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Boosting vector calculus with the graphical notation
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| Pengarang | : | - |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 89 (No. 2) |
| Halaman | : | 200-209 |
| Abstrak | : | Learning vector calculus techniques is one of the major hurdles faced by physics undergraduates. However, beginners report various difficulties dealing with the index notation due to its bulkiness. Meanwhile, there have been graphical notations for tensor algebra that are intuitive and effective in calculations and can serve as a quick mnemonic for algebraic identities. Although they have been introduced and applied in vector algebra in the educational context, to the best of our knowledge, there have been no publications that employ the graphical notation to three-dimensional Euclidean vector calculus, involving differentiation and integration of vector fields. Aiming for physics students and educators, we introduce such “graphical vector calculus,” demonstrate its pedagogical advantages, and provide a variety of exercises containing both purely mathematical identities and practical calculations in physics. The graphical notation can readily be utilized in the educational environment to not only lower the barriers in learning and practicing vector calculus but also make students interested and self-motivated to manipulate the vector calculus syntax and, on their own, heuristically comprehend the language of tensors. |
Length as a paradigm for understanding the classical limit
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 89 (No. 2) |
| Halaman | : | 185-199 |
| Abstrak | : | It is regrettable that the quantum length of an object is rarely if ever discussed, because it provides an ideal pedagogical paradigm for understanding how a physicist uses classical intuition to define quantum properties and how such quantum properties behave as one would expect in the classical limit. It also provides for a way to understand many-particle states, and leads to interesting quantum behavior that challenges our intuition about measurement. This exercise parallels the ways in which theories are developed, giving the student a concrete example of the thought process involved. |
Introducing SU(3) color charge in undergraduate quantum mechanics
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 89 (No. 2) |
| Halaman | : | 172-184 |
| Abstrak | : | We present a framework for investigating effective dynamics of SU(3) color charge. Two- and three-body effective interaction terms inspired by the Heisenberg spin model are considered. In particular, a toy model for a three-source “baryon” is constructed and investigated analytically and numerically for various choices of interactions. vpython is used to visualize the nontrivial color charge dynamics. The treatment should be accessible to undergraduate students who have taken a first course in quantum mechanics, and suggestions for independent student projects are proposed. |
d'Alembert digitized: A wave pulse method for visualizing electromagnetic waves in matter and for deriving the finite difference time domain method for numerically solving Maxwell's equations
| Pengarang | : | Ross Hyman |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 89 (No. 2) |
| Halaman | : | 164-171 |
| Abstrak | : | An alternative way of visualizing electromagnetic waves in matter and of deriving the Finite Difference Time Domain (FDTD) method for simulating Maxwell's equations for one-dimensional systems is presented. The method uses d'Alembert's splitting of waves into forward and backward pulses of arbitrary shape and allows for grid spacing and material properties that vary with the position. Constant velocity of waves in dispersionless dielectric materials, partial reflection and transmission at boundaries between materials with different indices of refraction, and partial reflection, transmission, and attenuation through conducting materials are derived without recourse to exponential functions, trigonometric functions, or complex numbers. Placing d'Alembert's method on a grid is shown to be equivalent to the FDTD method and allows for simple and visual proof that the FDTD method is exact for dielectrics when the ratio of the spatial and temporal grid spacing is the wave speed, a straightforward way to incorporate reflectionless boundary conditions and a derivation that the FDTD method retains second-order accuracy when the grid spacing varies with the position and the material parameters make sudden jumps across layer boundaries. |
Basics of sound in air: Correspondence with electromagnetic waves
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 89 (No. 2) |
| Halaman | : | 157-163 |
| Abstrak | : | The wave equation of sound is developed starting from the equations of fluid mechanics for the velocity and pressure variation in air. The equations for sound are analogous to Maxwell's equations for linear polarized electromagnetic plane waves: the pressure variation and the velocity correspond to the electric and magnetic fields, respectively. The order of magnitude of sound properties is explained in terms of the amplitude and energy density of the corresponding wave. |
Physically significant phase shifts in matter-wave interferometry
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 89 (No. 3) |
| Halaman | : | 324-332 |
| Abstrak | : | Many different formalisms exist for computing the phase of a matter-wave interferometer. However, it can be challenging to develop physical intuition about what a particular interferometer is actually measuring or about whether a given classical measurement provides equivalent information. Here, we investigate the physical content of the interferometer phase through a series of thought experiments. In low-order potentials, a matter-wave interferometer with a single internal state provides the same information as a sum of position measurements of a classical test object. In high-order potentials, the interferometer phase becomes decoupled from the motion of the interferometer arms, and the phase contains information that cannot be obtained by any set of position measurements on the interferometer trajectory. This phase shift in a high-order potential fundamentally distinguishes matter-wave interferometers from classical measuring devices. |
Active learning in a graduate quantum field theory course
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 89 (No. 3) |
| Halaman | : | 317-323 |
| Abstrak | : | This article describes how the author successfully adapted techniques drawn from the literature on active learning for use in a graduate-level course on quantum field theory. Students completed readings and online questions ahead of each class and spent class time working through problems that required them to practice the decisions and skills typical of a theoretical physicist. The instructor monitored these activities and regularly provided timely feedback to guide their thinking. Instructor-student interactions and student enthusiasm were similar to that encountered in one-on-one discussions with advanced graduate students. Course coverage was not compromised. The teaching techniques described here are well suited to other advanced courses. |
Quantum composer: A programmable quantum visualization and simulation tool for education and research
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 89 (No. 3) |
| Halaman | : | 307-316 |
| Abstrak | : | Interactive simulations and visualizations augment the teaching and learning of quantum mechanics by making equations and concepts come to life. However, graphical visualizations are nearly always limited to a set of hard-coded functionalities. Text-based codes can offer a higher degree of flexibility, but only at the expense of steep learning curves or time investments. We introduce Quantum Composer, which allows the user to build, expand, or explore quantum mechanical simulations by interacting with graphically connectable nodes, each corresponding to a physical concept, mathematical operation, or visualization. Quantum Composer eliminates numerical and programming details while retaining accessibility, emphasis on understanding, and rapid feedback mechanisms. We illustrate its open-ended applicability through a series of examples in introductory and advanced quantum mechanics courses, student projects, and research environments. |
Transport through quantum dots: An introduction via master equation simulations
| Pengarang | : | Robert A. Bush |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 89 (No. 3) |
| Halaman | : | 300-306 |
| Abstrak | : | In this work, we present a master equation approach to simulating DC transport through single electron transistors and quantum dots suitable for an upper-division undergraduate computational physics project. After introducing the basic theory describing transport through quantum dots, we present a simulation of the simple case of a metallic dot including effects due to the finite temperature of the leads. Building on this example, we simulate published data with orbital and spin states. We envision students building on these simulations to replicate other data in the published literature. Projects of this type would be suitable for an undergraduate independent study or computational project and will give students a strong introduction to the topic of transport through quantum dots without the need for expensive cryogenic and electrical measurement systems or device fabrication necessary for experimental work. |
How much electric surcharge fits on … a “white dwarf” star?
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | American Journal of Physics |
| Volume / Edisi | : | 89 (No. 3) |
| Halaman | : | 291-299 |
| Abstrak | : | The question of how much surplus of electric charge (“surcharge”) fits on an object is generally very difficult to answer. Here, it is shown that it is easy to answer when the object is a failed white dwarf star (a brown dwarf in its ground state) made of protons and electrons: Given the number of protons, how many electrons can there be? Surprisingly, the answer (in the form: as few as ? and as many as ??) is independent of the speed of light c and the Planck quantum h, even when the star is stabilized against collapse by relativistic quantum mechanics. |