Open Source and Open Access Resources for Quantum Physics Education, Lecture notes of Quantum Mechanics

Download Open Source and Open Access Resources for Quantum Physics Education and more Lecture notes Quantum Mechanics in PDF only on Docsity! Open Source and Open Access Resources for Quantum Physics Education Mario Belloni, Wolfgang Christian Department of Physics Davidson College Bruce Mason Homer L. Dodge Department of Physics & Astronomy University of Oklahoma Quantum mechanics is both a topic of great importance to modern science, engineering, and technology, and a topic with many inherent barriers to learning and understanding. Computational resources are vital tools for developing deep conceptual understanding of quantum systems for students new to the subject. This article outlines two projects that are taking an open source/open access approach to create and share teaching and learning resources for quantum physics. The Open Source Physics project provides program libraries, programming tools, example simulations, and pedagogical resources for instructors wishing to give a rich experience to their students. These simulations and student activities are, in turn, being integrated into a world-wide collection of teaching and learning resources available through the Quantum Exchange and the ComPADRE Portal to the National Science Digital Library. Both of these projects use technologies that encourage community development and collaboration. Using these tools, faculty can create learning experiences, share and discuss their content with others, and combine resources in new ways. Examples of the available content and tools are given, along with an introduction to accessing and using these resources. Since its inception in the 1920’s, quantum mechanics has been essential for advancements in fields that require an accurate description of atomic and sub-atomic phenomena. Advances in atomic, nuclear, and solid-state physics and most of chemistry are a direct result of our understanding and application of quantum mechanics. The laser is an obvious practical example. The ubiquitous solid state laser is based on simple quantum principles. These quantum systems now appear in grocery stores (scanners), operating rooms (laser surgery), entertainment devices (CD and DVD players), and even toys for pets. Similarly, the understanding of quantum theory is crucial for medicine with the advent and use of modern diagnostic techniques (e.g. PET scans and MRIs) based on quantum phenomena. Modern electronic devices, advanced nano-structure materials, and the latest in cryptography are all examples of how quantum theory is relevant to technology. The importance and relevance of quantum theory is reflected in the physics and chemistry curricula. Students see aspects of quantum mechanics in their introductory, intermediate, and advanced courses. The teaching of quantum mechanics at the introductory level is as important as the advanced level because of the audience. For many students, this will be the only time that they will have a chance to learn about the science responsible for many of the products that shape their lives. Of course, for the physics and chemistry students who will become the scientists of the future, understanding quantum theory is essential for making new fundamental and applied discoveries. Despite the importance of quantum theory, its teaching and learning is a difficult endeavor. To develop a conceptual “feel” for quantum systems, students must grasp the properties and time evolution of abstract objects and operators in an abstract vector space and then connect them to familiar measurable quantities such as position or energy. The most basic quantum dynamics is challenging because of the two very different and non-trivial time dependences of the theory; that of an isolated quantum system and that of “observations”. Providing help for students trying to understand these systems, and faculty teaching them, is the goal of the projects described in this article. I. Time-Dependence: A Learning Challenge In quantum courses, “time dependence” often refers to the deterministic evolution governed by the Schrödinger equation. This separable partial differential equation is usually solved by finding eigenstates of the spatial term, the time-independent Schrödinger equation (TISE), then incorporating the time-dependent phase ( ) ( ) tEi nn nertr −=Ψ ψ, . The solutions to the TISE, the energy eigenstates, contain the physics of the problem including boundary conditions, potentials, and interactions. Any arbitrary state can be constructed from a superposition of energy eigenstates: ( ) ( ) / 1 , ni E t n n n t c eψ ∞ − = Ψ =∑r r (1) A fundamentally different time dependence is the result of measurements on a quantum-mechanical system and is much more abstract. The canonical interpretation specialized nature of the teaching of quantum mechanics to more sophisticated students. This article describes Open Source Physics (OSP), an effort to improve pedagogical software for intermediate and advanced classes, and the resources for quantum mechanics education that are being developed as a part of this effort. The guiding paradigm is similar to much of modern software development; that an open community using and developing a code base is more efficient and effective than a small, closed group of programmers. This open access approach is as important for the reuse and repurposing of the pedagogical resources connected with the simulations as it is for the computer code. Students and teachers need resources that they can tailor to their needs and specific learning context. Physlets, scriptable java applets for introductory physics topics, provide an example of the power of tools designed for reuse [Christian and Belloni, 2001; Christian and Belloni, 2004]. Although not open source, the built-in javascript connections of these programs have made them a world- wide standard for the creation of simulation-enabled curricular pedagogies. The work of Duffy [Duffy, 2008] and Schneider [Schneider, 2008] are examples. To be effective, these resources must be disseminated as widely as possible to encourage access, sharing, and development. The ComPADRE Pathway of the National STEM Digital Library (NSDL) is helping in this effort. ComPADRE provides an online catalog of educational resources, almost all open access materials, where developers, educators, and students can share their work, rank and comment on materials in the catalog, and build personal collections that meet their own needs. Built on top of this catalog are specialized collections of materials designed for particular groups of users. The Quantum Exchange, an online collection of learning resources for quantum physics, makes use of many of the resources described here. The OSP quantum materials are an important part of the Quantum Exchange and, at the same time, ComPADRE and the Quantum Exchange can help disseminate the OSP results. Furthermore, through ComPADRE, the OSP resources are shared with the NSDL and connected to ComPADRE’s four professional society sponsors, the American Association of Physics Teachers, the American Physical Society, the American Astronomical Society, and the Society of Physics Students, providing greater opportunities for attracting attention to these high quality quantum education resources. In fact, ComPADRE has recently created a collection specifically for all OSP materials to take advantage of the library’s database, library, and dissemination tools. 3. Open Source Physics: Overview of Project, Library, Tools, and Examples The Open Source Physics (OSP) project promotes the innovative and effective uses of computation, computer-based curricular materials, and computer modeling through the integrated use of open source programs and models. Our material is based on (1) a consistent object-oriented Java library that is distributed under the GNU General Public License (GPL), (2) a computational physics textbook that uses a physics-first approach to motivate numerical algorithms and computer programming, and (3) high- level authoring and modeling tools that allow non-programmers to build, explore, edit, and distribute ready to run models. Although the modeling instruction method [Hestenes, 2008] can be used without computers, the use of computers allows students to study problems that are difficult and time consuming, to visualize their results, and to communicate their results with others. The combination of computer programming and modeling, theory, and experiment can achieve insight and understanding that cannot be achieved with only one approach. The OSP library is the basis for OSP curricular material. The library contains numerical methods, user-interface components, visualization tools, and an XML framework. The library’s source code and numerous examples are distributed under the GNU General Public License (GPL). An Introduction to Computer Simulation Methods by Harvey Gould, Jan Tobochnik, and Wolfgang Christian [Gould et al., 2007] uses this library to teach programming in the context of learning physics. This book does not discuss syntax or programming for its own sake but stresses the science and encourages student experimentation. The development of good programming habits is done by example. It is not necessary to become expert in programming to use the programs in An Introduction to Computer Simulation Methods. Compiled versions of these programs are available and run on any Java-enabled computer. In addition, we are currently modifying and adapting these programs for use in other contexts. For example, we have created a suite of programs based on algorithms described in the book to help students develop an understanding of the time dependence of quantum mechanical states. These programs are based on the superposition principle outlined in Eq. (1). The simplest is called QM Superposition and is shown in Figure 1. It displays the time evolution of the position-space wave function, ( )tx,Ψ , using an energy eigenstate expansion. In the example shown in Figure 1, the simulation shows a two-state superposition in a harmonic oscillator, ( ) 2/2xxV = . Because time-dependent wave functions are complex, the program must display complex functions. We depict the Figure 1: The QM Superposition program showing the wave function for a two- state superposition in a harmonic oscillator. The initial state and potential energy well can be customized to almost any state or well. The legend at the top maps the color of the wave function into phase of the complex functions. This program is available at: http://www.compadre.org/OSP/items/detail.cfm?ID=6798 . phase of the wave function with the colors shown on the color strip. The simulation is controlled by three buttons, play/pause, step, and reset. Figure 2: The QM Superposition control panel allows parameters to be changed and saved. The Initialize button switches the program to run mode where the buttons change to Run, Step, and New. The New button allows the user to enter new values and re-run the simulation. • Expectation P which adds a graph showing the expectation value of momentum • Carpet which adds a space-time graph of the wave function dynamics • FFT which adds a graph of the wave function in momentum space • Momentum Carpet which adds a momentum-time graph of the wave function dynamics • Wigner which adds a graph of the quasi-phase space (x-p) distribution A second approach to using OSP material is to use a high-level authoring and modeling tool that builds programs with minimal programming. Easy Java Simulations (EJS) [Esquembre, 2008] is an Open Source Physics application that enables both programmers and novices to quickly and easily prototype, test, and distribute packages of Java simulations. It is well suited for education because it is simple to use and combines authoring with powerful modeling tools. Its dynamic and highly interactive user interface greatly reduces the amount of programming required to implement an idea. Even experienced programmers find EJS useful, because it is faster and easier to: • Develop a prototype of an application in order to test an idea or algorithm, • Create user interfaces without programming, • Create models whose structure and algorithms non-programmers can inspect and understand, • Encourage students or colleagues (who may be new to Java) to create their own simulations, • Quickly prepare simulations to be distributed as applets or as standalone programs, • Create a package containing multiple programs and the associated curricular material. Figure 4 shows an EJS simulation of quantum mechanical tunneling through a Dirac delta function potential. Delta function tunneling can be described analytically but the resulting wave function is difficult to visualize without animation. The amplitude and phase of the state vary in both space and time. Well-designed computer models showing analytical solutions are useful because they reinforce the mathematics and provide feedback about the model's correctness and applicability. Figure 4: EJS model of quantum scattering due to a 1D delta function. The simulation provides visual illustrations of an analytically solvable model and allows students to explore the system in both time and the energy of the scattering state. This model is available at: http://www.compadre.org/osp/items/detail.cfm?ID=6990 4. Newly Accessible Physics The increasing power of computers allows simulations of more interesting and complicated phenomena. Wave packet dynamics is an excellent example. Schrödinger first proposed a localized wave packet solution to his wave equation to connect classical and quantum mechanics. Over the last 30 years quantum- mechanical wave packets and their revivals, the fact that certain bound-state wave packets reform to their original shape in a predictable way, have received considerable theoretical attention and experimental verification. Figure 5: The QM Wigner program showing the position-space wave function (top) and the Wigner quasi-phase-space representation (bottom). The initial state is a Gaussian (left). The middle images show the quantum bounce with the left wall of the well. The images on the right depict a fractional revival in which two mini wave packets appear on top of each other. The two wave packets can be observed separately with the Wigner function as shown. This program is available at: http://www.compadre.org/osp/items/detail.cfm?ID=6813 OSP tools allow the construction of wave packets, such as the one shown in Figure 5, and the exploration of their evolution in time. Theoretical research has most often focused on wave packets in the infinite square well because of the two well-defined time scales: the classical “bounce” time and the quantum revival time. The first bounce of the initially localized quantum state against one wall occurs at one quarter of the classical period of oscillation. Investigation of the bounce illustrates the similarities and differences between the classical and quantum-mechanical systems. On a considerably longer time scale, the well-known exact and fractional revivals occur in the infinite square well. At revivals, the wave packet or copies of the original wave packet, sometimes called mini-packets or clones, reform long after the original packet has spread throughout the well. Student exploration of this behavior can be an excellent exercise on the time dependence of quantum states and results of the expansion in energy eigenstates. The added visualization of the quasi-phase space to describe the materials and their use. Folders can be either private or public. The Quantum Exchange [Quantum Exchange, 2008] is the ComPADRE collection where the OSP quantum pedagogical resources are being incorporated. This collection is focused on providing resources that can help students understand the abstract, non- classical aspects of quantum theory described above. The OSP materials are featured in the collection because of their connection between pedagogy and interactive computer simulations. The catalog includes other simulations, more traditional pedagogical materials such as notes and homework problems, and references on effective teaching of quantum mechanics. Through the ComPADRE interface, these different resources can be connected where appropriate. For example, online lecture notes on spin physics and a simulation of the Stern-Gerlach experiment can be related through their ComPADRE records. In particular, every registered ComPADRE user has a personal filing cabinet to gather, organize, and annotate their favorite resources, such as shown in Figure 6. As with all ComPADRE collections, registered users are also able to recommend materials for consideration by the editor, comment on materials, and share folders from their filing cabinets with other users. Through modification in the OSP curricular resources and the ComPADRE submission process, faculty teaching different topics can easily share their work. The connection between OSP and ComPADRE is also being leveraged to help both projects grow. The connections between ComPADRE and other digital libraries provide avenues for spreading the word about the OSP resources. The harvesting of ComPADRE records by the NSDL is particularly important for this dissemination. Workshops and presentations by both groups publicize their materials and collaboration. Of course these connections also improve the rankings of resources by web search engines. In the near future an OSP-specific collection will be available on ComPADRE that will include all topics in physics. This will also provide a core set of quality pedagogical resources to develop new collections similar to the Quantum Exchange and expand existing collections. ComPADRE is also working with other curriculum developers to expand the depth and breadth of the library. We encourage all those involved in quantum education to try the resources described here. They are available through both the OSP web site on ComPADRE and the Quantum Exchange. The power of open source and open access resources is the feedback that comes from the community of users. Comments and suggestions are welcome and contributions are greatly appreciated. Most importantly, we hope that the materials presented here will help students develop a deeper understanding and appreciation of quantum mechanics. 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