Bulletin of the American Physical Society
75th Annual Meeting of the Southeastern Section of APS
Volume 53, Number 13
Thursday–Saturday, October 30–November 1 2008; Raleigh, North Carolina
Session GC: Mathematical Physics and Physics Education |
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Chair: Ronald Mickens, Clark Atlanta University Room: Holiday Inn Brownstone Lincoln |
Friday, October 31, 2008 8:30AM - 8:42AM |
GC.00001: Eigenvalue Spacings in the Asymmetric Infinite Square Well Todd Timberlake, Molly Nelson The distribution of eigenvalue spacings is an important tool in the study of quantum chaos. Quantum systems with chaotic classical counterparts exhibit eigenvalue spacings that follow random matrix statistics, while those with integrable counterparts generally follow Poisson statistics. One-dimensional quantum wells, which are always integrable under Newtonian mechanics, are expected to display uniform eigenvalue spacings (after unfolding). We will show that the asymmetric infinite square well (an infinite square well with a step) does not fit this expectation. The spacings between eigenvalues above the step height are not uniform after unfolding, but gradually approach uniformity at higher energies. Semiclassical analysis reveals that the departure from uniformity is due to the existence of non-Newtonian periodic orbits that reflect from the step even though their energies are above the step height. The sequence of eigenvalue spacings also displays some unusual features that are related to resonances in the classical (non-Newtonian) dynamics. [Preview Abstract] |
Friday, October 31, 2008 8:42AM - 8:54AM |
GC.00002: Nonassociative decomposition of angular momentum operator using complex octonions Jens Koeplinger, Vladimir Dzhunushaliev Non-associative octonion algebra has been proposed [e.g. V. Dzhunushaliev, J. Math. Phys. 49, 042108 (2008); arXiv:0712.1647] for description of a hidden structure in operator quantum mechanics, which contains traditional observables, as well as unobservable quantities that cannot be measured in principle. The approach allows to decompose the supersymmetric momentum operator, and also the angular momentum operator, as a bilinear combination of some (non-associative) operators [V. Dzhunushaliev, arXiv:0805.3221]. This talk presents the finding of a linear decomposition of the angular momentum operator, with use of complex octonion numbers. Potential implications and next steps will be outlined. [Preview Abstract] |
Friday, October 31, 2008 8:54AM - 9:06AM |
GC.00003: Information Content in Antenna Radiation Patterns Karan Mohan, M. Amir Khan, Amin Dharamsi We discuss the application of Shannon's Information Theory to the quantification of information lost in practical measurements of radiation patterns of antenna arrays. The radiation pattern of a uniform linear one-dimensional phased array of N dipole elements, with inter-element spacing of d and inter-element phase of $\psi $ is given by $P_N (\theta ,\phi )=P_{ind} \left[ {\sum\limits_{n=1}^N {I_0 \exp (i(kd_n Sin\theta Sin\phi +\psi _n ))} } \right]$, where P$_{ind}$ is the radiation pattern of an individual dipole element, I$_{0}$ is the intensity and k is the wavenumber. The probability density function, in space, of emitted photons is therefore given by $\xi _N (\theta ,\phi )=P_N (\theta ,\phi )/\int {P_N (\theta ,\phi )d\Omega } $. The entropy of this distribution is $H=-\int {\xi _N \ln \xi _N d\Omega } $. The information obtained about any change in the array is therefore $\Delta H=\left| {H_f -H_i } \right|$ where H$_{f}$ and H$_{i}$ correspond to the entropies of the final and initial distributions. This expression leads to a quantitative formulation for the information lost due to imprecision in detectors used to measure radiation patterns. The theoretical approach developed allows one to measure any changes in the antenna array that may occur in an optimal manner, given limitations such as noise, a finite number of detectors of finite precision, or access only to a limited section of the surrounding space. [Preview Abstract] |
Friday, October 31, 2008 9:06AM - 9:18AM |
GC.00004: Closed Analytic Solution for the Potential and Equations of Motion in the Presence of a Gravitating Oblate Spheroid William Atkinson A closed analytic solution for the potential due to a gravitating solid oblate spheroid, derived in oblate spheroidal coordinates in this paper, is shown to be much simpler than those obtained either in cylindrical coordinates (MacMillan) or in spherical coordinates (McCullough). The derivation in oblate spheroidal coordinates is also much simpler to follow than those of the MacMillan or McCullough. The potential solution is applied in exacting a closed solution for the equations of motion for an object rolling on the surface of the spheroid subjected only to the gravitational force component tangential to the surface of the spheroid. The exact solution was made possible by the fact that the force can be represented as separable functions of the coordinates only in oblate spheroidal coordinates. The derivation is a good demonstration of the use of curvilinear coordinates to problems in classical mechanics, potential theory, and mathematical physics for both undergraduate and graduate students. [Preview Abstract] |
Friday, October 31, 2008 9:18AM - 9:30AM |
GC.00005: Semi-classical Determination of the Energy Levels of a $x^{4/3}$ Potential Kale Oyedeji, Ronald Mickens Given a classical solution to a 1-dim in space system, for which all the solutions are periodic, the application of the modified Bohr-Sommerfeld quantization condition [1] allows a determination of semi- classical estimates for the energy levels of the associated quantum system. We consider a $x^{4/3}$ potential and use the methods of harmonic balance and iteration to calculate accurate approximations to the classical periodic solutions [2]. With these results, a general semi-classical energy spectrum can be determined. To judge the accuracy/validity of these calculations, we use a simple functional form for the ground state wave function in a variational calculation of the associated energy and compare this value with our semi-classical result. \\[3pt] [1] A. B. Migdal and V. P. Krainov, ``Approximation Methods in Quantum Mechanics (W. A. Benjamin, New York, 1969). \\[0pt] [2] R. E. Mickens, Journal of Sound and Vibration 292 (2006), 964-968. [Preview Abstract] |
Friday, October 31, 2008 9:30AM - 9:42AM |
GC.00006: A Theoretical Estimate for the Frequency of the TNL Oscillator $\ddot x + x + x^{1/3}=0$ Dorian Wilkenson, Ronald Mickens Truly nonlinear (TNL) oscillators have the property of having no linear approximation at the fixed-point of the modeling differential equation [1]. For a conservative oscillator this means that the fixed-point is a nonlinear center. Another feature of TNL oscillators is that none of the standard perturbation expansion procedures can be applied to calculate analytical approximations to the periodic or oscillator solutions [2]. Using the initial conditions $x(0)=A$ and $\dot x(0)=0$, we calculate the frequency $\omega (A)$ of the equation given in the title for small, $0 [Preview Abstract] |
Friday, October 31, 2008 9:42AM - 9:54AM |
GC.00007: On a Connection Between the Eigenvalues of the One-dimensional Finite Square Well and Number Theory Walter Jaronski The problem of the one-dimensional finite square well is a standard exercise in the introductory quantum mechanics course. As is well known, the energy eigenvalues cannot be determined in closed form. However, for a given range of the well parameter, the energies can be represented as an infinite series in this parameter. Although this series cannot be summed to give a common function, the coefficients in the series can be determined from a simple formula. In particular, for small values of the well parameter, the coefficients can be expressed in terms of generalized Euler numbers. This technique will be compared with standard numerical procedures for determining the energies, with regard to both computational ease and pedagogical benefits. The opportunity to introduce students to new mathematics will be emphasized. [Preview Abstract] |
Friday, October 31, 2008 9:54AM - 10:06AM |
GC.00008: Teaching Introductory Physics with Turning Point Software R. Seth Smith Physics professors attempt to engage students in the learning of physics by teaching this subject in a manner that students will perceive as relevant, interesting, fun, intriguing, and clear. Perhaps, the most valuable contribution that a professor can make to a student's education is to stimulate an interest in a particular discipline. To accomplish this, a professor has to be a performer. One doesn't necessarily have to entertain, but one must bring a certain level of enthusiasm and energy to a classroom in order to engage the students. If they are not engaged in the classroom, students will not learn effectively. Towards this end, Turning Point software was used to create new classroom presentations for FMU's introductory physics class in the Fall of 2008. Turning Point is essentially Power Point, but it provides one with the ability to embed interactive questions within a presentation. Students in the class respond to these questions by using radio frequency devices known as ``clickers.'' An analysis of the effectiveness of this approach, as well as a comparison to traditional chalk and blackboard methods, will be presented. [Preview Abstract] |
Friday, October 31, 2008 10:06AM - 10:18AM |
GC.00009: Investigation of 60Hz electromagnetic environmental noise John Howell, John Gaffney, Sanichiro Yoshida In addition to many other sources, alternating current causes electromagnetic noise. This noise is generated at close to 60Hz in the United States. Many different factors can influence the specifications of the noise, including mechanical load on the power company's generator. By using a simple antenna consisting of a capacitor and iron-core inductor along with gathering and processing programs (Labview and Matlab), we have characterized this 60Hz noise. Peak shifts as well as broadening and narrowing phenomena were common in measurements. By constructing a program script to step through the measured data in a set increment, we were able to determine the frequency and amplitude fluctuations of particular samples. Different times of day show different fluctuations as conditions such as seismic disturbance change and possibly electric load varies. [Preview Abstract] |
Friday, October 31, 2008 10:18AM - 10:30AM |
GC.00010: Dispositioning Unwanted Radioactive Materials - A South Carolina Perspective David Peterson The complex issue of the disposing of unwanted radioactive materials attracts significant public interest. As of July 2008, the use of the low-level radioactive waste repository in Barnwell, South Carolina, is restricted to the states of South Carolina, New Jersey, and Connecticut. The higher level radioactive materials at the Savannah River Site in Aiken, South Carolina are to be dispositioned over the next twenty to thirty years using the Salt-stone technique and the vitrification procedure, under a plan devised by the U.S. Department of Energy. [Preview Abstract] |
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