conceptual and mathematical foundation while promoting intuitive understanding and critical thinking. Students progress from basic concepts such as wave mechanics to more advanced topics within matrix mechanics. Complex concepts are presented in a diligent and understandable way, achieving a balance between conceptual insight and mathematical rigor without oversimplifying the subject. Thoroughly class-tested, this book is designed to support both classroom instruction and independent study. It will serve as a great tool for undergraduates studying introductory and intermediate quantum mechanics, as well as modern physics and quantum systems for engineers.