{"title":"The emergence of classical mixtures from an entangled quantum state","authors":"Mark G. Kuzyk","doi":"10.1119/5.0063636","DOIUrl":"https://doi.org/10.1119/5.0063636","url":null,"abstract":"","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":"135324251","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}
Theoretically, solutions of the damped harmonic oscillator asymptotically approach equilibrium, i.e., the zero energy state, without ever reaching it exactly, and the critically damped solution approaches equilibrium faster than the underdamped or the overdamped solution. Experimentally, the systems described with this model reach equilibrium when the system's energy has dropped below some threshold corresponding to the energy resolution of the measuring apparatus. We show that one can (almost) always find an optimal underdamped solution that will reach this energy threshold sooner than all other underdamped solutions, as well as the critically damped solution, no matter how small this threshold is. We also comment on one exception to this for a particular type of initial condition, when a specific overdamped solution reaches the equilibrium state sooner than all other solutions. We experimentally confirm some of our findings.
{"title":"Damped harmonic oscillator revisited: The fastest route to equilibrium","authors":"Karlo Lelas, Nikola Poljak, Dario Jukić","doi":"10.1119/5.0112573","DOIUrl":"https://doi.org/10.1119/5.0112573","url":null,"abstract":"Theoretically, solutions of the damped harmonic oscillator asymptotically approach equilibrium, i.e., the zero energy state, without ever reaching it exactly, and the critically damped solution approaches equilibrium faster than the underdamped or the overdamped solution. Experimentally, the systems described with this model reach equilibrium when the system's energy has dropped below some threshold corresponding to the energy resolution of the measuring apparatus. We show that one can (almost) always find an optimal underdamped solution that will reach this energy threshold sooner than all other underdamped solutions, as well as the critically damped solution, no matter how small this threshold is. We also comment on one exception to this for a particular type of initial condition, when a specific overdamped solution reaches the equilibrium state sooner than all other solutions. We experimentally confirm some of our findings.","PeriodicalId":7589,"journal":{"name":"American Journal of Physics","volume":"26 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":"135324539","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 present an overview of the thermal history of the Universe and the sequence of objects (e.g., protons, planets, and galaxies) that condensed out of the background as the Universe expanded and cooled. We plot (i) the density and temperature of the Universe as a function of time and (ii) the masses and sizes of all objects in the Universe. These comprehensive pedagogical plots draw attention to the triangular regions forbidden by general relativity and quantum uncertainty and help navigate the relationship between gravity and quantum mechanics. How can we interpret their intersection at the smallest possible objects: Planck-mass black holes (“instantons”)? Does their Planck density and Planck temperature make them good candidates for the initial conditions of the Universe? Our plot of all objects also seems to suggest that the Universe is a black hole. We explain how this depends on the unlikely assumption that our Universe is surrounded by zero density Minkowski space.
{"title":"All objects and some questions","authors":"Charles H. Lineweaver, Vihan M. Patel","doi":"10.1119/5.0150209","DOIUrl":"https://doi.org/10.1119/5.0150209","url":null,"abstract":"We present an overview of the thermal history of the Universe and the sequence of objects (e.g., protons, planets, and galaxies) that condensed out of the background as the Universe expanded and cooled. We plot (i) the density and temperature of the Universe as a function of time and (ii) the masses and sizes of all objects in the Universe. These comprehensive pedagogical plots draw attention to the triangular regions forbidden by general relativity and quantum uncertainty and help navigate the relationship between gravity and quantum mechanics. How can we interpret their intersection at the smallest possible objects: Planck-mass black holes (“instantons”)? Does their Planck density and Planck temperature make them good candidates for the initial conditions of the Universe? Our plot of all objects also seems to suggest that the Universe is a black hole. We explain how this depends on the unlikely assumption that our Universe is surrounded by zero density Minkowski space.","PeriodicalId":7589,"journal":{"name":"American Journal of Physics","volume":"28 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":"135324246","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 revisit the discussion about the boundary condition at the origin in the Schrödinger radial equation for central potentials. We give a transparent and convincing reason for demanding the radial part R(r) of the wave function to be finite at r = 0, showing that if R(0) diverges the complete wave function ψ does not satisfy the full Schrödinger equation. If R(r) is singular, we show that the corresponding ψ follows an equation similar to Schrödinger's, but with an additional term involving the Dirac delta function or its derivatives at the origin. Although, in general, understanding some of our arguments requires certain knowledge of the theory of distributions, the important case of a behavior R ∝ 1/r near r = 0, which gives rise to a normalizable ψ, is especially simple: The origin of the Dirac delta term is clearly demonstrated by using a slight modification of the usual spherical coordinates. The argument can be easily followed by undergraduate physics students.
{"title":"Acceptable solutions of the radial Schrödinger equation for a particle in a central potential","authors":"J. Etxebarria","doi":"10.1119/5.0141536","DOIUrl":"https://doi.org/10.1119/5.0141536","url":null,"abstract":"We revisit the discussion about the boundary condition at the origin in the Schrödinger radial equation for central potentials. We give a transparent and convincing reason for demanding the radial part R(r) of the wave function to be finite at r = 0, showing that if R(0) diverges the complete wave function ψ does not satisfy the full Schrödinger equation. If R(r) is singular, we show that the corresponding ψ follows an equation similar to Schrödinger's, but with an additional term involving the Dirac delta function or its derivatives at the origin. Although, in general, understanding some of our arguments requires certain knowledge of the theory of distributions, the important case of a behavior R ∝ 1/r near r = 0, which gives rise to a normalizable ψ, is especially simple: The origin of the Dirac delta term is clearly demonstrated by using a slight modification of the usual spherical coordinates. The argument can be easily followed by undergraduate physics students.","PeriodicalId":7589,"journal":{"name":"American Journal of Physics","volume":"16 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":"135324540","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}
One of the most accurate ways to measure the impedance of an electrical component is to place it in a bridge that is then balanced. The most familiar bridge in an undergraduate laboratory is the Wheatstone bridge, which can measure resistance to high precision. Other types are, however, required for reactive components. This paper describes the use of Anderson's bridge to measure inductance, allowing both the inductance and resistance of different inductors to be determined. The inductors are analysed with different cores: perspex, copper, and steel. Models for the inductance that include the effect of skin depth, winding proximity, eddy currents, and core effects are introduced and compared to measurements in the frequency range from 100 Hz to 100 kHz.
{"title":"An undergraduate physics experiment to measure the frequency-dependent impedance of inductors using an Anderson bridge","authors":"Andrew James Murray, Carl Hickman","doi":"10.1119/5.0148114","DOIUrl":"https://doi.org/10.1119/5.0148114","url":null,"abstract":"One of the most accurate ways to measure the impedance of an electrical component is to place it in a bridge that is then balanced. The most familiar bridge in an undergraduate laboratory is the Wheatstone bridge, which can measure resistance to high precision. Other types are, however, required for reactive components. This paper describes the use of Anderson's bridge to measure inductance, allowing both the inductance and resistance of different inductors to be determined. The inductors are analysed with different cores: perspex, copper, and steel. Models for the inductance that include the effect of skin depth, winding proximity, eddy currents, and core effects are introduced and compared to measurements in the frequency range from 100 Hz to 100 kHz.","PeriodicalId":7589,"journal":{"name":"American Journal of Physics","volume":"8 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":"135324243","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}
Chananya Groner, Timothy M. Minteer, Kirk T. McDonald
Ampère's force law for steady currents was not historically associated with a magnetic field, but it could have been. A magnetic field, inspired by work of Helmholtz in 1870, can be defined such that the double-differential form of Ampère's force law is a function of a double-differential of this field. We call this field the Ampère–Weber field, B, and show that its divergence is zero everywhere, as is that of the usual, but different, magnetic field B of Maxwellian electrodynamics. The curl of the Ampère–Weber field is nonzero everywhere in static examples, in contrast to that of the usual magnetic field B. We illustrate the field B for three examples, which exhibit patterns of field lines quite different from those of the usual magnetic field. As the Ampère–Weber field is based on Ampère's force law for steady currents, it does not extrapolate well to the Lorentz force on a moving charge in a magnetic field. That is, the Ampère–Weber field B, like Ampère's force law, is more of a curiosity than a viable alternative to the usual magnetic field B. If the Ampère–Weber field had been invented in the mid-1800s, it would have been a distraction more than a step toward a generally valid electromagnetic field theory.
{"title":"A magnetic field based on Ampère's force law","authors":"Chananya Groner, Timothy M. Minteer, Kirk T. McDonald","doi":"10.1119/5.0134722","DOIUrl":"https://doi.org/10.1119/5.0134722","url":null,"abstract":"Ampère's force law for steady currents was not historically associated with a magnetic field, but it could have been. A magnetic field, inspired by work of Helmholtz in 1870, can be defined such that the double-differential form of Ampère's force law is a function of a double-differential of this field. We call this field the Ampère–Weber field, B, and show that its divergence is zero everywhere, as is that of the usual, but different, magnetic field B of Maxwellian electrodynamics. The curl of the Ampère–Weber field is nonzero everywhere in static examples, in contrast to that of the usual magnetic field B. We illustrate the field B for three examples, which exhibit patterns of field lines quite different from those of the usual magnetic field. As the Ampère–Weber field is based on Ampère's force law for steady currents, it does not extrapolate well to the Lorentz force on a moving charge in a magnetic field. That is, the Ampère–Weber field B, like Ampère's force law, is more of a curiosity than a viable alternative to the usual magnetic field B. If the Ampère–Weber field had been invented in the mid-1800s, it would have been a distraction more than a step toward a generally valid electromagnetic field theory.","PeriodicalId":7589,"journal":{"name":"American Journal of Physics","volume":"49 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":"135324244","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 analyze a variety of Gaussian integrals with the aim of revisiting the derivation of the Hubbard–Stratonovich transformation as given in standard graduate-level textbooks and provide an overview of its applications. We pinpoint problematic steps in the usual discussions and propose careful derivations of the Hubbard–Stratonovich identity pertinent to a variety of situations relevant to statistical physics and quantum field theory. These derivations are based on direct use of either a resolution identity or a series expansion. A few homework problems for students are suggested.
{"title":"Generalized Gaussian integrals with application to the Hubbard–Stratonovich transformation","authors":"Krzysztof Byczuk, Paweł Jakubczyk","doi":"10.1119/5.0141045","DOIUrl":"https://doi.org/10.1119/5.0141045","url":null,"abstract":"We analyze a variety of Gaussian integrals with the aim of revisiting the derivation of the Hubbard–Stratonovich transformation as given in standard graduate-level textbooks and provide an overview of its applications. We pinpoint problematic steps in the usual discussions and propose careful derivations of the Hubbard–Stratonovich identity pertinent to a variety of situations relevant to statistical physics and quantum field theory. These derivations are based on direct use of either a resolution identity or a series expansion. A few homework problems for students are suggested.","PeriodicalId":7589,"journal":{"name":"American Journal of Physics","volume":"25 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":"135324249","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}
{"title":"Special issue in celebration of the International Year of Quantum Science and Technology","authors":"","doi":"10.1119/5.0173872","DOIUrl":"https://doi.org/10.1119/5.0173872","url":null,"abstract":"","PeriodicalId":7589,"journal":{"name":"American Journal of Physics","volume":"31 10 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":"135324254","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}
Studying worked examples has been shown by extensive research to be an effective method for learning to solve well-structured problems in physics and mathematics. The effectiveness of learning with worked examples has been demonstrated and documented in many research projects. In this work, we propose a new four-step approach for teaching with worked examples that includes writing explanations and finding and correcting errors. This teaching method can even be implemented in courses in which homework performance constitutes part of the grading system. This four-step approach is illustrated in the context of Lagrangian mechanics, which is ideal for the application of worked examples due to its universal approach to solve problems.
{"title":"Why and how to implement worked examples in upper division theoretical physics","authors":"Philipp Scheiger, Holger Cartarius, Ronny Nawrodt","doi":"10.1119/5.0105612","DOIUrl":"https://doi.org/10.1119/5.0105612","url":null,"abstract":"Studying worked examples has been shown by extensive research to be an effective method for learning to solve well-structured problems in physics and mathematics. The effectiveness of learning with worked examples has been demonstrated and documented in many research projects. In this work, we propose a new four-step approach for teaching with worked examples that includes writing explanations and finding and correcting errors. This teaching method can even be implemented in courses in which homework performance constitutes part of the grading system. This four-step approach is illustrated in the context of Lagrangian mechanics, which is ideal for the application of worked examples due to its universal approach to solve problems.","PeriodicalId":7589,"journal":{"name":"American Journal of Physics","volume":"13 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":"135324247","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}
A simple model coupling a one-dimensional beam particle to a one-dimensional harmonic oscillator is used to explore complementarity and entanglement. This model, well-known in the inelastic scattering literature, is presented under three different conceptual approaches, with both analytical and numerical techniques discussed for each. In a purely classical approach, the final amplitude of the oscillator can be found directly from the initial conditions. In a partially quantum approach, with a classical beam and a quantum oscillator, the final magnitude of the quantum-mechanical amplitude for the oscillator's first excited state is directly proportional to the oscillator's classical amplitude of vibration. Nearly the same first-order transition probabilities emerge in the partially and fully quantum approaches, but conceptual differences emerge. The two-particle scattering wavefunction clarifies these differences and allows the consequences of quantum entanglement to be explored.
{"title":"Complementarity and entanglement in a simple model of inelastic scattering","authors":"David Kordahl","doi":"10.1119/5.0141389","DOIUrl":"https://doi.org/10.1119/5.0141389","url":null,"abstract":"A simple model coupling a one-dimensional beam particle to a one-dimensional harmonic oscillator is used to explore complementarity and entanglement. This model, well-known in the inelastic scattering literature, is presented under three different conceptual approaches, with both analytical and numerical techniques discussed for each. In a purely classical approach, the final amplitude of the oscillator can be found directly from the initial conditions. In a partially quantum approach, with a classical beam and a quantum oscillator, the final magnitude of the quantum-mechanical amplitude for the oscillator's first excited state is directly proportional to the oscillator's classical amplitude of vibration. Nearly the same first-order transition probabilities emerge in the partially and fully quantum approaches, but conceptual differences emerge. The two-particle scattering wavefunction clarifies these differences and allows the consequences of quantum entanglement to be explored.","PeriodicalId":7589,"journal":{"name":"American Journal of Physics","volume":"62 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":"135324248","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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