Teaching Quantum Mechanics Research Articles

USING ANDROID-BASED EQUATION PLOTTERS AS SUPPORTING TOOLS FOR TEACHING AND LEARNING ATOMIC ORBITALS

Chemistry undergraduate students are required to master the theory of quantum mechanics as the basic requirement to study the properties of the submicroscopic particle. However, many studies showed the students’ difficulties to bring the quantum mechanical concept to the macroscopic world and vise-versa. The use of computer visualization tool will very useful for teaching and learning the quantum concept, especially electron and its related atomic orbitals. This technological report paper demonstrates some of the Android-based applications i.e FuncXY, TriPlot 3D Graphing, and Graphing calculator to effectively, quickly, and accurately convert wave functions into various hydrogen atomic orbital visualizations. By selecting spherical coordinate and entering wave equation, these smartphone graphical devices visualize three-dimensional atomic orbitals. Users are also easy to rotate, zoom and manipulate the object. Finally, these user-friendly plotters are very recommended to assist chemistry educators in teaching quantum mechanics.

A Holistic Picture of Physics Student Conceptions of Energy Quantization, the Photon Concept, and Light Quanta Interference

A detailed investigation of student conceptions of quantum phenomena is needed, both to characterize student understanding of quantum concepts and to inform how we might teach quantum mechanics (QM). In this vein, in-depth semi-structured interviews were conducted with 35 students who majored in physics and received university-level QM instructions. Interview protocols were used and based on three quantum contexts: the quantization of energy in explaining the blackbody radiation, the photon concept in explaining the photoelectric effect, and light quanta in explaining the gradual formation of an interference pattern in the cases of low-intensity light beam. By applying a developmental phenomenographic analysis of the interview responses, three description categories (i.e., general patterns behind the conceptual understandings used in explaining each quantum contexts) were identified. These categories of descriptions revealed that most students’ thinking regarding foundational concepts in QM ranged from naive and deficient descriptions based on classical ontologies to simple hybrid and/or mixed models of classical and quantum conceptions. Regarding learning QM, the study found that the perspective of naive and classical ontologies in explaining quantum phenomena influenced students’ responses; they made incorrect generalizations and/or inappropriate links to the concepts learned in classical physics; and patterns of incorrect notions of QM are analogous to those that were documented. Besides, the study confirmed that students’ conceptual difficulties with QM are real, stable over time, and cross-cultural. It seems that the challenge to make QM interesting, effective, and relevant for physics students is a universal concern that knows no boundaries.

Quantum Nonlocality and Quantum Correlations in the Stern-Gerlach Experiment.

The Stern–Gerlach experiment (SGE) is one of the foundational experiments in quantum physics. It has been used in both the teaching and the development of quantum mechanics. However, for various reasons, some of its quantum features and implications are not fully addressed or comprehended in the current literature. Hence, the main aim of this paper is to demonstrate that the SGE possesses a quantum nonlocal character that has not previously been visualized or presented before. Accordingly, to show the nonlocality into the SGE, we calculate the quantum correlations by redefining the Banaszek–Wódkiewicz correlation in terms of the Wigner operator, that is , where is the Wigner operator, is the Pauli spin operator in an arbitrary direction and is the quantum state given by an entangled state of the external degree of freedom and the eigenstates of the spin. We show that this correlation function for the SGE violates the Clauser–Horne–Shimony–Holt Bell inequality. Thus, this feature of the SGE might be interesting for both the teaching of quantum mechanics and to investigate the phenomenon of quantum nonlocality.

Teaching Quantum Mechanics with Blended Learning

Quantum mechanics is very difficult for the beginners to learn, because of its high degree of abstraction and the high request to the advanced mathematical knowledge. For undergraduate students majoring in materials physical at the University of Jinan, the quantum theory is particularly difficult. The main reason is their fragile mathematical basis and students' indifference. The aim of our developmental work is to supply the mathematical knowledge reasonable and inspire the students learning interest in the limited class hour, based on the blended learning. We share here the concrete examples of how blended learning has been used into quantum mechanics teaching. Our blended learning practice provides ideas for developing new ways of quantum mechanics teaching. Whether camera-ready or not, please follow the guidelines.

Playing with Quantum Toys: Julian Schwinger’s Measurement Algebra and the Material Culture of Quantum Mechanics Pedagogy at Harvard in the 1960s

In the early 1960s, a PhD student in physics, Costas Papaliolios, designed a simple—and playful—system of Polaroid polarizer filters with a specific goal in mind: explaining the core principles behind Julian Schwinger’s quantum mechanical measurement algebra, developed at Harvard in the late 1940s and based on the Stern-Gerlach experiment confirming the quantization of electron spin. Papaliolios dubbed his invention “quantum toys.” This article looks at the origins and function of this amusing pedagogical device, which landed half a century later in the Collection of Historical Scientific Instruments at Harvard University. Rendering the abstract tangible was one of Papaliolios’s demonstration tactics in reforming basic teaching of quantum mechanics. This article contends that Papaliolios’s motivation in creating the quantum toys came from a renowned endeavor aimed, inter alia, at reforming high-school physics training in the United States: Harvard Project Physics. The pedagogical study of these quantum toys, finally, compels us to revisit the central role playful discovery performs in pedagogy, at all levels of training and in all fields of knowledge.

Undergraduate quantum mechanics: lost opportunities for engaging motivated students?

Quantum mechanics is widely recognised as an important and difficult subject, and many studies have been published focusing on students’ conceptual difficulties. However, the sociocultural aspects of studying such an emblematic subject have not been researched to any large extent. This study explores students’ experiences of undergraduate quantum mechanics using qualitative analysis of semi-structured interview data. The results inform discussions about the teaching of quantum mechanics by adding a sociocultural dimension. Students pictured quantum mechanics as an intriguing subject that inspired them to study physics. The study environment they encountered when taking their first quantum mechanics course was however not always as inspiring as expected. Quantum mechanics instruction has commonly focused on the mathematical framework of quantum mechanics, and this kind of teaching was also what the interviewees had experienced. Two ways of handling the encounter with a traditional quantum mechanics course were identified in the interviews; either students accept the practice of studying quantum mechanics in a mathematical, exercise-centred way or they distance themselves from these practices and the subject. The students who responded by distancing themselves experienced a crisis and disappointment, where their experiences did not match the way they imagined themselves engaging with quantum mechanics. The implications of these findings are discussed in relation to efforts to reform the teaching of undergraduate quantum mechanics.

Relativistic particle in a box: Klein–Gordon versus Dirac equations

The problem of a particle in a box is probably the simplest problem in quantum mechanics which allows for significant insight into the nature of quantum systems and thus is a cornerstone in the teaching of quantum mechanics. In relativistic quantum mechanics this problem allows also to highlight the implications of special relativity for quantum physics, namely the effect that spin has on the quantised energy spectra. To illustrate this point, we solve the problem of a spin zero relativistic particle in a one- and three-dimensional box using the Klein–Gordon equation in the Feshbach–Villars formalism. We compare the solutions and the energy spectra obtained with the corresponding ones from the Dirac equation for a spin one-half relativistic particle. We note the similarities and differences, in particular the spin effects in the relativistic energy spectrum. As expected, the non-relativistic limit is the same for both kinds of particles, since, for a particle in a box, the spin contribution to the energy is a relativistic effect.

Pinturas de Salvador Dalí para introduzir conceitos de Mecânica Quântica no Ensino Médio

Poucos trabalhos têm apresentado resultados do ensino de mecânica quântica no ensino médio. Como um alto nível de abstração é exigido, é necessário reconsiderar os pressupostos teóricos, abordagens e metodologias, a fim de reduzir as dificuldades na sua inserção. Este artigo apresenta os resultados de uma pesquisa, que é uma proposta didática para ensino médio compreendendo discussões sobre mecânica quântica e elementos que relacionam física e algumas pinturas de Salvador Dalí. Abraçar as relações entre a física e a arte nos permite aproximar e expressar as relações entre a educação científica e a cultura, com formas alternativas para enriquecer o significado do conhecimento, também buscando proporcionar aos alunos uma visão mais ampla sobre a construção do conhecimento científico, em um diálogo inteligente com o mundo. Para isso, a proposta apresenta uma metodologia operatória, aproximando a ciência como conhecimento histórico e social para ensinar mecânica quântica, e visa tornar o aluno capaz de desenvolver hipóteses, conceitos, situar explicações científicas no tempo, em suma, operar de acordo com as ferramentas fornecidas. Trabalhar com as formas pelas quais a física, introduzida como um componente cultural, influencia as interpretações dos fenômenos macro e microscópicos contribui para a compreensão da física moderna, parte da física do século XX. Portanto, adicionar elementos culturais ao mundo da física parece ser cada vez mais necessário, uma vez que parece ressignificá-la ao reumanizar seu conhecimento. Além disso, usar o contexto histórico parece nos dar uma maior amplitude ao conhecimento, uma vez que ele também começa a fazer parte de uma cultura mais ampla, além de seu significado científico.

Planck’s radiation law, the light quantum, and the prehistory of indistinguishability in the teaching of quantum mechanics

Planck’s law for black-body radiation marks the origin of quantum theory and is discussed in all introductory (or advanced) courses on this subject. However, the question whether Planck really implied quantisation is debated among historians of physics. We present a simplified account of this debate which also sheds light on the issue of indistinguishability and Einstein’s light quantum hypothesis. We suggest that the teaching of quantum mechanics could benefit from including this material beyond the question of historical accuracy.

How diverse are physics instructors’ attitudes and approaches to teaching undergraduate level quantum mechanics?

Understanding instructors’ attitudes and approaches to teaching undergraduate-level quantum mechanics can be helpful in developing effective instructional tools to help students learn quantum mechanics. Here we discuss the findings from a survey in which 12 university faculty members reflected on various issues related to undergraduate-level quantum mechanics teaching and learning. Topics included faculty members’ thoughts on the goals of a college quantum mechanics course, general challenges in teaching the subject matter, students’ preparation for the course, views about foundational issues and the difficulty in teaching certain topics, reflection on their own learning of quantum mechanics when they were students versus how they teach it to their students and the extent to which they incorporate contemporary topics into their courses. The findings related to instructors’ attitudes and approaches discussed here can be useful in improving teaching and learning of quantum mechanics.

Insights into teaching quantum mechanics in secondary and lower undergraduate education

This study presents a review of the current state of research on teaching quantum mechanics in secondary and lower undergraduate education. A conceptual approach to quantum mechanics is being implemented in more and more introductory physics courses around the world. Because of the differences between the conceptual nature of quantum mechanics and classical physics, research on misconceptions, testing, and teaching strategies for introductory quantum mechanics is needed. For this review, 74 articles were selected and analyzed for the misconceptions, research tools, teaching strategies and multimedia applications investigated. Outcomes were categorized according to their contribution to the various subtopics of quantum mechanics. Analysis shows that students have difficulty relating quantum physics to physical reality. It also shows that the teaching of complex quantum behavior, such as time dependence, superposition and the measurement problem, has barely been investigated for the secondary and lower undergraduate level. At the secondary school level, this review shows a need to investigate student difficulties concerning wave functions and potential wells. Investigation of research tools shows the necessity for the development of assessment tools for secondary and lower undergraduate education, which cover all major topics and are suitable for statistical analysis. Furthermore, this review shows the existence of very diverse ideas concerning teaching strategies for quantum mechanics and a lack of research into which strategies promote understanding. This review underlines the need for more empirical research into student difficulties, teaching strategies, activities and research tools intended for a conceptual approach for quantum mechanics.

工学部における量子力学教育 : ケース・スタディー

工学部における量子力学教育 : ケース・スタディー

A full quantum analysis of the Stern–Gerlach experiment using the evolution operator method: analyzing current issues in teaching quantum mechanics

To the quantum mechanics specialists community it is a well-known fact that the famous original Stern–Gerlach experiment (SGE) produces entanglement between the external degrees of freedom (position) and the internal degree of freedom (spin) of silver atoms. Despite this fact, almost all textbooks on quantum mechanics explain this experiment using a semiclassical approach, where the external degrees of freedom are considered classical variables, the internal degree is treated as a quantum variable, and Newton's second law is used to describe the dynamics. In the literature there are some works that analyze this experiment in its full quantum mechanical form. However, astonishingly, to the best of our knowledge the original experiment, where the initial states of the spin degree of freedom are randomly oriented coming from the oven, has not been analyzed yet in the available textbooks using the Schrödinger equation (to the best of our knowledge there is only one paper that treats this case: Hsu et al (2011 Phys. Rev. A 83 012109)). Therefore, in this contribution we use the time-evolution operator to give a full quantum mechanics analysis of the SGE when the initial state of the internal degree of freedom is completely random, i.e. when it is a statistical mixture. Additionally, as the SGE and the development of quantum mechanics are heavily intermingled, we analyze some features and drawbacks in the current teaching of quantum mechanics. We focus on textbooks that use the SGE as a starting point, based on the fact that most physicist do not use results from physics education research, and comment on traditional pedagogical attitudes in the physics community.

WavePacket: A Matlab package for numerical quantum dynamics. I: Closed quantum systems and discrete variable representations

WavePacket is an open-source program package for the numerical simulation of quantum-mechanical dynamics. It can be used to solve time-independent or time-dependent linear Schrödinger and Liouville–von Neumann-equations in one or more dimensions. Also coupled equations can be treated, which allows to simulate molecular quantum dynamics beyond the Born–Oppenheimer approximation. Optionally accounting for the interaction with external electric fields within the semiclassical dipole approximation, WavePacket can be used to simulate experiments involving tailored light pulses in photo-induced physics or chemistry. The graphical capabilities allow visualization of quantum dynamics ‘on the fly’, including Wigner phase space representations. Being easy to use and highly versatile, WavePacket is well suited for the teaching of quantum mechanics as well as for research projects in atomic, molecular and optical physics or in physical or theoretical chemistry. The present Part I deals with the description of closed quantum systems in terms of Schrödinger equations. The emphasis is on discrete variable representations for spatial discretization as well as various techniques for temporal discretization. The upcoming Part II will focus on open quantum systems and dimension reduction; it also describes the codes for optimal control of quantum dynamics. The present work introduces the MATLAB version of WavePacket 5.2.1 which is hosted at the Sourceforge platform, where extensive Wiki-documentation as well as worked-out demonstration examples can be found. Program summaryProgram Title: WavePacketProgram Files doi:http://dx.doi.org/10.17632/tc5mrhx9sb.1Licensing provisions: GPLv3Programming language:MatlabNature of problem:Schrödingerś equations are of fundamental importance in non-relativistic quantum mechanics of distinguishable particles. The solutions of the time-independent equation (TISE) are wavefunctions in coordinate space, the absolute squares of which are usually interpreted as probability density. The time-dependent equation (TDSE) describes the dynamics of a quantum system evolving in time. It plays a crucial role for the simulation, understanding, and prediction of modern experiments in atomic, molecular and optical physics where systems are driven by temporally shaped external fields.Solution method:All numerical methods compiled in WavePacket are based on discrete variable representations. Currently implemented are Gauss–Hermite, Gauss–Legendre and FFT-based schemes. The TISE is solved either by direct diagonalization or by propagation in imaginary time. For the TDSE there is a choice between second order differencing, operator splitting and Chebychev polynomial methods.Additional comments including Restrictions and Unusual features:The WavePacket program package is rather easy and intuitive to use, providing visualization of quantum dynamics ‘on the fly’. It is mainly intended for low-dimensional systems, typically not exceeding three to five degrees of freedom. Detailed user guides and reference manuals are available through numerous Wiki pages hosted at the SourceForge platform where also a large number of well documented demonstration examples can be found.

Computations in quantum mechanics made easy

Convenient and simple numerical techniques for performing quantum computations based on matrix representations of Hilbert space operators are presented and illustrated by various examples. The applications include the calculations of spectral and dynamical properties for one-dimensional and two-dimensional single-particle systems as well as bosonic many-particle and open quantum systems. Due to their technical simplicity these methods are well suited as a tool for teaching quantum mechanics to undergraduates and graduates. Explicit implementations of the presented numerical methods in Matlab are given.

About the wave’s concept in quantum mechanics and how university student understand its aspects. Algerian students’ case

The most important basic aspect of Quantum Mechanics description is the wave, but there are many different interpretations between different schools and scientists about the reality of this wave. Till now there are more than thirteen important schools of interpretation for the basic concepts of quantum mechanics. In the case of the wave, some interpretations consider it as a real wave, while others consider it as only a mathematics toll. Through our experience in teaching quantum theory, we conclude that teaching quantum mechanics is not only just deriving beautiful and maybe logical equations and relations related to the subject. We believe that each step of deriving those equations or relations should be followed by deep discussions of concepts and meanings. Because of the students difficulty for understand the wave concept as an example of general basic concepts of quantum mechanics. For this purpose and in order to clarify and focus on the problematic, a questionnaire sheet was presented to homogenous sample of Algerian third year physics students related to the subject. But we concluded from the results of students’ survey, that the basic concepts of quantum mechanics should have a profound educational revision.

Enhancing student learning of two-level quantum systems with interactive simulations

The QuVis Quantum Mechanics Visualization project aims to address challenges of quantum mechanics instruction through the development of interactive simulations for the learning and teaching of quantum mechanics. In this article, we describe the evaluation of simulations focusing on two-level systems developed as part of the Institute of Physics Quantum Physics resources. Simulations are research-based and have been iteratively refined using student feedback in individual observation sessions and in-class trials. We give evidence that these simulations are helping students learn quantum mechanics concepts at both the introductory and advanced undergraduate level, and that students perceive simulations to be beneficial to their learning.

Numerical determination of the eigenenergies of the Schrödinger equation in one dimension

In most approaches to teaching quantum mechanics, discrete energy levels are introduced through the example of the infinite square well. Here, the boundary conditions at the walls, namely a vanishing amplitude of the wave function, lead to quantized energy levels. This contribution extends the discussion to more realistic, smooth potentials in one dimension, including non-symmetric potentials. A numerical solution algorithm for the corresponding time-independent Schrödinger equation illustrates quantization in an intuitive manner by use of simple forward-shooting and bisection methods. It also demonstrates how computational physics can go beyond finding approximate solutions to a specific problem and truly aid the understanding of a basic concept in physics.

Assessing Advanced High School and Undergraduate Students’ Thinking Skills: The Chemistry—From the Nanoscale to Microelectronics Module

Chemistry students in Israel have two options for studying chemistry: basic or honors (advanced placement). For instruction in high school honors chemistry courses, we developed a module focusing on abstract topics in quantum mechanics: Chemistry—From the Nanoscale to Microelectronics. The module adopts a visual–conceptual approach, which replaces mathematical derivations and includes interdisciplinary, real-life applications. The module has also been used as enrichment material for an undergraduate mathematically oriented quantum chemistry course. We assessed the comprehension of quantum mechanical concepts and the thinking skills of high school honors students and undergraduate chemistry students who studied this module. These skills included visual and textual chemical understanding, graphing, and far transfer of learning. Participants included over 100 high school honors students in high and intermediate levels, and over 60 undergraduate students, with students separated into groups of those exposed to the module and those who received mathematics enrichment instead. The questionnaires revealed that both high school honors students and undergraduate students improved their scores. High-level academic students outperformed their undergraduate peers, and undergraduate students who had been exposed to the module outperformed undergraduates who had studied the topic with mathematical enrichment. The research contributes to the field of teaching quantum mechanics and thinking skills, suggesting that high school honors students and undergraduate students could benefit from studying quantum mechanics using a visual–conceptual approach along with real-life applications. This contribution is part of a special issue on teaching introductory chemistry in the context of the advanced placement (AP) chemistry course redesign.

Quantum mechanics teaching resources from the Institute of Physics

In December 2013, the Institute of Physics (IOP) launched a set of freely available resources at quantumphysics.iop.org for the teaching and learning of quantum mechanics. The website includes about 80 short articles written by experts in the field and 17 interactive simulations with accompanying activities for an introductory course in quantum mechanics starting from two-level systems. The articles are arranged according to five themes, including a focus on quantum information, interpretations of quantum mechanics, the mathematical structure of the theory, physics applications and historical experiments. The resources make topics such as entanglement, hidden variables and quantum information theory accessible to introductory-level students. They can be used flexibly for a variety of instructional aims at both the introductory and more advanced level. The website includes links to pre-readings, suggestions for further reading, a glossary of technical terms and allows users to rate their understanding of articles.Sharing of these resources is encouraged, with all usage under the Creative Commons CC BY-NC-ND licence. Solutions to problems and activities are available for instructors by emailing quantumphysics@iop.org. Instructors interested in evaluating these resources with their students in order to help us further develop and optimize the site are requested to contact the corresponding author.