122777413
NEW TYPE SHIFT OPERATORS FOR THREE-DIMENSIONAL INFINITE WELL POTENTIAL
New type shift operators for three-dimensional infinite well potential are identified to connect those quantum systems with different radials R but with the same energy spectrum. It should be pointed out that these shift operators depend on all variables contained in wave functions. Thus they establish a novel relation between wave functions ψlm(r) and ψ(l±1)(m±1)(r).
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.
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.
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.
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 $E=\pm M$.
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.
118387861
Q in Other Solar Systems
A significant fraction of the hot Jupiters with final circularized orbital periods of less than 5 days are thought to form through the channel of high-eccentricity migration. Tidal dissipation at successive periastron passages removes orbital energy of the planet, which has the potential for changes in semi-major axis of a factor of ten to a thousand. In the equilibrium tide approximation we show that, in order for high-eccentricity migration to take place, the relative level of tidal dissipation in Jupiter analogues must be at least 10 times higher than the upper-limit attributed to the Jupiter-Io interaction. While this is not a severe problem for high-e migration, it contradicts the results of several previous calculations. We show that these calculations of high-e migration inadvertently over-estimated the strength of tidal dissipation by three to four orders of magnitude. These discrepancies were obscured by the use of various parameters, such as lag time \tau, tidal quality factor Q and viscous time t_V. We provide the values of these parameters required for the Jupiter-Io interaction, tidal circularization and high-e migration. Implications for tidal theory as well as models of the inflated radii of hot Jupiters are discussed. Though the tidal Q is not, in general, well-defined, we derive a formula for it during high-eccentricity migration where Q is approximately constant throughout evolution.
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.
118452553
The Effect of Differential Limb Magnification on Abundance Analysis of Microlensed Dwarf Stars
Finite source effects can be important in observations of gravitational microlensing of stars. Near caustic crossings, for example, some parts of the source star will be more highly magnified than other parts. The spectrum of the star is then no longer the same as when it is unmagnified, and measurements of the atmospheric parameters and abundances will be affected. The accuracy of abundances measured from spectra taken during microlensing events has become important recently because of the use of highly magnified dwarf stars to probe abundance ratios and the abundance distribution in the Galactic bulge. In this paper, we investigate the importance of finite source effects on spectra by using magnification profiles motivated by two events to synthesize spectra for dwarfs between 5000 K and 6200 K at solar metallicity. We adopt the usual techniques for analyzing the microlensed dwarfs, namely, spectroscopic determination of temperature, gravity, and microturbulent velocity, relying on equivalent widths. We find that ignoring the finite source effects for the more extreme case results in errors in T eff <45 K, in log g of <0.1 dex, and in ξ of <0.1 km s–1. In total, changes in equivalent widths lead to small changes in atmospheric parameters and changes in abundances of <0.06 dex, with changes in [Fe I/H] of <0.03 dex. For the case with a larger source-lens separation, the error in [Fe I/H] is <0.01 dex. This latter case represents the maximum effect seen in events whose light curves are consistent with a point-source lens, which includes the majority of microlensed bulge dwarfs published so far.
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.