American Journal of Physics

Published by American Association of Physics Teachers

Online ISSN: 1943-2909


Print ISSN: 0002-9505


Predicting charmonium and bottomonium spectra with a quark harmonic oscillator
  • Article

December 1986


167 Reads

J W Norbury


F F Badavi


We present a simple application of the three-dimensional harmonic oscillator which should provide a very nice particle physics example to be presented in introductory undergraduate quantum mechanics course. The idea is to use the nonrelativistic quark model to calculate the spin-averaged mass levels of the charmonium and bottomonium spectra.

Resource Letter ETC-1: Extraterrestrial civilization

February 1989


182 Reads

This Resource Letter provides a guide to the literature about intelligent life beyond the human sphere of exploration. It offers a starting point for professionals and academics interested in participating in the debate about the existence of other technological civilizations or in the search for extraterrestrial intelligence (SETI). It can also serve as a reference for teaching. This Letter is not intended as an exhaustive bibliography, but several extensive bibliographies have been cited. The letter E after an item indicates elementary, nontechnical material of general interest to persons becoming informed in the field. Intermediate level material, of a somewhat more specialized nature, is indicated by the Letter I. The annotation A indicates advanced, technical material. An asterisk (*) precedes items to be included in an accompanying Reprint Book.

A cost-efficient frequency-domain photoacoustic imaging system

September 2013


133 Reads

Photoacoustic (PA) imaging techniques have recently attracted much attention and can be used for noninvasive imaging of biological tissues. Most PA imaging systems in research laboratories use the time domain method with expensive nanosecond pulsed lasers that are not affordable for most educational laboratories. Using an intensity modulated light source to excite PA signals is an alternative technique, known as the frequency domain method, with a much lower cost. In this paper, we describe a simple frequency domain PA system and demonstrate its imaging capability. The system provides opportunities not only to observe PA signals in tissue phantoms, but also to acquire hands-on skills in PA signal detection. It also provides opportunities to explore the underlying mechanisms of the PA effect.

FIG. 1: Colloidal free expansion setup to illustrate diffusion involving small numbers of particles. (a) Schematic of experimental setup (see text for details.) (b) Several snapshots from the experiment. (c) Normalized histogram of particle positions during the experiment. The solution to the diffusion equation for the microfluidic “free expansion” experiment is superposed for comparison. 
FIG. 2: Schematic of the simple dog-flea model. (a) State of the system at time t , (b) a particular microtrajectory in which two-fleas jump from the dog on the left and one jumps from the dog on the right, (c) occupancies of the dogs at time t + ∆ t . 
FIG. 3: Schematic of the distribution of fluxes for different time points as the system approaches equilibrium. 
FIG. 4: Schematic of which trajectories are potent and which are impotent. The shaded region corresponds to the impotent trajectories for which m 1 and m 2 are either equal or approximately equal and hence make relatively small change in the macrostate. The unshaded region corresponds to potent trajectories. 
FIG. 5: Illustration of the potency of the microtrajectories associated with different distributions of N particles on the two dogs. The total number of particles N 1 + N 2 = N = 100. 


Teaching the Principles of Statistical Dynamics
  • Article
  • Full-text available

February 2006


1,243 Reads



Mandar M Inamdar




Rob Phillips
We describe a simple framework for teaching the principles that underlie the dynamical laws of transport: Fick's law of diffusion, Fourier's law of heat flow, the Newtonian viscosity law, and the mass-action laws of chemical kinetics. In analogy with the way that the maximization of entropy over microstates leads to the Boltzmann distribution and predictions about equilibria, maximizing a quantity that E. T. Jaynes called "caliber" over all the possible microtrajectories leads to these dynamical laws. The principle of maximum caliber also leads to dynamical distribution functions that characterize the relative probabilities of different microtrajectories. A great source of recent interest in statistical dynamics has resulted from a new generation of single-particle and single-molecule experiments that make it possible to observe dynamics one trajectory at a time.

Resource Letter: LBOT-1: Laser-based optical tweezers

April 2003


100 Reads

This Resource Letter provides a guide to the literature on optical tweezers, also known as laser-based, gradient-force optical traps. Journal articles and books are cited for the following main topics: general papers on optical tweezers, trapping instrument design, optical detection methods, optical trapping theory, mechanical measurements, single molecule studies, and sections on biological motors, cellular measurements and additional applications of optical tweezers. (C) 2003 American Association of Physics Teachers.

Active Mass Under Pressure

May 2005


88 Reads

After a historical introduction to Poisson's equation in Newtonian gravity, we review its analog for static gravitational fields in Einstein's theory. The source of the potential, which we call the active mass density, comprises not only all possible sources of energy, but also the pressure term 3P/c 2. In the Hamburg seminar on relativity in the 1950s we discussed whether this term due to Fermi pressure in different atomic nuclei could be detected in Cavendish-type experiments. Our reasoning contained an instructive mistake that we are now able to resolve. We conclude that this term should not lead to discrepancies for different materials in a Cavendish-type experiment, although it is important in the early universe and collapsing stellar cores.

Understanding Heisenberg's 'Magical' Paper of July 1925: a New Look at the Calculational Details

April 2004


1,847 Reads

In July 1925 Heisenberg published a paper [Z. Phys. 33, 879-893 (1925)] which ended the period of `the Old Quantum Theory' and ushered in the new era of Quantum Mechanics. This epoch-making paper is generally regarded as being difficult to follow, perhaps partly because Heisenberg provided few clues as to how he arrived at the results which he reported. Here we give details of calculations of the type which, we suggest, Heisenberg may have performed. We take as a specific example one of the anharmonic oscillator problems considered by Heisenberg, and use our reconstruction of his approach to solve it up to second order in perturbation theory. We emphasize that the results are precisely those obtained in standard quantum mechanics, and suggest that some discussion of the approach - based on the direct computation of transition amplitudes - could usefully be included in undergraduate courses in quantum mechanics. Comment: 24 pages, no figures, Latex, submitted to Am. J. Phys

The hydrogen atom in electric and magnetic fields : Pauli's 1926 article

December 2002


1,124 Reads

The results obtained by Pauli, in his 1926 article on the hydrogen atom, made essential use of the dynamical so(4) symmetry of the bound states. Pauli used this symmetry to compute the perturbed energy levels of an hydrogen atom in a uniform electric field (Stark effect) and in uniform electric and magnetic fields. Although the experimental check of the single Stark effect on the hydrogen atom has been studied experimentally, Pauli's results in mixed fields have been studied only for Rydberg states of rubidium atoms in crossedfields and lithium atoms in parallel fields. Comment: 11 pages, latex file, 2 figures

Comment on "Fun and frustration with quarkonium in a 1+1 dimension," by R. S. Bhalerao and B. Ram [Am. J. Phys. 69 (7), 817-818 (2001)]

October 2001


43 Reads

The Dirac equation in a 1+1 dimension with the Lorentz scalar potential g|x| is approached. It is claimed that the eigenfunctions are proportional to the parabolic cylinder functions instead Hermite polynomials. Numerical evaluation of the quantization condition does not result in frustration. Comment: Submitted to American Journal of Physics

2007 Gyroscope precession in special and general relativity from basic principles Am

September 2007


103 Reads

In special relativity a gyroscope that is suspended in a torque-free manner will precess as it is moved along a curved path relative to an inertial frame S. We explain this effect, which is known as Thomas precession, by considering a real grid that moves along with the gyroscope, and that by definition is not rotating as observed from its own momentary inertial rest frame. From the basic properties of the Lorentz transformation we deduce how the form and rotation of the grid (and hence the gyroscope) will evolve relative to S. As an intermediate step we consider how the grid would appear if it were not length contracted along the direction of motion. We show that the uncontracted grid obeys a simple law of rotation. This law simplifies the analysis of spin precession compared to more traditional approaches based on Fermi transport. We also consider gyroscope precession relative to an accelerated reference frame and show that there are extra precession effects that can be explained in a way analogous to the Thomas precession. Although fully relativistically correct, the entire analysis is carried out using three-vectors. By using the equivalence principle the formalism can also be applied to static spacetimes in general relativity. As an example, we calculate the precession of a gyroscope orbiting a static black hole. In an addendum the general reasoning is extended to include also rotating reference frames.

Quantum fields are not fields. Comment on "There are no particles, there are only fields," by Art Hobson [Am. J. Phys. 81(3), 211-223 (2013)]

April 2012


353 Reads

We comment on a recent paper by Hobson, explaining that quantum "fields" are no more fields than quantum "particles" are particles, so that the replacement of a particle ontology by an all-field ontology cannot solve the typical interpretational problems of quantum mechanics.

Fig. 3: Dynamical structure of the brain's response to input data.
Fig. 4: The cognitive/socio-cultural grain-size staircase.
Fig. 5: Student responses to nearly equivalent questions.
Oersted Lecture 2013: How should we think about how our students think?

August 2013


414 Reads

Physics Education Research (PER) applies a scientific approach to the question, "How do our students think about and learn physics?" PER allows us to explore such intellectually engaging questions as, "What does it mean to understand something in physics?" and, "What skills and competencies do we want our students to learn from our physics classes?" To address questions like these, we need to do more than observe student difficulties and build curricula. We need a theoretical framework -- a structure for talking about, making sense of, and modeling how one thinks about, learns, and understands physics. In this paper, I outline some aspects of the Resources Framework, a structure that some of us are using to create a phenomenology of physics learning that ties closely to modern developments in neuroscience, psychology, and linguistics. As an example of how this framework gives new insights, I discuss epistemological framing -- the role of students' perceptions of the nature of the knowledge they are learning and what knowledge is appropriate to bring to bear on a given task. I discuss how this foothold idea fits into our theoretical framework, show some classroom data on how it plays out in the classroom, and give some examples of how my awareness of the resources framework influences my approach to teaching.

Optimal Pacing for Running 400 m and 800 m Track Races

April 2012


182 Reads

Physicists seeking to understand complex biological systems often find it rewarding to create simple "toy models" that reproduce system behavior. Here a toy model is used to understand a puzzling phenomenon from the sport of track and field. Races are almost always won, and records set, in 400 m and 800 m running events by people who run the first half of the race faster than the second half, which is not true of shorter races, nor of longer. There is general agreement that performance in the 400 m and 800 m is limited somehow by the amount of anaerobic metabolism that can be tolerated in the working muscles in the legs. A toy model of anaerobic metabolism is presented, from which an optimal pacing strategy is analytically calculated via the Euler-Lagrange equation. This optimal strategy is then modified to account for the fact that the runner starts the race from rest; this modification is shown to result in the best possible outcome by use of an elementary variational technique that supplements what is found in undergraduate textbooks. The toy model reproduces the pacing strategies of elite 400 m and 800 m runners better than existing models do. The toy model also gives some insight into training strategies that improve performance.

Answer to question Nr. 55

September 1997


76 Reads

We answer to question Nr. 55 [Are there pictorial examples that distinguish covariant and contravariant vectors ?] posed by D. Neuenschwander, Am. J. Phys. 65 (1), 11 (1997) Comment: 3 pages, LaTeX, 2 figures, Am. J. Phys. in print

FIG. 1: Electric field of an accelerated charge 
An ab initio derivation of the electromagnetic fields of a point charge in arbitrary motion

January 2011


664 Reads

Electromagnetic fields of an accelerated charge are derived from the first principles using Coulomb's law and the relativistic transformations. The electric and magnetic fields are derived first for an instantaneous rest frame of the accelerated charge, without making explicit use of Gauss's law, an approach different from that available in the literature. Thereafter we calculate the electromagnetic fields for an accelerated charge having a non-relativistic motion. The expressions for these fields, supposedly accurate only to a first order in velocity $\beta$, surprisingly yield all terms exactly for the acceleration fields, only missing a factor $1-\beta^2$ in the velocity fields. The derivation explicitly shows the genesis of various terms in the field expressions, when expressed with respect to the time retarded position of the charge. A straightforward transformation from the instantaneous rest frame, using relativistic Doppler factors, yields expressions of the electromagnetic fields for the charge moving with an arbitrary velocity. The field expressions are derived without using Li\'{e}nard-Wiechert potentials, thereby avoiding evaluation of any spatial or temporal derivatives of these potentials at the retarded time.

FIG. 2: A plot of the auxiliary function w(θ).  
FIG. 6: The function θ(s) for an accelerating observer that is initially at ten light years distance from an object with a radius of one light year. We have chosen a moderate acceleration of a = 1g.
Relativistic Aberration for Accelerating Observers

August 2007


78 Reads

We investigate the effects of the aberration of light for a uniformly accelerating observer. The observer we consider is initially at rest with respect to a luminous spherical object--a star, say--and then starts to move away with constant acceleration. The main results we derive are the following: (i) The observer always sees an initial increase of the apparent size of the object; (ii) The apparent size of the object approaches a non-zero value as the proper time of the observer goes to infinity. (iii) There exists a critical value of the acceleration such that the apparent size of the object is always increasing when the acceleration is super-critical. We show that, while (i) is a purely non-relativistic effect, (ii) and (iii) are effects of the relativistic aberration of light and are intimately connected with the Lorentzian geometry of Minkowksi spacetime. Finally, the examples we present illustrate that, while more or less negligible in everyday life, the three effects can be significant in the context of space-flight. Comment: 7 figures; subject: special relativity; pedagogical article; replaced to match version appearing in Am. J. Phys

Reduced-order Abraham-Lorentz-Dirac equation and the consistency of classical electromagnetism

February 2014


143 Reads

It is widely believed that classical electromagnetism is either unphysical or inconsistent, owing to pathological behaviour when self-force and radiation reaction are non-negligible. We argue that there is no inconsistency as long as it is recognized that certain types of charge distribution are simply impossible, such as, for example, a point particle with finite charge and finite inertia. This is owing to the fact that negative inertial mass is an unphysical concept in classical physics. It remains useful to obtain an equation of motion for small charged objects that describes their motion to good approximation without requiring knowledge of the charge distribution within the object. We give a simple method to achieve this, leading to a reduced-order form of the Abraham-Lorentz-Dirac equation, essentially as proposed by Eliezer, Landau and Lifshitz.

Propagating and evanescent waves in absorbing media

June 2002


175 Reads

We compare the behavior of propagating and evanescent light waves in absorbing media with that of electrons in the presence of inelastic scattering. The imaginary part of the dielectric constant results primarily in an exponential decay of a propagating wave, but a phase shift for an evanescent wave. We then describe how the scattering of quantum particles out of a particular coherent channel can be modeled by introducing an imaginary part to the potential in analogy with the optical case. The imaginary part of the potential causes additional scattering which can dominate and actually prevent absorption of the wave for large enough values of the imaginary part. We also discuss the problem of maximizing the absorption of a wave and point out that the existence of a bound state greatly aids absorption. We illustrate this point by considering the absorption of light at the surface of a metal. Comment: Brief Review, to appear in the American Journal of Physics,

Measurement of mutual inductance from frequency dependence of impedance of AC coupled circuit using digital dual-phase lock-in amplifier

July 2006


1,017 Reads

We present a simple method to determine the mutual inductance $M$ between two coils in a coupled AC circuit by using a digital dual-phase lock-in amplifier. The frequency dependence of the real and imaginary parts is measured as the coupling constant is changed. The mutual inductance $M$ decreases as the distance $d$ between the centers of coils is increased. We show that the coupling constant is proportional to $d^{-n}$ with an exponent $n$ ($\approx$ 3). This coupling is similar to that of two magnetic moments coupled through a dipole-dipole interaction.

FIG. 4: (color online) The running current J(t) = x(t)/t vs t for deterministic system (1) and the dependence of the current J (15) on the number of basic states Z (20). The other parameters are the same as in Fig.3.
AC-driven Brownian motors: A Fokker-Planck treatment

October 2008


382 Reads

We consider a primary model of ac-driven Brownian motors, i.e., a classical particle placed in a spatial-time periodic potential and coupled to a heat bath. The effects of fluctuations and dissipations are studied by a time-dependent Fokker-Planck equation. The approach allows us to map the original stochastic problem onto a system of ordinary linear algebraic equations. The solution of the system provides complete information about ratchet transport, avoiding such disadvantages of direct stochastic calculations as long transients and large statistical fluctuations. The Fokker-Planck approach to dynamical ratchets is instructive and opens the space for further generalizations.

FIG. 2: A summary of the forces and powers involved in the example motion considered in the text, each plotted as a function of time. The top graph shows the applied force (full line) and resulting ma (dashed line). The next graph shows df /dt; the self force is proportional to this. The third graph shows the particle speed v(t). The last graph shows two contributions to the power delivered to the electromagnetic field: the radiated power (full line) and the power delivered to the bound field (dashed line). When f is changing most of the power goes to the bound field. When f is constant but nonzero the two contributions are equal and opposite. The total area under the dotted curve is zero. Remark. The essential insight here is not new, but the author has not been able to find a presentation like the above in prior work. Pointers to books or papers where the same idea is set out would be appreciated and will be added to the list of references as appropriate.
A pair of small charged objects with the same final state of motion; one has accelerated, one has not.
Tracking the radiation reaction energy when charged bodies accelerate

August 2014


193 Reads

We address some questions related to radiation and energy conservation in classical electromagnetism. We first treat the well-known problem of energy accounting during radiation from a uniformly accelerating particle. We present the problem in the form of a paradox, and then answer it using a modern treatment of radiation reaction and self-force, as it appears in the expression due to Eliezer and Ford and O'Connell. We clarify the influence of the Schott force and the total radiated power, which differs from Larmor's formula. Finally, we present a simple and highly visual argument which enables one to track the radiated energy without the need to appeal to the far field in the distant future (the 'wave zone').

Simplified derivation of the Hawking-Unruh temperature for an accelerated observer in vacuum

January 2004


241 Reads

A detector undergoing uniform acceleration $a$ in a vacuum field responds just as though it were immersed in thermal radiation of temperature $T=\hbar a/2\pi k c$. A simple, intuitive derivation of this result is given for the case of a scalar field in one spatial dimension. The approach is then extended to treat the case where the field seen by the accelerated observer is a spin-1/2 Dirac field. Comment: 11 pages, no figures, (written in REVTEX4). Submitted to Am. J. Phys, 26Jan04. - Accepted Am.J.Phys (to appear Nov 2004). 15 pages, fixed confusing typo in Eq.(8), expanded references citng related previous works, discussion of parameter domain of integrals used, and relationship of Minkowski to Rindler vacuum

FIG. 1: The hyperbolic trajectory r a (τ ) given by Eq. (1). The retarded time τret associated with a given point (ct, x, y, z) corresponds to the (unique) intersection of the past light-cone of (ct, x, y, z) with the trajectory r a (τ ). For instance, Q ′ = (c 2 /g)(sinh(gτ Q ret /c), 0, 0, cosh(gτ Q ret /c)) and P ′ = (c 2 /g)(sinh(gτ P ret /c), 0, 0, cosh(gτ P ret /c)) define, respectively, the retarded times τ Q ret and τ P ret associated with the points Q and P. The future light-cone is the boundary of the causal future of a given point. Thus, any event occurring, for instance, in the spacetime point R will affect only the region enclosed by its future light-cone, with the light-cone surface reserved only to signals moving with velocity c. Note that only regions I and II are affected by the fields due to a charged particle with a worldline given by Eq. (1).
FIG. 2: The lines of constant ξ and τ according to Eqs. (10) and (16), respectively, for the regions I and II. In region I, the Rindler frame where ξ > 0, the identified lines correspond to ξ1 < ξ2 < ξ3 and τ1 < τ2 < τ3. Lines of constant ξ (the hyperbola) are timelike. On the other hand, for region II, known as the Milne frame where ξ < 0, the lines of constant τ are timelike. The identified lines in II correspond to the cases τ1 < τ2 < τ3 and ξ1 > ξ2 > ξ3. The horizon, the boundary ct = z between I and II, corresponds to one half of the degenerated hyperbola corresponding to ξ = 0.
The radiation of a uniformly accelerated charge is beyond the horizon: A simple derivation

July 2005


475 Reads

We show, by exploring some elementary consequences of the covariance of Maxwell's equations under general coordinate transformations, that, despite inertial observers can indeed detect electromagnetic radiation emitted from a uniformly accelerated charge, comoving observers will see only a static electric field. This simple analysis can help understanding one of the most celebrated paradoxes of last century.

Y The equivalence principle, uniformly accelerated reference frames, and the uniform gravitational field

March 2010


625 Reads

The relationship between uniformly accelerated reference frames in flat spacetime and the uniform gravitational field is examined in a relativistic context. It is shown that, contrary to previous statements in the pages of this journal, equivalence does not break down in this context. No restrictions to Newtonian approximations or small enclosures are necessary.

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