Volume 81, Issue 3, March 2013
 PAPERS


Computational problems in introductory physics: Lessons from a bead on a wire
View Description Hide DescriptionWe have found that incorporating computer programming into introductory physics requires problems suited for numerical treatment while still maintaining ties with the analytical themes in a typical introductorylevel university physics course. In this paper, we discuss a numerical adaptation of a system commonly encountered in the introductory physics curriculum: the dynamics of an object constrained to move along a curved path. A numerical analysis of this problem that includes a computer animation can provide many insights and pedagogical avenues not possible with the usual analytical treatment. We present two approaches for computing the instantaneous kinematic variables of an object constrained to move along a path described by a mathematical function. The first is a pedagogical approach, appropriate for introductory students in the calculusbased sequence. The second is a more generalized approach, suitable for simulations of more complex scenarios.

Observation of phase transition in a simple mechanical system
View Description Hide DescriptionIf a quantummechanical Hamiltonian is symmetric, there are two possibilities: either all of the eigenvalues are real, in which case the Hamiltonian is said to be in an unbroken symmetric phase, or else some of the eigenvalues are real and some are complex, in which case the Hamiltonian is said to be in a broken symmetric phase. As one varies the parameters of the Hamiltonian, one can pass through the phase transition that separates the unbroken and broken phases. This transition has recently been observed in a variety of laboratory experiments. This paper explains the phase transition in a simple and intuitive fashion and then describes an elementary experiment in which the phase transition is easily observed.

The LorentzDirac and LandauLifshitz equations from the perspective of modern renormalization theory
View Description Hide DescriptionThis paper uses elementary techniques drawn from renormalization theory to derive the LorentzDirac equation for the relativistic classical electron from the MaxwellLorentz equations for a classical charged particle coupled to the electromagnetic field. I show that the resulting effective theory, valid for electron motions that change over distances large compared to the classical electron radius, reduces naturally to the LandauLifshitz equation. No familiarity with renormalization or quantum field theory is assumed.

Noether's theorem and the workenergy theorem for a charged particle in an electromagnetic field
View Description Hide DescriptionNoether's theorem is based on two fundamental ideas. The first is the extremum of the action and the second is the invariance of the action under infinitesimal continuous transformations in space and time. The first gives Hamilton's principle of least action, which results in the Euler–Lagrange equations. The second gives the Rund–Trautman identity for the generators of infinitesimal transformations in space and time. We apply these ideas to a charged particle in an external electromagnetic field. A solution of the Rund–Trautman identity for the generators is obtained by solving generalized Killing equations. The Euler–Lagrange equations and the Rund–Trautman identity are combined to give Noether's theorem for a conserved quantity. When we use the Lagrangian and the generators of infinitesimal transformations for a charged particle in an external electromagnetic field, we obtain the workenergy theorem.

Transmission resonances and Bloch states for a periodic array of delta function potentials
View Description Hide DescriptionThe relationship between transmission resonances and Bloch states for a periodic array consisting of N delta function potentials is discussed. The transmission resonances are derived for matter waves incident on the periodic array, while the Bloch states are calculated using periodic boundary conditions for the array. It is shown that approximately half of the transmission resonances map into pairs of degenerate Bloch states. Wave functions are shown for both the transmission resonances and the Bloch states for arrays of five and six delta function potentials. The origin of the band structure of the Bloch states is interpreted in terms of the wave functions and eigenenergies for a particle confined to move on a ring, subjected to a periodic array of delta function potentials on the ring.

Bound charges and currents
View Description Hide DescriptionBound charges and currents are among the conceptually challenging topics in advanced courses on electricity and magnetism. It may be tempting for students to believe that they are merely computational tools for calculating electric and magnetic fields in matter, particularly because they are usually introduced through abstract manipulation of integral identities, with the physical interpretation provided a posteriori. Yet these charges and currents are no less real than free charges and currents and can be measured experimentally. A simpler and more direct approach to introducing this topic, suggested by the ideas in the classic book by Purcell and emphasizing the physical origin of these phenomena, is proposed.

Elucidating Fermi's golden rule via boundtobound transitions in a confined hydrogen atom
View Description Hide DescriptionWe demonstrate an effective method for calculating boundtocontinuum crosssections by examining transitions to bound states above the ionization energy that result from placing the system of interest within an infinite spherical well. Using photoionization of the hydrogen atom as an example, we demonstrate convergence between this approach for a large volume of confinement and an exact analytical alternate approach that uses energynormalized continuum wavefunctions, which helps to elucidate the implementation of Fermi's golden rule. As the radius of confinement varies, the resulting changes in physical behavior of the system are presented and discussed. The photoionization crosssections from a variety of atomic states with principal quantum number n are seen to obey particular scaling laws.

There are no particles, there are only fields
View Description Hide DescriptionQuantum foundations are still unsettled, with mixed effects on science and society. By now it should be possible to obtain consensus on at least one issue: Are the fundamental constituents fields or particles? As this paper shows, experiment and theory imply that unbounded fields, not bounded particles, are fundamental. This is especially clear for relativistic systems, implying that it's also true of nonrelativistic systems. Particles are epiphenomena arising from fields. Thus, the Schrödinger field is a spacefilling physical field whose value at any spatial point is the probability amplitude for an interaction to occur at that point. The field for an electron is the electron; each electron extends over both slits in the twoslit experiment and spreads over the entire pattern; and quantum physics is about interactions of microscopic systems with the macroscopic world rather than just about measurements. It's important to clarify this issue because textbooks still teach a particles and measurementoriented interpretation that contributes to bewilderment among students and pseudoscience among the public. This article reviews classical and quantum fields, the twoslit experiment, rigorous theorems showing particles are inconsistent with relativistic quantum theory, and several phenomena showing particles are incompatible with quantum field theories.

Simulation of a Brownian particle in an optical trap
View Description Hide DescriptionAn optically trapped Brownian particle is a sensitive probe of molecular and nanoscopic forces. An understanding of its motion, which is caused by the interplay of random and deterministic contributions, can lead to greater physical insight into the behavior of stochastic phenomena. The modeling of realistic stochastic processes typically requires advanced mathematical tools. We discuss a finite difference algorithm to compute the motion of an optically trapped particle and the numerical treatment of the white noise term. We then treat the transition from the ballistic to the diffusive regime due to the presence of inertial effects on short time scales and examine the effect of an optical trap on the motion of the particle. We also outline how to use simulations of optically trapped Brownian particles to gain understanding of nanoscale force and torque measurements, and of more complex phenomena, such as Kramers transitions, stochastic resonant damping, and stochastic resonance.
