Volume 52, Issue 4, April 2011
Index of content:
- Quantum Mechanics (General and Nonrelativistic)
52(2011); http://dx.doi.org/10.1063/1.3570817View Description Hide Description
Conventional nondegenerate perturbation theory for some nth state starts with the corresponding unperturbed state. The present formulation yields recursively perturbation expansions for any bound state using the sole information of the unperturbed ground state. Logarithmic perturbation theory is exploited along with supersymmetric quantum mechanics to achieve this end. As the method involves ground-state perturbations of a series of supersymmetric Hamiltonians, concern about nodal shifts of targeted excited states arises only at the ultimate step, thus, minimizing considerably the labor of clumsy computations involved in dealing with excited states.
52(2011); http://dx.doi.org/10.1063/1.3566359View Description Hide Description
We derive the spectra of D = 2, SU(3) supersymmetric Yang–Mills quantum mechanics in all fermionic sectors. Moreover, we provide exact expressions for the corresponding eigenvectors in the sectors with none and one fermionic quantum. We also generalize our results obtained in a cut Fock space to the infinite cut-off limit.
52(2011); http://dx.doi.org/10.1063/1.3571969View Description Hide Description
We study the Green's function of the time-dependent 3D space-fractional Schrödinger equation for the scattering problem in the fractional quantum mechanics. The Green's function is expressed in terms of Fox's H-function and in a computable series form. We get the asymptotic formula of the Green's function, and apply it to obtain the approximate wave function for the fractional quantum scattering problem.
Riccati equation and the problem of decoherence II: Symmetry and the solution of the Riccati equation52(2011); http://dx.doi.org/10.1063/1.3574889View Description Hide Description
In this paper we revisit the problem of decoherence by applying the block operator matrices analysis. The Riccati algebraic equation associated with the Hamiltonian describing the process of decoherence is studied. We prove that if the environment responsible for decoherence is invariant with respect to the antilinear transformation then the antilinear operator solves the Riccati equation in question. We also argue that this solution leads to neither a linear nor an antilinear operator similarity matrix. Therefore, we cannot use the standard procedure for solving a linear differential equation (e.g., Schrödinger equation). Furthermore, the explicit solution of the Riccati equation is found for the case where the environmental operators commute with each other. We discuss the connection between our results and the standard description of decoherence (one that uses the Kraus representation). We show that the reduced dynamics we obtain does not have the Kraus representation if the initial correlations between the system and its environment are present. However, for any initial state of the system (even when the correlations occur) reduced dynamics can be written in a manageable way.
52(2011); http://dx.doi.org/10.1063/1.3576189View Description Hide Description
We study the time-space fractional Schrödinger equation with a nonlocal potential. By the method of Fourier transform and Laplace transform, the Green function, and hence the wave function, is expressed in terms of H-functions. Graphical analysis demonstrates that the influence of both the space-fractal parameter α and the nonlocal parameter ν on the fractional quantum system is strong. Indeed, the nonlocal potential may act similar to a fractional spatial derivative as well as fractional time derivative.
52(2011); http://dx.doi.org/10.1063/1.3574886View Description Hide Description
For each integer n ⩾ 1, we demonstrate that a (2n + 1)-dimensional generalized MICZ-Kepler problem has a Spin(2, 2n + 2) dynamical symmetry which extends the manifest Spin(2n + 1) symmetry. The Hilbert space of bound states is shown to form a unitary highest weight Spin(2, 2n + 2)-module with the minimal positive Gelfand–Kirillov dimension. As a byproduct, we obtain a simple geometric realization for such a unitary highest weight Spin(2, 2n + 2)-module.
52(2011); http://dx.doi.org/10.1063/1.3577961View Description Hide Description
We find both the electromagnetic TE and TM transverse eigenmodes of an infinite cylindrical sample of two-level atoms. The TM modes are similar to those obtained in the “scalar photon” theory, while the TE series is new. The TE series possesses “anomalous modes,” which are absent in the TM series. We find the metric for the scalar product for the eigenfunctions in both series and numerically show the completeness of both these series. Using Fourier-like expansions of the initial excitation state of the atoms in these eigenfunctions, we are able to compute to arbitrary accuracy the polarization of the atomic state at an arbitrary time. We compare these accurate results with those obtained from the eikonal–slowly varying envelope approximation in space and find remarkable agreement in the results for a system which is initially weakly excited.
Exact solution for a noncentral electric dipole ring-shaped potential in the tridiagonal representation52(2011); http://dx.doi.org/10.1063/1.3578348View Description Hide Description
The Schrödinger equation with noncentral electric dipole ring-shaped potential is investigated by working in a complete square integrable basis that supports an infinite tridiagonal matrix representation of the wave operator. The three-term recursion relations for the expansion coefficients of both the angular and radial wavefunctions are presented. The discrete spectrum for the bound states is obtained by the diagonalization of the radial recursion relation. Some potential applications of this system in different fields are discussed.
52(2011); http://dx.doi.org/10.1063/1.3580273View Description Hide Description
We consider the fine grading of induced by tensor product of generalized Pauli matrices in the paper. Regard as the inner automorphism group of . Based on the classification of maximal abelian subgroups of consisting of diagonalizable automorphisms by Havlicek et al., we prove that any such subgroup K of is a symplectic abelian group and its Weyl group, which describes the symmetry of the fine grading induced by the action of K, is just the isometry group of the symplectic abelian group K. For a finite symplectic abelian group, it is also proved that its isometry group is always generated by the transvections contained in it.
- Quantum Information and Computation
52(2011); http://dx.doi.org/10.1063/1.3575568View Description Hide Description
We study the asymptotic position distribution of general quantum walks on a lattice, including walks with a random coin, which is chosen from step to step by a general Markov chain. In the unitary (i.e., nonrandom) case, we allow any unitary operator which commutes with translations and couples only sites at a finite distance from each other. For example, a single step of the walk could be composed of any finite succession of different shift and coin operations in the usual sense, with any lattice dimension and coin dimension. We find ballistic scaling and establish a direct method for computing the asymptotic distribution of position divided by time, namely as the distribution of the discrete time analog of the group velocity. In the random case, we let a Markov chain (control process) pick in each step one of finitely many unitary walks, in the sense described above. In ballistic order, we find a nonrandom drift which depends only on the mean of the control process and not on the initial state. In diffusive scaling, the limiting distribution is asymptotically Gaussian, with a covariance matrix (diffusion matrix) depending on momentum. The diffusion matrix depends not only on the mean but also on the transition rates of the control process. In the nonrandom limit, i.e., when the coins chosen are all very close or the transition rates of the control process are small, leading to long intervals of ballistic evolution, the diffusion matrix diverges. Our method is based on spatial Fourier transforms, and the first and second order perturbation theory of the eigenvalue 1 of the transition operator for each value of the momentum.
52(2011); http://dx.doi.org/10.1063/1.3581879View Description Hide Description
We obtain explicit expressions for the Choi–Jamiolkowski (CJ) forms and operators defining a general bosonic Gaussian channel. The four principal cases are considered in Sec. III; in Sec. IV, we give a decomposition of Gaussian CJ form into product of these four principal types and provide a necessary and sufficient condition for the existence of the bounded CJ operator.
52(2011); http://dx.doi.org/10.1063/1.3578015View Description Hide Description
We show that the linear group of automorphism of Hermitian matrices which preserves the set of separable states is generated by natural automorphisms: change of an orthonormal basis in each tensor factor, partial transpose in each tensor factor, and interchanging two tensor factors of the same dimension. We apply our results to the preservers of the product numerical range.
- Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)
52(2011); http://dx.doi.org/10.1063/1.3579983View Description Hide Description
There exists a relation between the Klein–Gordon and the Dirac equations with scalar and vector potentials of equal magnitude and the Schrödinger equation. We obtain the relativistic energy spectrum for the four relativistic quantum Smorodinsky–Winternitz systems from their quasi-Hamiltonian and the quadratic algebras studied by Daskaloyannis in the nonrelativistic context. We also apply the quadratic algebra approach directly to the initial Dirac equation for these four systems and show that the quadratic algebras obtained are the same than those obtained from the quasi-Hamiltonians. We point out how results obtained in context of quantum superintegrable systems and their polynomialalgebras can be applied to the quantum relativistic case.
- General Relativity and Gravitation
52(2011); http://dx.doi.org/10.1063/1.3567157View Description Hide Description
We consider a spatially homogeneous and isotropic cosmological model where Dirac spinors are coupled to classical gravity. For the Dirac spinors we choose a Hartree–Fock ansatz where all one-particle wave functions are coherent and have the same momentum. If the scale function is large, the universe behaves like the classical Friedmann dust solution. If however the scale function is small, quantum effects lead to oscillations of the energy-momentum tensor. It is shown numerically and proven analytically that these quantum oscillations can prevent the formation of a big bang or big crunch singularity. The energy conditions are analyzed. We prove the existence of time-periodic solutions which go through an infinite number of expansion and contraction cycles.
52(2011); http://dx.doi.org/10.1063/1.3567409View Description Hide Description
In this paper, we consider the Einstein–Gauss–Bonnet gravity in the presence of power Maxwell invariant field and present a class of rotating magnetic solutions. These solutions are nonsingular and horizonless, and satisfy the so-called flare-out condition at r = r + and may be interpreted as traversable wormhole near r = r +. In order to have a vanishing electromagnetic field at spatial infinity, we restrict the nonlinearity parameter to s > 1/2. Investigation of the energy conditions shows that these solutions satisfy the null, week, and strong energy conditions simultaneously for s > 1/2, which means that there is no exotic matter near the throat. We also calculate the conserved quantities of the wormhole such as mass, angular momentum, and electric charge density, and show that the electric charge depends on the rotation parameters and the static wormhole does not have a net electric charge density. In addition, we show that for s = (n + 1)/4, the energy–momentum tensor is traceless and the solutions are conformally invariant, in which the expression of the Maxwell field does not depend on the dimensions and its value coincides with the Reissner–Nordström solution in four dimensions. Finally, we produce higher dimensional BTZ-like wormhole solutions for s = n/2, in which in this case the electromagnetic field F ψr ∝r −1.
52(2011); http://dx.doi.org/10.1063/1.3579486View Description Hide Description
A simple algebraic global isometric embedding is presented for the nonrotating Bañados, Teitelboim, and Zanelli black hole and its counterpart of Euclidean signature. The image of the embedding, in Minkowski space of two extra dimensions, is the intersection of two quadric hypersurfaces. Furthermore an embedding into AdS 4 or H 4 is also obtained, showing that the spacetime is of embedding class one with respect to maximally symmetric space of negative curvature. The rotating solution of Euclidean signature is also shown to admit a quadratic algebraic embedding, but seemingly requires more than two extra dimensions.
- Dynamical Systems
52(2011); http://dx.doi.org/10.1063/1.3567957View Description Hide Description
We establish long time soliton asymptotics for the nonlinear system of Maxwell equations coupled to a charged particle. The coupled system has a six-dimensional manifold of soliton solutions. We show that in the long time approximation, any solution, with an initial state close to the solitary manifold, is a sum of a soliton and a dispersive wave which is a solution of the free Maxwell equations. It is assumed that the charge density satisfies the Wiener condition. The proof further develops the general strategy based on the symplectic projection in Hilbert space onto the solitary manifold, modulation equations for the parameters of the projection, and decay of the transversal component.
52(2011); http://dx.doi.org/10.1063/1.3573598View Description Hide Description
This paper is devoted to investigate a class of stochastic Volterra–Levin equations with Poisson jumps. To the best of the authors’ knowledge, till now, the stability problem for this class of new systems has not yet been solved since Poisson jumps are considered. The main objective of this paper is to fill the gap. By using the fixed point theory, we first study the existence and uniqueness of the solution as well as the pth moment exponential stability for the considered system. Then based on the well known Borel–Cantelli lemma, we prove that the solution is almost surely pth moment exponentially stable. Our results improve and generalize those given in the previous literature. Finally, two simple examples are provided to illustrate the effectiveness of the obtained results.
52(2011); http://dx.doi.org/10.1063/1.3574630View Description Hide Description
The stochastic 2D Navier–Stokes equations with finite memory are studied. For the differentiable memory function, the almost sure exponential stability of the weak solution is shown by employing a non-negative semimartingale convergence argument. For the nondifferentiable memory function, the exponential stability in mean square for the weak solution is proved by using the differentiability property of the moment function.
- Classical Mechanics and Classical Fields
52(2011); http://dx.doi.org/10.1063/1.3562929View Description Hide Description
The notion of a 2-point susceptibility kernel used to describe linear electromagnetic responses of dispersive continuous media in nonrelativistic phenomena is generalized to accommodate the constraints required of a causal formulation in spacetimes with background gravitational fields. In particular the concepts of spatial material inhomogeneity and temporal nonstationarity are formulated within a fully covariant spacetime framework. This framework is illustrated by recasting the Maxwell–Vlasovequations for a collisionless plasma in a form that exposes a 2-point electromagnetic susceptibility kernel in spacetime. This permits the establishment of a perturbative scheme for nonstationary inhomogeneous plasma configurations. Explicit formulae for the perturbed kernel are derived in both the presence and absence of gravitation using the general solution to the relativistic equations of motion of the plasma constituents. In the absence of gravitation this permits an analysis of collisionless damping in terms of a system of integral equations that reduce to standard Landau damping of Langmuir modes when the perturbation refers to a homogeneous stationary plasma configuration. It is concluded that constitutive modeling in terms of a 2-point susceptibility kernel in a covariant spacetime framework offers a natural extension of standard nonrelativistic descriptions of simple media and that its use for describing linear responses of more general dispersive media has wide applicability in relativistic plasma modeling.