15491159
Exact Solutions of the Two-Dimensional Schrödinger Equation with Certain Central Potentials
By applying an ansatz to the eigenfunction, an exact closed-form solution of theSchrödinger equation in two dimension is obtained with the potentials V(r) =ar2 + br4 + cr6,V(r) = ar + br2 + cr−1,and V(r) = ar2 + br−2+ cr−4 + dr−6,respectively. The restrictions on the parameters of the given potential andthe angular momentum m are obtained.
120777661
ANALYTICAL APPROXIMATIONS TO THE SCHRÖDINGER EQUATION FOR A SECOND PÖSCHL–TELLER-LIKE POTENTIAL WITH CENTRIFUGAL TERM
The bound state solutions of the Schrodinger equation for a second Poschl–Teller-like potential with the centrifugal term are obtained approximately. It is found that the solutions can be expressed in terms of the hypergeometric functions 2F1(a, b; c; z). To show the accuracy of our results, we calculate the eigenvalues numerically for arbitrary quantum numbers n and l. It is found that the results are in good agreement with those obtained by other method for short-range potential. Two special cases for l = 0 and V1 = V2 are also studied briefly.
122483032
Wave Equations in Higher Dimensions
Part I (Introduction).- Part II (Theory). - 2. Special orthogonal groups (Introduction Abstract groups Orthogonal group SO(n) Tensor representations of the SO(n) \Gamma matrix groups Spinor representations of the SO(n) Concluding remarks).- 3. Rotational symmetry and Schrodinger equation in N-dimensional space (Introduction Rotation operator Orbital angular momentum operators The linear momentum operators Radial momentum operator Spherical harmonic polynomials Schrodinger equation for a two-body system Concluding remarks).- 4. Dirac equation in higher dimensions (Introduction Dirac equation in N+1 dimensions The radial equation Application to hydrogen atom Concluding remarks).- 5. Klein-Gordon equation in higher dimensions (Introduction The Radial equation Application to hydrogen atom Concluding remarks).- Part III (Application in Non-relativistic Quantum Mechanics).- 6. Harmonic oscillator (Introduction Exact solutions of harmonic oscillator Recurrence relations for the radioal function Realization of dynamic group SU(1, 1) Generalization to pseudoharmonic ooscillator Position and momentum information entropy Conclusions).- 7. Coulomb potential (Introduction Exact solution Shift operators Mapping between Coulumb and harmonic oscillator radial functions Realization of dynamic of dynamic group SU (1, 1) Generalization to Kratzer potential Concluding remarks).- 8. Wave function ansatz method (Introduction Sextic potential Singular one-fraction power potential Mixture potential Non-polynomial potential Screened Coulomb potential Morse potential Conclusions).- 9. Levinson theorem for Schrodinger equation (Introduction Scattering states and phase shifts Bound states Sturm--Liouville theorem Levinson theorem Discussions Conclusions).- 10. Generalized hypervirial theorem for Schrodinger equation (Introduction Generalized Blanchard's and Kramers' recurrence relations Applications to central potentials Conclusions).- 11. Exact quantization rule and Langer modification (Introduction WKB approximation Exact quantization rule Application to trigonometric Rosen-Morse potential Proper quantization rule Illustrations of proper quantization rule Langer modification in D dimensions Calculations of logarithmic derivatives of wavefunction Conclusions).- 12. Schrodinger equation with position-dependent mass (Introduction Formalism Applications to harmonic oscillator and Coulomb potential Conclusions).- Part IV (Application in Relativistic Quantum Mechanics).- 13. Dirac equation with Coulomb potential (Introduction Exact solutions of hydrogen-like atoms Analysis of eigenvalues Generalization to the Dirac equation with Coulomb potential plus scalar potential Concluding remarks).- 14. Klein-Gordon equation with Coulomb potential (Introduction Eigenfunctions and eigenvalues Analysis of eigenvalues Generalization: Klein-Gordon equation with Coulomb plus scalar potential Comparison theorem Conclusions).- 15. Levinson theorem for Dirac equation (Introduction Generalization Sturm-Liouville theorem Number of bound states Relativistic Levinson theorem Discussions Friedel Theorem Comparison theorem Conclusions).- 16. Generalized hypervirial theorem for Dirac equation (Introduction Relativistic recurrence relation Diagonal case Conclusions).- 17. Kaluza-Klein theory (Introduction (4+D) -dimensional Kaluza-Klein theories Paritcle spectrum of Kaluza-Klein theories for ferminions Warped extra dimensions Conclusions).- PART V (Conclusions and Outlooks).- 18. Conclusions and outlooks.- Appendices.- References.- Index.
121351229
The relativistic bound and scattering states of the Manning-Rosen potential with an improved new approximate scheme to the centrifugal term
The approximately analytical bound and scattering state solutions of the arbitrary l-wave Klein-Gordon equation for the mixed Manning-Rosen potentials are carried out by an improved new approximation to the centrifugal term. The normalized analytical radial wave functions of the l-wave Klein-Gordon equation with the mixed Manning-Rosen potentials are presented and the corresponding energy equations for bound states and phase shifts for scattering states are derived. It is shown that the energy levels of the continuum states, reduce to the bound states of those at the poles of the scattering amplitude. Some useful figures are plotted to show the improved accuracy of our results and the special case for wave is studied briefly.
121592624
The Quantum Spectrum of the 2D Dirac Equation with a Coulomb Potential: Power Series Approach
The two-dimensional Dirac equation with a Coulomb potential has been studied by the power series expansion method. The eigenfunctions expressed by the confluent hypergeometric functions are analytically obtained. The eigenvalues and their fine structures of this system are also studied.
117217126
Quantum Spectrum of Some Anharmonic Central Potentials: Wave Functions Ansatz
Based on the ansatz to the wave functions, the quasi-exact solutions of the 2D Schrödinger equation with some anharmonic potentials are reviewed and analyzed if admitting restrictions on the parameters of the potential and the angular momentum m. These potentials are taken as the screened Coulomb potential V(r)=a/r+b/(r+λ), the singular one-fraction power one V(r)=ar−1/2+br−3/2 and the singular two-fraction one V(r)=ar2/3+br−2/3+cr−4/3. The latter one is found that the hidden symmetry exists if substituting r→−ir. It will reverse the signs of E and c of quantum system, leaving the remaining parameters invariant.
7872671
The Demographics of Extrasolar Planets Beyond the Snow Line with Ground-based Microlensing Surveys
In the currently-favored paradigm of planet formation, the location of the snow line in the protoplanetary disk plays a crucial role. Determining the demographics of planets beyond the snow line of stars of various masses is thus essential for testing this model. Microlensing is sensitive to planets that are generally inaccessible to other methods, and in particular is most sensitive to cool planets at or beyond the snow line, including very low-mass (i.e. terrestrial) planets. Hence, microlensing is uniquely suited and so essential for a comprehensive study of this region. Microlensing is also sensitive to planets orbiting low-mass stars, free-floating planets, planets in the Galactic bulge and disk, and even planets in external galaxies. These planets can also provide critical constraints on models of planet formation. Although microlensing searches have so far detected only a handful of planets, these have already changed our understanding of planet formation beyond the snow line. Next generation microlensing surveys, which would be sensitive to tens of "cold Earths" in this region, are well advanced in design conception and are starting initial practical implementation.
119233296
THE STATISTICS OF MULTI-PLANET SYSTEMS
We describe statistical methods for measuring the exoplanet multiplicity function (the fraction of host stars containing a given number of planets) and inclination distribution from transit and radial-velocity (RV) surveys. The analysis is based on the approximation of separability-that the distribution of planetary parameters in an n-planet system is the product of identical 1-planet distributions. We review the evidence that separability is a valid approximation for exoplanets and conclude that it captures many, but not all, of the known characteristics of multi-planet systems. We show how to relate the observable multiplicity function in surveys with similar host-star populations but different sensitivities. We also show how to correct for geometrical selection effects to derive the multiplicity function from transit surveys if the distribution of relative inclinations is known. Applying these tools to the Kepler transit survey and to RV surveys, we find that (1) the Kepler data alone do not constrain the mean inclination of multi-planet systems; even spherical distributions are allowed by the data but only if a small fraction of host stars contain large planet populations ({approx}> 30); (2) comparing the Kepler and RV surveys shows that the mean inclination of multi-planet systems is less than 5 Degree-Sign ;more » and (3) the multiplicity function of the Kepler planets is not well determined by the present data.« less
20954769
MOA-2009-BLG-387Lb: A massive planet orbiting an M dwarf
Aims. We report the discovery of a planet with a high planet-to-star mass ratio in the microlensing event MOA-2009-BLG-387, which exhibited pronounced deviations over a 12-day interval, one of the longest for any planetary event. The host is an M dwarf, with a mass in the range 0.07 M_⊙ < M_(host) < 0.49 M_⊙ at 90% confidence. The planet-star mass ratio q = 0.0132 ± 0.003 has been measured extremely well, so at the best-estimated host mass, the planet mass is m_p = 2.6 Jupiter masses for the median host mass, M = 0.19 M_⊙. Methods. The host mass is determined from two “higher order” microlensing parameters. One of these, the angular Einstein radius θ_E = 0.31 ± 0.03 mas has been accurately measured, but the other (the microlens parallax πE, which is due to the Earth’s orbital motion) is highly degenerate with the orbital motion of the planet. We statistically resolve the degeneracy between Earth and planet orbital effects by imposing priors from a Galactic model that specifies the positions and velocities of lenses and sources and a Kepler model of orbits. Results. The 90% confidence intervals for the distance, semi-major axis, and period of the planet are 3.5 kpc < D_L < 7.9 kpc, 1.1 AU < a < 2.7 AU, and 3.8 yr < P < 7.6 yr, respectively.
118852244
Exponential growth of eccentricity in secular theory
The Kozai mechanism for exponentially exciting eccentricity of a Keplerian orbit by a distant perturber is extended to a general perturbing potential. In particular, the case of an axisymmetric potential is solved analytically. The analysis is applied to orbits around an oblate central object with a distant perturber. If the equatorial plane of the central object is aligned with the orbit of the distant perturber (axisymmetric potential), a single instability zone, in which eccentricity grows exponentially, is found between two critical inclinations; if misaligned (non-axisymmetric potential), a rich set of critical inclinations separating stable and unstable zones is obtained (Vashkoviak 1974). The analysis is also applied to a general quadratic potential. Similarly, for non-axisymmetric cases, multiple stability and instability zones are obtained. Here eccentricity can reach very high values in the instability zones even when the potential's deviation from axisymmetry is small.
119217995
SUB-SATURN PLANET MOA-2008-BLG-310Lb: LIKELY TO BE IN THE GALACTIC BULGE*
We report the detection of sub-Saturn-mass planet MOA-2008-BLG-310Lb and argue that it is the strongest candidate yet for a bulge planet. Deviations from the single-lens fit are smoothed out by finite-source effects and therefore are not immediately apparent from the light curve. Nevertheless, we find that a model in which the primary has a planetary companion is favored over the single-lens model by Δχ2 ~ 880 for an additional 3 degrees of freedom. Detailed analysis yields a planet/star mass ratio q = (3.3 ± 0.3) × 10–4 and an angular separation between the planet and star within 10% of the angular Einstein radius. The small angular Einstein radius, θ E = 0.155 ± 0.011 mas, constrains the distance to the lens to be DL >6.0 kpc if it is a star (ML >0.08 M ). This is the only microlensing exoplanet host discovered so far that must be in the bulge if it is a star. By analyzing VLT NACO adaptive optics images taken near the baseline of the event, we detect additional blended light that is aligned to within 130 mas of the lensed source. This light is plausibly from the lens, but could also be due to a companion to the lens or source, or possibly an unassociated star. If the blended light is indeed due to the lens, we can estimate the mass of the lens, ML = 0.67 ± 0.14 M , planet mass m = 74 ± 17 M ⊕, and projected separation between the planet and host, 1.25 ± 0.10 AU, putting it right on the "snow line." If not, then the planet has lower mass, is closer to its host and is colder. To distinguish among these possibilities on reasonable timescales would require obtaining Hubble Space Telescope images almost immediately, before the source-lens relative motion of U=5 mas yr -1 causes them to separate substantially.