Showing papers in "American Journal of Physics in 1989"

Demonstration of single‐electron buildup of an interference pattern

Abstract: The wave–particle duality of electrons was demonstrated in a kind of two‐slit interference experiment using an electron microscope equipped with an electron biprism and a position‐sensitive electron‐counting system. Such an experiment has been regarded as a pure thought experiment that can never be realized. This article reports an experiment that successfully recorded the actual buildup process of the interference pattern with a series of incoming single electrons in the form of a movie.

Fisher information as the basis for the Schrödinger wave equation

Abstract: It is shown that the assumption that nature acts to make optimum estimates of position maximally in error leads to the Schrodinger (energy) wave equation (SWE). In this way, the SWE follows from a simple statement of uncertainty. The approach grows out of probability estimation theory, in particular the principle of minimum Fisher information (or maximum Cramer–Rao bound). The minimized Fisher information turns out to be proportional to the mean kinetic energy of the particle. This approach is an attractive supplement to conventional ways of introducing quantum mechanics to students, since it avoids the use of imperfect physical models (such as vibrating strings) or the immediate need for complex momentum operators, and follows from a plausible assumption as to how nature works.

Linear least‐squares fits with errors in both coordinates

Abstract: York’s solution to the problem of linear least‐squares fits with errors in both coordinates [D. York, Can. J. Phys. 44, 1079 (1966)] is shown to be exact and not subject to the erroneous results that attempts to modify standard least‐squares algorithms can produce. Detailed examples of the use of York’s method are given; a fortran implementation suitable for use on personal computers is available to interested parties.

Gamow vectors and decaying states

Abstract: Gamow vectors are generalized eigenvectors of the Hamiltonian with complex eigenvalues that describe exponentially decaying (or growing) states. The energy wavefunctions corresponding to Gamow vectors have a pole immediately below (or above) the real axis in the complex energy plane. Although complex energy values were introduced more than half a century ago for the theory of alpha decay, they have become disreputable and have been banished from quantum mechanics textbooks because of mathematical problems and incorrect physical interpretation. Developments in modern mathematics have now provided a mathematical foundation that has led to the correct physical interpretation. In this article, energy eigenvectors with complex eigenvalues are first introduced by explicitly considering a specific simple decaying system. Then, an elementary account of the properties of Gamow vectors is given. The results from the decaying system provide a motivation for the introduction of Gamow vectors and some of the backgroun...

Experimental Techniques in Condensed Matter Physics at Low Temperatures

Abstract: Cooling and cryogenic equipment (Dewars and magnets, cryostats for storage Dewars, dilution refrigerators, pumps and plumbing, magnetic cooling) cryogenic design aids (recipes, special materials, vibration isolation, electric and magnetic isolation, cooling the sample) experimental techniques and special devices (bridges, torsional oscillators, cryogenic electronics, nuclear magnetic resonance, ultrasound, high-frequency methods, pressure measurements, electromagnetic compatibility) thermometry (primary thermometry, magnetic thermometry, other secondary thermometers).

Evolution of the modern photon

Abstract: The term ‘‘photon’’ represents at least four distinct models and carries different connotations for students and for practicing physicists. This reflects the long and complex historical evolution of the concept and its association with the largely misinterpreted principle of duality. The unsatisfactory nature of the corpuscular and wave packet models is discussed, and the pedagogical desirability urged of replacing them with a semiclassical approach in elementary presentations. Derivations of the photoelectric (PE) effect without photons are cited and a vector analysis is given, demonstrating that the PE effect cannot be considered as simply the interaction of a photon and electron.

But is it science? : the philosophical question in the creation/evolution controversy

Abstract: Is 'creation science' really scientific? What criteria, if any, exist for identifying a discipline as a science? This collection of essays answers these questions and many more about the creation/evolution controversy and the very nature of science itself.

Gauge kinematics of deformable bodies

Abstract: The treatment of the motion of deformable bodies requires a specification of axes for each shape. A natural kinematic formulation of this problem is presented in terms of a gauge structure over the space of shapes that the body may assume. Within this framework, one may compute the net change in orientation of a body with angular momentum zero, resulting from a given sequence of deformations.

Nonlinear and chaotic string vibrations

Abstract: The nonlinear vibrations of an elastic string are studied. A string is capable of showing nonlinear phenomena including periodic, quasiperiodic, and chaotic motions. Hysteresis is found using the method of slowly varying amplitudes, and chaotic vibrations are predicted for an experimentally accessible regime.

Correlations and squeezing of two‐mode oscillations

Abstract: In a squeezed state, the variance in one canonical variable may be suppressed below that normally associated with either the ground state or a coherent state, at the expense of an expansion in the variance of the conjugate variable. Squeezed states are usually discussed in the context of quantized light fields. The principal properties of squeezed states are demonstrated using the motion of quantum mechanical simple harmonic oscillators. The motion of a single oscillator is discussed to introduce the key concepts of squeezing, including quadrature operators, and the error contours of the Wigner function describing the quantum quasiprobability distributions in phase space of the oscillator motion. The main topic of two‐mode squeezing is then addressed, in which the fluctuations in a system of two oscillators are tightly correlated. The existence of squeezing is demonstrated in normal mode coordinates representing motion of superpositions of the motion of the two oscillators. The fluctuations within a single oscillator are shown to increase when squeezing increases in the normal modes, generating thermal noise in individual mode subsystems.

The quantum mechanical few‐body problem

Abstract: A solution to quantum mechanical problems of two, three, and four bodies is discussed from the point of view of two‐body harmonic oscillator forces. An exact solution to the three‐ and four‐body problems is found for harmonic oscillator forces, assuming equal masses. The aim of the discussion is to make students aware of the real difficulties of the few‐body problem.

Nonequivalence of a uniformly accelerating reference frame and a frame at rest in a uniform gravitational field

Abstract: A general expression is obtained for the space‐time interval between neighboring events in a one‐dimensional space in which it is possible to set up a rigid reference frame. Particular expressions are then obtained for the interval for the special cases of a rigid frame at rest in a uniform gravitational field and a rigid frame uniformly accelerating in field‐free space. The two expressions are not equivalent and are used to show why, how, and to what extent observations made in a rigid enclosure at rest in a gravitational field are not equivalent to observations made in a rigid enclosure that is uniformly accelerating in field‐free space. Two facts of particular interest that are demonstrated in the course of the analysis are the following: (i) Two spatially separated particles that are simultaneously released from rest and allowed to fall freely in a uniform gravitational field will not remain at rest with respect to one another. (ii) Uniformly accelerating reference frames and inertial frames are the only possible one‐dimensional rigid frames in flat space‐time.

Kramers-Kronig in two lines

Abstract: A short derivation of the Kramers–Kronig relations is presented. A short derivation of the Kramers–Kronig relations is presented.

Derivation of reciprocity relations for a beam splitter from energy balance

Abstract: It is shown that the usual amplitude and phase relations connecting the reflectance and transmittance of a stratified or continuous, nonabsorbing beam splitter can be derived by a simple energy balance argument relating to a Michelson interferometer.

Maximum power point characteristics of heat engines as a general thermodynamic problem

Abstract: The average power versus efficiency characteristics of common selected engine cycles and a few irreversibility mechanisms are analyzed toward assessing the universal nature of assorted equations that have been derived recently for efficiency at maximum power point. Most results derived to date are shown to be special cases of one general thermodynamic problem of linearly irreversible heat engines, which explains the unifying basis for seemingly fortuitous similarities in their maximum power point characteristics.

The adiabatic theorem and Berry’s phase

Abstract: A study is presented of Berry’s observation that when a quantum‐mechanical system is transported on a closed adiabatic journey, a topological phase arises in addition to the usual dynamical phase expected from the adiabatic theorem. Consequences are explored in the case of a simple magnetic moment–magnetic field interaction and are shown to lead, among other things, to Dirac’s famous relation between electric and magnetic charges.

The dynamics of a falling chain: I

Abstract: ‘‘A chain is suspended, both ends at the same elevation, and then one end is released.’’ This familiar problem is studied and the real dynamical behavior is found to be very different from the textbook answer. Results are presented that show that the real chain is largely an energy‐conserving system. This gives rise to acceleration of the free end greater than g, and to very large tension at the fixed end.

Wind energy as a solar‐driven heat engine: A thermodynamic approach

Abstract: An upper bound on annual average energy in the Earth’s winds is calculated via the formalism of finite‐time thermodynamics. The Earth’s atmosphere is viewed as the working fluid of a heat engine where the heat input is solar radiation, the heat rejection is to the surrounding universe, and the work output is the energy in the Earth’s winds. The upper bound for the annual average power in the Earth’s winds is found to be 17 W/m2, which can be contrasted with the actual estimated annual average wind power of 7 W/m2. Our thermodynamic model also predicts the average extreme temperatures of the Earth’s atmosphere and can be applied to wind systems on other planets.

The case of the identically accelerated twins

Abstract: A variation on the ‘‘twin paradox’’ of special relativity is presented wherein twins undergo the same acceleration for the same length of time, yet they age differently. Although this problem is simple to solve, it gets to the heart of the behavior of clocks in special relativity and, hopefully, will help to dispel the notion students develop that the acceleration experienced by a relativistic traveler is directly related to the rate at which that traveler ages.

Multiple scattering approach to one‐dimensional potential problems

Abstract: By calculating the reflection and transmission coefficients for a square potential barrier in terms of a series of scatterings at step potential discontinuities, a multiple scattering approach to potential problems is presented. This approach is applied to WKB potential problems by interpreting the WKB connection formulas in terms of reflection and transmission coefficients at each turning point. An algorithm for the systematic construction of the energy Green’s function is given. The significance of complex coordinate turning points is demonstrated. The equivalence of this approach and that of the Gaussian approximation of the path integral is mentioned briefly.

Electrodynamics of moving dipoles: The case of the missing torque

Abstract: In a recent article, Bedford and Krumm [‘‘On the Origin of Magnetic Dynamics,’’ Am. J. Phys. 5 4, 1036 (1986)], examine in detail the interaction between a moving line of charges and a magnetic dipole consisting of a conducting ring of current from the point of view of different frames of reference. One particular orientation of the magnetic dipole leads to an apparently paradoxical situation of a torque acting in one inertial frame but not in another. These authors explain the situation by considering the rate of change of mass of the charge carriers and a wall force that prevents them from accelerating in the direction of their motion. They also hint at an intriguing analogy with some kind of inertial or fictitious force such as occurs in noninertial frames. First, the general expression is derived for the torque on a moving magnetic dipole in any orientation and the term representing the missing torque is explicitly revealed. Then the physical origin of this torque is investigated in the particular case considered by Bedford and Krumm, where the magnetic dipole consists of a small current‐carrying conducting loop. It appears that, in this case, the elusive torque arises from the interaction between the current carriers in the magnetic dipole loop and the magnetic field due to the surfacecurrentgenerated by the motion of the induced charges on the surface of the conducting loop.

Alvarez: Adventures of a Physicist

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Thermodynamics as the continuum limit of statistical mechanics

Abstract: The thermodynamic limit in statistical mechanics is often equivalent to a properly defined continuum limit, in which Boltzmann’s constant k vanishes together with the microscopic scales of length and time. In this continuum limit, which is the missing link between statistical mechanics and the historical development of thermodynamics, all microscopic fluctuations are suppressed, just as the quantum mechanical uncertainties are in the classical limit; these two limits are similar, but can be taken independently. The continuum limit is to be preferred above the thermodynamic limit when macroscopic dependences on space or time are present and may help to solve conceptual problems. As an example, capillary phenomena are discussed with particular attention to Kelvin’s equation. The enforced behavior of coupling constants in the continuum limit stresses the phenomenological and approximate status of thermodynamics. The relation with other reduction schemes of macroscopic physics is indicated, and possible consequences for ergodic theory are sketched.

Energy density and stress: A new approach to teaching electromagnetism

Abstract: By introducing the electromagnetic field in the customary way, ideas are promoted that do not correspond to those of contemporary physics: on the one hand, ideas that stem from pre‐Maxwellian times when interactions were still conceived as actions at a distance and, on the other hand, ideas that can be understood only from the point of view that the electromagnetic field is carried by a medium. A part of a course in electromagnetism is sketched in which, from the beginning, the electromagnetic field is presented as a system in its own right and the local quantities energy density and stress are put into the foreground. In this way, justice is done to the views of modern physics and, moreover, the field becomes conceptually simpler.

An experiment on the physical reality of edge‐diffracted waves

Abstract: An experiment that deals with the physical reality of the edge‐diffracted wave in the boundary wave theory of diffraction is described in this article. The basic philosophy is simple and the experimental evidence is strong.