Session J42: General Theory, Methods, Education and Relativity

Using a new Delta Function Representation, QFT Disconnected Diagrams Evaluate to Zero

When evaluating Quantum Field Theory terms using diagram based perturbation expansions (Feynman Diagrams), those diagrams that have subgraphs that are not connected to incoming or outgoing particles are ignored. There are convincing physical reasons why these particular diagrams should be ignored. However, when the numerical value of a typical such term is evaluated, the calculated value is conventionally divergent. In this paper, these disconnected diagrams are evaluated using a new representation of the Dirac delta function. By constraining the choices of the delta function representations, the value of any QFT disconnected diagram can be made to be precisely zero. This approach gives a mathematical justification for ignoring such terms without having to invoke infinite renormalization.   

New Analytic Solutions of Schr\"{o}dinger's Equation in Time and Space

The Schr\"{o}dinger equation is a pillar of modern science. Numerous methods and techniques have been developed to find an exact or an approximate solution of the SE such as perturbation theory, variational methods, and diagram methods. One widely used approximation is the WKB method and variants. The WKB approximation has proven its efficiency to solve Schr\"{o}dinger-like equations. Nevertheless the WKB method is limited to an adiabatic potential where the variation of the potential energy at the distance of the order of the de Broglie wavelength is small in comparison to the kinetic energy. In the first part, we present an analytic solution beyond adiabatic approximation by transferring the 1D Schr\"{o}dinger equation into the Ricatti equation. Then we show that our solution is more accurate than WKB approximation. The generalization of the approach to 3D is suggested. In the second part, we present a new analytic treatment of the detuned atom-field interaction beyond the rotating wave approximation; we find a new approximate but very accurate analytic solution for population transfer. Finally the connection between the two parts will be discussed.   

Differential geometry and the design of optimization algorithms

The problem of parameter estimation by nonlinear least-squares minimization can be recast as a problem in differential geometry by interpreting the set of model predictions as a manifold embedded within the space of data. The minimization problem is then to find the point on the manifold closest to the data. It is well known that problems with large numbers of parameters can be extremely difficult, often because of sloppiness, or an extreme degeneracy in the parameters. Geometrically, this is interpreted as the manifold forming a narrow hyper-ribbon. Standard algorithms fail when they approach the boundaries of the manifold. Differential geometry suggests improvements to standard algorithms to avoid this pitfall. By including a geodesic acceleration correction and allowing uphill moves we dramatically improve the success rate and the convergence speed of the standard Levenberg-Marquardt algorithm.   

Continuum dislocation dynamics: analogies to fluid turbulence?

The dislocations which mediate plastic flow in crystals are described in the continuum with a nine-component tensor field. We study a nonlinear evolution law for this dislocation density, which shows several intriguing analogies to fully developed turbulence. (a)~As in the infinite Reynolds number limit, where vortex singularities are conjecured to form in finite time, our equation form wall singularites related to those in Burger's equation\footnote{S. Limkumnerd and J. P. Sethna, Phys. Rev. Letters \textbf{96}, 095503 (2006)}. To resolve these walls accurately we apply a central upwind scheme\footnote{A. Kurganov and E. Tadmor, J. of Comp. Phys. \textbf{160}, 1, 241-282 (2000)}. (b)~As in turbulence, we find self-similarity and scaling in the resulting cell wall morphologies when dislocation climb is forbidden. When climb is allowed (i.e., high temperatures) we form non-fractal walls representing grain boundaries. (c)~As in turbulence, where chaos allows only statistical convergence at long times, our two-dimensional simulations appear to have no weak solutions in the continuum limit -- the cascade of structure to short length scales appears to be sensitively dependent on the ultraviolet cutoff.   

Angular Momentum decomposition of nucleon

We calculate the orbital angular momentum of the electron in the QED as well as that of the quark in scalar diquark model. We compare the orbital angular momentum obtained from the Jaffe- Manohar decomposition to that obtained from the Ji relation and estimate the importance of the vector potential in the definition of orbital angular momentum.   

Level correlations in an integrable potential system: a modified Kepler problem

We investigate level correlations in the semiclassical energy spectrum of a modified Kepler problem. The latter features the exponential distribution of the nearest level spacings typical of ``generic'' (that is without extra degeneracies) integrable systems. Numerically, moving up in spectrum, we observe jumps of saturation level rigidity at certain values of energy. Also in saturation regime, we observe large oscillations of the level number variance over an energy interval as a function of the interval width; these have fixed period and amplitude and the variance reaches near zero values with the same periodicity. These unusual spectral properties are explained using both the periodic orbit theory and a derivation based directly on the quantum mechanical spectrum in the semiclassical limit. Numerical and analytical results are in excellent agreement.   

Study of stability and dynamical properties of Rosenau-Hyman compactons

We use Pade approximants to study numerically the stability and dynamical properties of K(2,2) Rosenau-Hyman compactons. We present a systematic derivation of Pade approximants for calculating the derivatives of smooth functions on an uniform grid and we illustrate our finding by improving upon traditional fourth-order finite-differences formulas. This study is intended as a stepping block towards a systematic numerical study of soliton solutions with a compact support of generalized Korteweg-de Vries equations.   

Energy Loss by Photons and Charged Particles in Metals to Insulators

We have used composite sets of optical data to compare the energy-loss of photons and charged particles in metals, semiconductors and insulators. For small momentum transfers in a dielectric-continuum, these losses are proportional to Im \textit{$\varepsilon $}(\textit{$\omega $}) and Im[-1/\textit{$\varepsilon $}(\textit{$\omega $})], respectively. The first involves excitation of transverse E-M modes; the second excitation of longitudinal collective modes. Both involve dissipation \textit{via}. electric-dipole transitions, but their absorption spectra are remarkably different. Low-frequency absorptions prominent in optical spectra are suppressed in charged-particle energy loss, whereas high-frequency transitions are strongly enhanced. In metals the latter appear as plasmon absorptions. The total oscillator strength is found to be the same for both transverse and longitudinal modes, verifying the f-sum rule. The spectral shift and strength redistribution are essentially inertial effects. This is apparent as dispersion in the experimental collective-mode screening functions: At low frequencies, electrons follow the charge's field responding in phase to \textit{reduce} the charge's field, whereas above the collective-mode resonance, their motion falls behind the field by 180$^{\circ}$ \textit{enhancing} the charge's field.   

On the quantumdynamics of measurement with geometric algebra

In the spirit of Special Relativity, a notion for the equivalence of quantum observers is given by denying the existence of an absolute quantum mechanical description for the wavefunction of a particular quantum observer. In fact, the language of geometric algebra frees us from any particular representation so that we may construct a spacetime invariant, called the proper observable, between quantum observers from within the framework of Special Relativity. Since we wish to capture the seemingly nonlocal effects of Quantum Mechanics, we establish a description of equivalence in the geometric algebra of the observables of Quantum Mechanics, providing a couple of examples with their consequences. Then we discuss the implications for the equivalence of quantum observers with regard towards Bell's inequality and the measurement of spin.   

Lowest Order Relativistic Corrections of the Helium Atom and the Hydrogen Molecule Computed Using Monte Carlo Methods

We have calculated the lowest order relativistic effects for the three lowest states of the helium atom with symmetry $^{1}$S, $^{1}$P, $^{1}$D, $^{3}$S, $^{3}$P and $^{3}$D using variational Monte Carlo methods and compact, explicitly correlated trial wavefunctions. Our results are in excellent agreement with those of Drake and Yan (Phys. Rev. A \textbf{46}, 2378, 1992). We have also calculated the lowest order relativistic effects for the ground state of the hydrogen molecule with symmetry X${ }^1\Sigma _{\mbox{g}}^{\mbox{+}}$, B${ }^1\Sigma _{\mbox{u}}^{\mbox{+}}$, a${ }^3\Sigma _{\mbox{g}}^{\mbox{+}}$, b${ }^3\Sigma _{\mbox{u}}^{\mbox{+}}$, I${ }^1\Pi _{\mbox{g}}$, C${ }^1\Pi _{\mbox{u}}$, i${ }^3\Pi _{\mbox{g}}$, c${ }^3\Pi _{\mbox{u}}$, J${ }^1\Delta _{\mbox{g}}$ and j${ }^3\Delta _{\mbox{g}}$ using these same methods and a new set of compact, explicitly correlated trial wavefunctions. Our values are in excellent agreement with earlier calculations on the X${ }^1\Sigma _{\mbox{g}}^{\mbox{+}}$ and B${ }^1\Sigma _{\mbox{u}}^{\mbox{+}}$ states. For the other states, our work provides the first evaluation of these properties. Finally, we discuss the extension of these methods to larger atoms and molecule.   

Computational Physics in Undergraduate Solid State

Computational physics is of essential importance in the undergraduate curriculum. In solid state physics the need to visualize complicated concepts and the need to understand difficult formulae demand the use of computers. Here, one of the most challenging tasks is to calculate the density of states of a solid, for example, where the band structure is needed. Integrations over the bands in k-space are quite challenging. I have developed an approach to this task and demonstrate the steps to carry out its computation suitable for undergraduate student use. The approach is applied to the simple cubic structures. The method used is based on employing the band structure's Green's function and employing the k-space Brillouin-zone ray approach [1] combined with a complex integration method [2]. Because the Green's function contains information about the system's spectrum, the density of states can naturally be used for this purpose. The number of occupied electron states up to a certain energy is obtained using Romberg's method and the results are shown for the above example structures. Other solid state properties that can be illustrated will be discussed. \\[4pt] [1] An-Ban Chen, Phys. Rev. B, Vol. 16, 3291 (1977). [2] Hasbun, Javier, http://meetings.aps.org/link/BAPS.2009.MAR.L29.12   

Computer Based Collaborative Problem Solving for Introductory Courses in Physics ~

We discuss collaborative problem solving computer-based recitation style. The course is designed by Lee [1], and the idea was proposed before by Christian, Belloni and Titus [2,3]. The students find the problems on a web-page containing simulations (physlets) and they write the solutions on an accompanying worksheet after discussing it with a classmate. Physlets have the advantage of being much more like real-world problems than textbook problems. We also compare two protocols for web-based instruction using simulations in an introductory physics class [1]. The inquiry protocol allowed students to control input parameters while the worked example protocol did not. We will discuss which of the two methods is more efficient in relation to Scientific Discovery Learning and Cognitive Load Theory. 1. Lee, Kevin M., Nicoll, Gayle and Brooks, Dave W. (2004). ``A Comparison of Inquiry and Worked Example Web-Based Instruction Using Physlets'', Journal of Science Education and Technology 13, No. 1: 81-88. 2. Christian, W., and Belloni, M. (2001). Physlets: Teaching Physics With Interactive Curricular Material, Prentice Hall, Englewood Cliffs, NJ. 3. Christian,W., and Titus,A. (1998). ``Developing web-based curricula using Java Physlets.'' Computers in Physics 12: 227--232.   

Proposed Experimental Test of Gall's Predicted Isochromatic Black Body Displacement Law

The test of any Black Body Distribution function is how well it satisfies: Stefan-Boltzmann law $\left( I=\sigma T^{4}\right) $; Wien's isothermal displacement law $\left( \lambda _{m} T=b\right) $ and the maximum isothermal emitted intensity condition $\left( I_{\lambda _{m}}\propto T^{5}\right) $. Gall's function $\left( I_{\lambda }=\sigma \frac{T^{6}}{b^{2}} \lambda e^{-\frac{\lambda T}{b}}\right) $ satisfies these conditions exactly and unlike all previous candidates employs the original empirical constants $\left( \sigma ,b\right) $ in its formulation. Distinct from Planck and all others, it predicts an isochromatic displacement law: $\lambda T_{m}=6b$, where $T_{m}$ is the temperature of maximum emitted intensity for a given $\lambda $. The associated maximum isochromatic emitted intensity should satisfy $I_{T_{m}}\propto \lambda ^{-5} $. At wavelengths of 20, 25 and 29 $\mu m$, $T_{m}$ is calculated to be 870, 696 and 600 K respectively. In this range $\left( T_{m}<1000\,K\right) $ , temperatures can be measured without using the colour temperature. Gall's exact distribution function seriously questions Planck's inexact function. This proposed test is imperative as the existence of an isochromatic maximum intensity at $T_{m}$ would affirm Gall's prediction of a crossover wavelength above which a colder body would emit with greater intensity than a hotter one. Its non-existence would reassert support for Planck's traditional notion that a hotter body always emits more intensely than a colder one throughout the entire EMR spectrum (http://sites.google.com/site/purefieldphysics).