Catálogo de publicaciones - revistas

<jats:p>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.</jats:p>

<jats:p>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.</jats:p>

<jats:p>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.</jats:p>

<jats:p>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.</jats:p>

<jats:p>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.</jats:p>

<jats:p>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.</jats:p>

<jats:p>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.</jats:p>

<jats:p>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.</jats:p>

<jats:p>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.</jats:p>