Solution of the relativistic Schrödinger equation for the δ ′ -Function potential in one dimension using cutoff regularization

We study the relativistic version of the Schrödinger equation for a point particle in one dimension with the potential of the first derivative of the delta function. The momentum cutoff regularization is used to study the bound state and scattering states. The initial calculations show that the reciprocal of the bare coupling constant is ultraviolet divergent, and the resultant expression cannot be renormalized in the usual sense, where the divergent terms can just be omitted. Therefore, a general procedure has been developed to derive different physical properties of the system. The procedure is used first in the nonrelativistic case for the purpose of clarification and comparisons. For the relativistic case, the results show that this system behaves exactly like the delta function potential, which means that this system also shares features with quantum filed theories, like being asymptotically free. In addition, in the massless limit, it undergoes dimensional transmutation, and it possesses an infrared conformal fixed point. The comparison of the solution with the relativistic delta function potential solution shows evidence of universality.

Numerical corrections to the strong coupling effective Polyakov-line action for finite T Yang-Mills theory

We consider a three-dimensional effective theory of Polyakov lines derived previously from lattice Yang-Mills theory and QCD by means of a resummed strong coupling expansion. The effective theory is useful for investigations of the phase structure, with a sign problem mild enough to allow simulations also at finite density. In this work we present a numerical method to determine improved values for the effective couplings directly from correlators of 4d Yang-Mills theory. For values of the gauge coupling up to the vicinity of the phase transition, the dominant short range effective coupling are well described by their corresponding strong coupling series. We provide numerical results also for the longer range interactions, Polyakov lines in higher representations as well as four-point interactions, and discuss the growing significance of non-local contributions as the lattice gets finer. Within this approach the critical Yang-Mills coupling β c is reproduced to better than one percent from a one-coupling effective theory on N τ = 4 lattices while up to five couplings are needed on N τ = 8 for the same accuracy.

The Gronwall form of the Schrodinger Equation for Helium

The 1937 paper of Gronwall which concerns an alternative form for the Schrodinger Equation of the 2-electron Helium problem is re-derived in a (hopefully) transparent (possibly pedestrian) manner.

Frobenius Solution for Legendre's Equation, Rodrigue's Formula and Normalization

The Frobenius solution to Legendre/s equation is developed in detail as is Rodrigue's formula, which is employed to normalize Legendre polynomials.

Continued Fraction Solutions to Hermite's, Legendre's and Laguerre's Differential Equation

The continued fraction method for solving differential equations is illustrated using three famous differential equations used in quantum chemistry.

The Wave Equation for Stringed Instruments

An elementary derivation of the wave equation as applied to violin strings is given.

Guessing Solutions to the H-atom Schrodinger Equation

In teaching elementary quantum chemistry, the concept of eigenfunctionality is explored using the H-atom's Hamiltonian and various guessed functions. This is done in Cartesian coordinates, in Spherical Polar coordinates, and in Confocal Elliptical coordinates.

Yang, Figurine, Profil, in: Daniel-François-Esprit Auber: Le Cheval de Bronze

Signatur des Originals: S 36/G10193

Consumers mediate the effects of experimental ocean acidification and warming on primary producers

It is well known that ocean acidification can have profound impacts on marine organisms. However, we know little about the direct and indirect effects of ocean acidification and also how these effects interact with other features of environmental change such as warming and declining consumer pressure. In this study, we tested whether the presence of consumers (invertebrate mesograzers) influenced the interactive effects of ocean acidification and warming on benthic microalgae in a seagrass community mesocosm experiment. Net effects of acidification and warming on benthic microalgal biomass and production, as assessed by analysis of variance, were relatively weak regardless of grazer presence. However, partitioning these net effects into direct and indirect effects using structural equation modeling revealed several strong relationships. In the absence of grazers, benthic microalgae were negatively and indirectly affected by sediment-associated microalgal grazers and macroalgal shading, but directly and positively affected by acidification and warming. Combining indirect and direct effects yielded no or weak net effects. In the presence of grazers, almost all direct and indirect climate effects were nonsignificant. Our analyses highlight that (i) indirect effects of climate change may be at least as strong as direct effects, (ii) grazers are crucial in mediating these effects, and (iii) effects of ocean acidification may be apparent only through indirect effects and in combination with other variables (e.g., warming). These findings highlight the importance of experimental designs and statistical analyses that allow us to separate and quantify the direct and indirect effects of multiple climate variables on natural communities.

Putting prey and predator into the CO2 equation-qualitative and quantitative effects of ocean acidification on predator-prey interactions

Little is known about the impact of ocean acidification on predator-prey dynamics. Herein, we examined the effect of carbon dioxide (CO(2)) on both prey and predator by letting one predatory reef fish interact for 24 h with eight small or large juvenile damselfishes from four congeneric species. Both prey and predator were exposed to control or elevated levels of CO(2). Mortality rate and predator selectivity were compared across CO(2) treatments, prey size and species. Small juveniles of all species sustained greater mortality at high CO(2) levels, while large recruits were not affected. For large prey, the pattern of prey selectivity by predators was reversed under elevated CO(2). Our results demonstrate both quantitative and qualitative consumptive effects of CO(2) on small and larger damselfish recruits respectively, resulting from CO(2)-induced behavioural changes likely mediated by impaired neurological function. This study highlights the complexity of predicting the effects of climate change on coral reef ecosystems.

Geometric Integrability of the Camassa-Holm Equation. II

It is known that the Camassa–Holm (CH) equation describes pseudo-spherical surfaces and that therefore its integrability properties can be studied by geometrical means. In particular, the CH equation admits nonlocal symmetries of “pseudo-potential type”: the standard quadratic pseudo-potential associated with the geodesics of the pseudo-spherical surfaces determined by (generic) solutions to CH, allows us to construct a covering π of the equation manifold of CH on which nonlocal symmetries can be explicitly calculated. In this article, we present the Lie algebra of (first-order) nonlocal π-symmetries for the CH equation, and we show that this algebra contains a semidirect sum of the loop algebra over sl(2,R) and the centerless Virasoro algebra. As applications, we compute explicit solutions, we construct a Darboux transformation for the CH equation, and we recover its recursion operator. We also extend our results to the associated Camassa–Holm equation introduced by J. Schiff.

Equation of state for hot dense matter using a relativistic screened hydrogenic model

The study of matter under conditions of high density, pressure, and temperature is a valuable subject for inertial confinement fusion (ICF), astrophysical phenomena, high-power laser interaction with matter, etc. In all these cases, matter is heated and compressed by strong shocks to high pressures and temperatures, becomes partially or completely ionized via thermal or pressure ionization, and is in the form of dense plasma. The thermodynamics and the hydrodynamics of hot dense plasmas cannot be predicted without the knowledge of the equation of state (EOS) that describes how a material reacts to pressure and how much energy is involved. Therefore, the equation of state often takes the form of pressure and energy as functions of density and temperature. Furthermore, EOS data must be obtained in a timely manner in order to be useful as input in hydrodynamic codes. By this reason, the use of fast, robust and reasonably accurate atomic models, is necessary for computing the EOS of a material.

Equation of State and Opacities for Warm Dense Matter.

We will present recent developments in the calculation of opacity and equation of state tables suitable for including in the radiation hydrodynamic code ARWEN [1] to study processes like ICF or X-ray secondary sources. For these calculations we use the code BiG BART to compute opacities in LTE conditions, with self-consistent data generated with the Flexible Atomic Code (FAC) [2]. Non-LTE effects are approximately taken into account by means of the improved RADIOM model [3], which makes use of existing LTE data tables. We use the screened-hydrogenic model [4] to derive the Equation of State using the population and energy of the levels avaliable from the atomic data

New reynolds equation for line contact based on the carreau model modification by bair

This paper presents a new form of the one-dimensional Reynolds equation for lubricants whose rheological behaviour follows a modified Carreau rheological model proposed by Bair. The results of the shear stress and flow rate obtained through a new Reynolds–Carreau equation are shown and compared with the results obtained by other researchers.