The motion of a pendulum is derived using Fourier series and perturbation analysis at levels appropriate for undergraduate physics students. Instead of using the elliptic integral of the first kind, higher order terms of the Taylor-expanded differential equation are considered, leading to increasingly accurate corrections to the period in terms of a single expansion parameter. The relation between the expansion parameter and the initial conditions is not fixed, allowing many solutions to the motion in terms of the expansion parameter but a unique solution in terms of the initial conditions.
{"title":"An analysis of the large amplitude simple pendulum using Fourier series","authors":"Brennen Black, Vetri Vel","doi":"10.1119/5.0130943","DOIUrl":"https://doi.org/10.1119/5.0130943","url":null,"abstract":"The motion of a pendulum is derived using Fourier series and perturbation analysis at levels appropriate for undergraduate physics students. Instead of using the elliptic integral of the first kind, higher order terms of the Taylor-expanded differential equation are considered, leading to increasingly accurate corrections to the period in terms of a single expansion parameter. The relation between the expansion parameter and the initial conditions is not fixed, allowing many solutions to the motion in terms of the expansion parameter but a unique solution in terms of the initial conditions.","PeriodicalId":7589,"journal":{"name":"American Journal of Physics","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135324245","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"教育学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We review an explicit approach to obtaining numerical solutions of the Schrödinger equation that is conceptionally straightforward and capable of significant accuracy and efficiency. The method and its efficacy are illustrated with several examples. Because of its explicit nature, the algorithm can be readily extended to systems with a higher number of spatial dimensions. We show that the method also generalizes the staggered-time approach of Visscher and allows for the accurate calculation of the real and imaginary parts of the wave function separately.
{"title":"On numerical solutions of the time-dependent Schrödinger equation","authors":"Wytse van Dijk","doi":"10.1119/5.0159866","DOIUrl":"https://doi.org/10.1119/5.0159866","url":null,"abstract":"We review an explicit approach to obtaining numerical solutions of the Schrödinger equation that is conceptionally straightforward and capable of significant accuracy and efficiency. The method and its efficacy are illustrated with several examples. Because of its explicit nature, the algorithm can be readily extended to systems with a higher number of spatial dimensions. We show that the method also generalizes the staggered-time approach of Visscher and allows for the accurate calculation of the real and imaginary parts of the wave function separately.","PeriodicalId":7589,"journal":{"name":"American Journal of Physics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136079918","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"教育学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Scattering of non-interacting, identical bosons or fermions by a one-dimensional Dirac delta function barrier potential underlines the importance of the role of statistics (that is, whether the particles obey Fermi–Dirac or Bose–Einstein statistics) in the scattering. We consider an initial wave function for the system that corresponds to one particle incident from the left and one from the right of the potential barrier. For bosons, both particles are scattered either to the left or to the right if the intensity reflection coefficient is 1/2, provided the left and right propagating wave packets fully overlap in the scattering region. For fermions, the particles “pass through” one another, provided the left and right propagating wave packets fully overlap in the scattering region, with zero probability that both particles are scattered to the left or right, consistent with the Pauli exclusion principle.
{"title":"Scattering of identical particles by a one-dimensional Dirac delta function barrier potential: The role of statistics","authors":"P. R. Berman, Alberto G. Rojo","doi":"10.1119/5.0089907","DOIUrl":"https://doi.org/10.1119/5.0089907","url":null,"abstract":"Scattering of non-interacting, identical bosons or fermions by a one-dimensional Dirac delta function barrier potential underlines the importance of the role of statistics (that is, whether the particles obey Fermi–Dirac or Bose–Einstein statistics) in the scattering. We consider an initial wave function for the system that corresponds to one particle incident from the left and one from the right of the potential barrier. For bosons, both particles are scattered either to the left or to the right if the intensity reflection coefficient is 1/2, provided the left and right propagating wave packets fully overlap in the scattering region. For fermions, the particles “pass through” one another, provided the left and right propagating wave packets fully overlap in the scattering region, with zero probability that both particles are scattered to the left or right, consistent with the Pauli exclusion principle.","PeriodicalId":7589,"journal":{"name":"American Journal of Physics","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135324250","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"教育学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The coefficient of sliding friction is usually regarded as a constant for any given surfaces although it is sometimes known to vary with sliding speed. An experiment is described where the coefficient of friction for a billiard ball decreases to a very small value as the sliding speed decreases to zero, after which the ball starts rolling without sliding. A simple theoretical explanation is provided.
{"title":"A note on combined sliding and rolling friction","authors":"Rod Cross","doi":"10.1119/5.0149826","DOIUrl":"https://doi.org/10.1119/5.0149826","url":null,"abstract":"The coefficient of sliding friction is usually regarded as a constant for any given surfaces although it is sometimes known to vary with sliding speed. An experiment is described where the coefficient of friction for a billiard ball decreases to a very small value as the sliding speed decreases to zero, after which the ball starts rolling without sliding. A simple theoretical explanation is provided.","PeriodicalId":7589,"journal":{"name":"American Journal of Physics","volume":"147 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135324242","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"教育学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Dunkan Martínez, Yuriko Baba, Francisco Domínguez-Adame
Introductory textbooks in solid state physics present solvable models for illustrating the occurrence of allowed bands and forbidden gaps in the energy spectrum of Bloch electrons. However, the quantum mechanical description of electrons in non-periodic solids, such as amorphous materials, is beyond the scope of introductory courses because of its intrinsic complexity. The tight-binding approximation can account for such a scenario by letting the atomic levels vary at random from lattice site to site. We theoretically tackle the study of the average properties of the energy spectrum by introducing a transfer matrix method that allows us to obtain closed expressions for the so-called coherent potential. The coherent potential is energy-dependent and constant in space. It replaces the actual atomic random potential, thus generating a periodic effective medium with the same average properties as the non-periodic solid. We demonstrate that the average density of states can be calculated within this framework without relying on heavy mathematical machinery. Thus, our approach is suitable for introductory courses in solid state physics and materials science.
{"title":"Treating disorder in introductory solid state physics","authors":"Dunkan Martínez, Yuriko Baba, Francisco Domínguez-Adame","doi":"10.1119/5.0133701","DOIUrl":"https://doi.org/10.1119/5.0133701","url":null,"abstract":"Introductory textbooks in solid state physics present solvable models for illustrating the occurrence of allowed bands and forbidden gaps in the energy spectrum of Bloch electrons. However, the quantum mechanical description of electrons in non-periodic solids, such as amorphous materials, is beyond the scope of introductory courses because of its intrinsic complexity. The tight-binding approximation can account for such a scenario by letting the atomic levels vary at random from lattice site to site. We theoretically tackle the study of the average properties of the energy spectrum by introducing a transfer matrix method that allows us to obtain closed expressions for the so-called coherent potential. The coherent potential is energy-dependent and constant in space. It replaces the actual atomic random potential, thus generating a periodic effective medium with the same average properties as the non-periodic solid. We demonstrate that the average density of states can be calculated within this framework without relying on heavy mathematical machinery. Thus, our approach is suitable for introductory courses in solid state physics and materials science.","PeriodicalId":7589,"journal":{"name":"American Journal of Physics","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135324252","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"教育学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Joseph C. Amato, Harvey Gould, Jesse Kinder, Raina Olsen, Beth Parks, B. Cameron Reed, Todd Springer, Jan Tobochnik
{"title":"In this issue: October 2023","authors":"Joseph C. Amato, Harvey Gould, Jesse Kinder, Raina Olsen, Beth Parks, B. Cameron Reed, Todd Springer, Jan Tobochnik","doi":"10.1119/5.0173606","DOIUrl":"https://doi.org/10.1119/5.0173606","url":null,"abstract":"","PeriodicalId":7589,"journal":{"name":"American Journal of Physics","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135324253","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"教育学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Measuring the relative humidity of air is an important challenge for meteorological measurements, food conservation, building design, and evaporation control, among other applications. Relative humidity can be measured with a psychrometer, which is a hygrometer composed of two identical thermometers. The bulb of one thermometer is covered by a wick soaked with water so that evaporative cooling makes it indicate a lower temperature than the dry-bulb thermometer; it is possible to determine the relative humidity from the difference between these readings. We describe both a model and an experimental setup to illustrate the principle of a psychrometer for a pedagogical laboratory. The science of psychrometry could be more broadly taught at the undergraduate level to help introduce students to aspects of measurement techniques, fluid mechanics, heat transfer, and non-equilibrium thermodynamics.
{"title":"Measuring relative humidity from evaporation with a wet-bulb thermometer: The psychrometer","authors":"Marie Corpart, Frédéric Restagno, F. Boulogne","doi":"10.1119/5.0154559","DOIUrl":"https://doi.org/10.1119/5.0154559","url":null,"abstract":"Measuring the relative humidity of air is an important challenge for meteorological measurements, food conservation, building design, and evaporation control, among other applications. Relative humidity can be measured with a psychrometer, which is a hygrometer composed of two identical thermometers. The bulb of one thermometer is covered by a wick soaked with water so that evaporative cooling makes it indicate a lower temperature than the dry-bulb thermometer; it is possible to determine the relative humidity from the difference between these readings. We describe both a model and an experimental setup to illustrate the principle of a psychrometer for a pedagogical laboratory. The science of psychrometry could be more broadly taught at the undergraduate level to help introduce students to aspects of measurement techniques, fluid mechanics, heat transfer, and non-equilibrium thermodynamics.","PeriodicalId":7589,"journal":{"name":"American Journal of Physics","volume":"68 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139340702","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"教育学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We discuss contemporary ideas in Monte Carlo algorithms in the simplified setting of the one-dimensional anharmonic oscillator. After reviewing the connection between molecular dynamics and Monte Carlo, we introduce the Metropolis and the factorized Metropolis algorithms and lifted non-reversible Markov chains. We, furthermore, illustrate the concept of thinning, where moves are accepted by simple bounding potentials rather than the harmonic and quartic contributions to the anharmonic oscillator. We point out the multiple connections of our example algorithms with real-world sampling problems. This paper is self-contained, and Python implementations are provided.
{"title":"Concepts in Monte Carlo sampling","authors":"Gabriele Tartero, Werner Krauth","doi":"10.1119/5.0176853","DOIUrl":"https://doi.org/10.1119/5.0176853","url":null,"abstract":"We discuss contemporary ideas in Monte Carlo algorithms in the simplified setting of the one-dimensional anharmonic oscillator. After reviewing the connection between molecular dynamics and Monte Carlo, we introduce the Metropolis and the factorized Metropolis algorithms and lifted non-reversible Markov chains. We, furthermore, illustrate the concept of thinning, where moves are accepted by simple bounding potentials rather than the harmonic and quartic contributions to the anharmonic oscillator. We point out the multiple connections of our example algorithms with real-world sampling problems. This paper is self-contained, and Python implementations are provided.","PeriodicalId":7589,"journal":{"name":"American Journal of Physics","volume":"127 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139342233","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"教育学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article describes a learner-centered, tutorial-based approach to climate modeling. Using these tutorials, students in a general education course in global climate change taught in the physics department engaged with compelling, “real-world” questions by drawing on foundational physics concepts along with their prior experiences and conceptual resources in the process of knowledge construction. Students used simulations, energy tracking diagrams, and algebraic analysis to construct climate models that address challenging quantitative questions and reveal key concepts in climate science. Special emphasis was given to the learning goals, pre-requisite knowledge, and conceptual challenges associated with these activities. Instructional materials are provided for educators who wish to adapt these tutorials for use in their own educational context. The supplementary materials are appropriate for both non-science majors and upper-level physics majors.
{"title":"Tutorials in climate modeling","authors":"L. Seeley","doi":"10.1119/5.0134144","DOIUrl":"https://doi.org/10.1119/5.0134144","url":null,"abstract":"This article describes a learner-centered, tutorial-based approach to climate modeling. Using these tutorials, students in a general education course in global climate change taught in the physics department engaged with compelling, “real-world” questions by drawing on foundational physics concepts along with their prior experiences and conceptual resources in the process of knowledge construction. Students used simulations, energy tracking diagrams, and algebraic analysis to construct climate models that address challenging quantitative questions and reveal key concepts in climate science. Special emphasis was given to the learning goals, pre-requisite knowledge, and conceptual challenges associated with these activities. Instructional materials are provided for educators who wish to adapt these tutorials for use in their own educational context. The supplementary materials are appropriate for both non-science majors and upper-level physics majors.","PeriodicalId":7589,"journal":{"name":"American Journal of Physics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44125227","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"教育学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
S. Toffaletti, M. Di Mauro, M. Malgieri, T. Rosi, E. Tufino, P. Onorato, S. Oss
We describe a collection of relatively simple experiments and laboratory demonstrations devoted to the physics of Earth's energy balance. Many of the experiments also address fundamental aspects of physics at the undergraduate level. Results of classroom testing of this sequence of activities are presented and discussed.
{"title":"A teaching-learning sequence about climate change: From theory to practice","authors":"S. Toffaletti, M. Di Mauro, M. Malgieri, T. Rosi, E. Tufino, P. Onorato, S. Oss","doi":"10.1119/5.0137089","DOIUrl":"https://doi.org/10.1119/5.0137089","url":null,"abstract":"We describe a collection of relatively simple experiments and laboratory demonstrations devoted to the physics of Earth's energy balance. Many of the experiments also address fundamental aspects of physics at the undergraduate level. Results of classroom testing of this sequence of activities are presented and discussed.","PeriodicalId":7589,"journal":{"name":"American Journal of Physics","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63423019","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"教育学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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