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.
121783508
Quantum states of a sextic potential: hidden symmetry and quantum monodromy
A known terminating polynomial ansatz for a finite sub-set of states of the sextic central potential V(a,b,c;r) = ar2 + br4 + cr6, c>0, subject to integer related parameter constraints, is confirmed to apply in two as well as three dimensions. A hidden symmetry relating the eigenstates of V(a,b,c;r) to those of V(a,-b,c;r) is used to establish the boundary between the ansatz sub-set and the infinite set of higher states. Connections between the radial wavefunctions Rn(a,b,c;r) and Rn'(a,-b,c;r) also provide semiclassical insight into the origin of the parameter constraint. Finally the ansatz eigenvalue dispositions are related to the phenomenon of quantum monodromy.
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.
8332580
The Exact Solution to the Schr\"{o}dinger Equation with the Octic Potential
The Schr\"{o}dinger equation with the central potential is first studied in the arbitrary dimensional spaces and obtained an analogy of the two-dimensional Schr\"{o}dinger equation for the radial wave function through a simple transformation. As an example, applying an ${\it ansatz}$ to the eigenfunctions, we then arrive at an exact closed form solution to the modified two-dimensional Schr\"{o}dinger equation with the octic potential, $V(r)=ar^2-br^4+cr^6-dr^4+er^{10}$.
119543557
LEVINSON'S THEOREM FOR THE KLEIN-GORDON EQUATION IN TWO DIMENSIONS
The two-dimensional Levinson theorem for the Klein-Gordon equation with a cylindrically symmetric potential $V(r)$ is established. It is shown that $N_{m}\pi=\pi (n_{m}^{+}-n_{m}^{-})= [\delta_{m}(M)+\beta_{1}]-[\delta_{m}(-M)+\beta_{2}]$, where $N_{m}$ denotes the difference between the number of bound states of the particle $n_{m}^{+}$ and the ones of antiparticle $n_{m}^{-}$ with a fixed angular momentum $m$, and the $\delta_{m}$ is named phase shifts. The constants $\beta_{1}$ and $\beta_{2}$ are introduced to symbol the critical cases where the half bound states occur at $E=\pm M$.
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.
118732424
Exploring a Stream of Highly-Eccentric Binaries with Kepler
With 16-month Kepler data, 14 long-period (40 d - 265 d) eclipsing binaries on highly eccentric orbits (minimum e between 0.5 and 0.85) are recognized from their closely separated primary and secondary eclipses (\Delta t_I,II = 3 d - 10 d). These systems confirm the existence of a previously hinted binary population situated near a constant angular momentum track at P(1-e^2)^(3/2) ~ 15 d, close to the tidal circularization period P_circ. They may be presently migrating due to tidal dissipation and form a steady-state stream (~1% of stars) feeding the close-binary population (few percent of stars). If so, future Kepler data releases will reveal a growing number (dozens) of systems at longer periods, following dN/dlgP \propto P^(1/3) with increasing eccentricities reaching e -> 0.98 for P -> 1000d. Radial-velocity follow up of long-period eclipsing binaries with no secondary eclipses could offer a significantly larger sample. Orders of magnitude more (hundreds) may reveal their presence from periodic "eccentricity pulses", such as tidal ellipsoidal variations, near pericenter passages. Several new few-day-long eccentricity-pulse candidates with long period (P = 25 d - 80 d) are reported.
18317896
Long-term cycling of Kozai-Lidov cycles: extreme eccentricities and inclinations excited by a distant eccentric perturber.
The very long-term evolution of the hierarchical restricted three-body problem is calculated analytically for high inclinations. The Kozai-Lidov Cycles (KLCs) slowly evolve due to the octupole term in the perturber's potential and exhibit striking features, including extremely high eccentricities and the generation of retrograde orbits with respect to the perturber. These features were found in recent numerical experiments of the nonrestricted three-body problem and were attributed inaccurately to the comparable and low masses of the two orbiting companions. Our calculation is done by averaging for the first time the double averaged secular equations of motion over the KLCs and finding a new constant of the motion. These very long-term effects are likely to be important in various astrophysical systems thought to involve KLCs, such as hot Jupiters, irregular moons of planets, and many others.
18017754
The absolute magnitude of RRC variables from statistical parallax
We present the first definitive measurement of the absolute magnitude of RR Lyrae c-type variable stars (RRc) determined purely from statistical parallax. We use a sample of 242 RRc variables selected from the All Sky Automated Survey for which high-quality light curves, photometry, and proper motions are available. We obtain high-resolution echelle spectra for these objects to determine radial velocities and abundances as part of the Carnegie RR Lyrae Survey. We find that M V, RRc = 0.59 ? 0.10 at a mean metallicity of [Fe/H] = ?1.59. This is to be compared with previous estimates for RRab stars (M V, RRab = 0.76 ? 0.12) and the only direct measurement of an RRc absolute magnitude (RZ Cephei, M V, RRc = 0.27 ? 0.17). We find the bulk velocity of the halo relative to the Sun to be (W ?, W ?, Wz ) = (12.0, ?209.9, 3.0) km s?1 in the radial, rotational, and vertical directions with dispersions . For the disk, we find (W ?, W ?, Wz ) = (13.0, ?42.0, ?27.3) km s?1 relative to the Sun with dispersions . Finally, as a byproduct of our statistical framework, we are able to demonstrate that UCAC2 proper-motion errors are significantly overestimated as verified by UCAC4.
119219662
The rate of WD-WD head-on collisions may be as high as the SNe Ia rate
We show that a White Dwarf-White Dwarf (WD-WD) binary with semi-major axis a=1-300 AU, which is orbited by a stellar mass outer perturber with a moderate pericenter r_{p, out} \sim 3-10 x a, has a few percent chance of experiencing a head-on collision within ~5 Gyr. Such a perturber is sufficiently distant to allow the triple system to remain intact for millions of orbits while efficiently exchanging angular momentum with the WD-WD binary. In ~ 5% of the initial orientations, the inner orbit efficiently scans the (equal energy) phase space in the region of zero angular momentum. In these systems, the binary experiences increasingly closer, stochastic, pericenter approaches r_p ~ a/2N with the increasing number (N) of orbits elapsed. Within N~10^5(a/30AU) orbits, a collision is likely to occur. This is shown by performing \simten thousand 3-body integrations and is explained by simple analytic arguments. The collisions are conservatively restricted to "clean" collisions in which all passages prior to the collision are greater than 4R_WD=4x10^9cm. In particular, within the last single orbit, the pericenter changes from r_p>4R_WD to a collision value of r_p 3000 km/s and the collision is likely to detonate the WDs leading to a type Ia SNe. If a significant fraction of WDs reside in such triples, the rate of such collisions is as high as the SNe Ia rate, and it is possible that some or all type Ia SNe occur in this way. Such SNe have a unique gravitational wave signature, which will allow a decisive identification in the future.
118316787
An exact integral relation between the Ni56 mass and the bolometric light curve of a type Ia supernova
An exact relation between the Ni56 mass and the bolometric light curve of a type Ia supernova can be derived as follows, using the following excellent approximations: 1. the emission is powered solely by Ni56-> Co56 ->Fe56; 2. each mass element propagates at a non-relativistic velocity which is constant in time (free coasting); and 3. the internal energy is dominated by radiation. Under these approximations, the energy E(t) carried by radiation in the ejecta satisfies: dE/dt=-E(t)/t-L(t)+Q(t), where Q(t) is the deposition of energy by the decay which is precisely known and L(t) is the bolometric luminosity. By multiplying this equation by time and integrating over time we find: E(t)*t=\int_0^t Q(t')t'dt' -\int_0^t L(t')t'dt'. At late time, t>> t_peak, the energy inside the ejecta decreases rapidly due to its escape, and thus we have \int_0^t Q(t')t'dt'=\int_0^t L(t')t'dt'. This relation is correct regardless of the opacities, density distribution or Ni56 deposition distribution in the ejecta and is very different from "Arnett's rule", L_peak ~ Q(t_peak). By comparing \int_0^t Q(t')t'dt' with \int_0^t L(t')t'dt' at t~40 day after the explosion, the mass of Ni56 can be found directly from UV, optical and infrared observations with modest corrections due to the unobserved gamma-rays and due to the small residual energy in the ejecta, E(t)*t>0.