Volume 51, Issue 12, December 2010
Index of content:
- Quantum Mechanics (General and Nonrelativistic)
51(2010); http://dx.doi.org/10.1063/1.3511330View Description Hide Description
We show that it is possible to obtain numerical solutions to quantum mechanical problems involving a fractional Laplacian, using a collocation approach based on little sinc functions, which discretizes the Schrödinger equation on a uniform grid. The different boundary conditions are naturally implemented using sets of functions with the appropriate behavior. Good convergence properties are observed. A comparison with results based on a Wentzel–Kramers–Brillouin analysis is performed.
51(2010); http://dx.doi.org/10.1063/1.3503775View Description Hide Description
We consider discrete spectra of bound states for nonrelativistic motion in attractive potentials . For these potentials the quasiclassical approximation for n → ∞ predicts quantized energy levels of a bounded spectrum varying as . We construct collective quantum states using the set of wavefunctions of the discrete spectrum assuming this asymptotic behavior. We give examples of states that are normalizable and satisfy the resolution of unity, using explicit positive functions. These are coherent states in the sense of Klauder and their completeness is achieved via exact solutions of Hausdorff moment problems, obtained by combining Laplace and Mellin transform methods. For σ in the range 0 < σ ⩽ 2/3 we present exact implementations of such states for the parametrization σ = 2(k − l)/(3k − l) with k and l positive integers satisfying .
51(2010); http://dx.doi.org/10.1063/1.3503472View Description Hide Description
Given an arbitrary Lagrangian function on and a choice of classical path, one can try to define Feynman's path integral supported near the classical path as a formal power series parameterized by “Feynman diagrams,” although these diagrams may diverge. We compute this expansion and show that it is (formally, if there are ultraviolet divergences) invariant under volume-preserving changes of coordinates. We prove that if the ultraviolet divergences cancel at each order, then our formal path integral satisfies a “Fubini theorem” expressing the standard composition law for the time evolution operator in quantum mechanics. Moreover, we show that when the Lagrangian is inhomogeneous quadratic in velocity such that its homogeneous-quadratic part is given by a matrix with constant determinant, then the divergences cancel at each order. Thus, by “cutting and pasting” and choosing volume-compatible local coordinates, our construction defines a Feynman-diagrammatic “formal path integral” for the nonrelativistic quantum mechanics of a charged particle moving in a Riemannian manifold with an external electromagnetic field.
51(2010); http://dx.doi.org/10.1063/1.3520507View Description Hide Description
We consider the hydrogenic atom in a space of the form , where may be a generalized manifold obeying certain properties. We separate the solution to the governing time-independent Schrödinger equation into a component over and a component over . Upon obtaining a solution to the relevant eigenvalue problems, we recover both the wave functions and energyspectrum for the hydrogenic atom over . We consider some specific examples of , including the fairly simple D-dimensional torus and the more complicated Kähler conifold in order to illustrate the method. In the examples considered, we see that the corrections to the standard energyspectrum for the hydrogen atom due to the addition of higher dimensions scale as a constant times , where L denotes the size of the additional dimensions. Thus, under the assumption of small compact extra dimensions, even the first energy corrections to the standard spectrum will be quite large.
51(2010); http://dx.doi.org/10.1063/1.3527268View Description Hide Description
Let n ⩾ 2 be a positive integer. To each irreducible representation σ of U(1), a U(1)-Kepler problem in dimension (2n − 1) is constructed and analyzed. This system is superintegrable and when n = 2 it is equivalent to a MICZ-Kepler problem. The dynamical symmetry group of this system is , and the Hilbert space of bound states is the unitary highest weight representation of with the minimal positive Gelfand-Kirillov dimension. Furthermore, it is shown that the correspondence between (the dual of σ) and is the theta-correspondence for dual pair .
51(2010); http://dx.doi.org/10.1063/1.3527069View Description Hide Description
The Riemann surface approach to bound and resonant states is extended to the case of a separable nonlocal potential that is constant on a certain domain of the inner region and vanishes in the rest of the domain. The approach consists in the construction of the Riemann surface of the S-matrix pole function k = k(g) over the g-plane, where g is the strength of the complex nonlocal potential. On the Riemann surface the pole function k = k(g) is single-valued and analytic. The branch points of the pole function k = k(g) and their k-plane images are determined and analyzed as a function of the position of the region of nonlocality. The Riemann surface of the S-matrix pole function is constructed. According to the Riemann surface approach to each bound or resonant state a sheet of the Riemann surface is associated. All the natural modes (bound and resonant states) of the system are identified and treated in a unified way. The nonlocal potential generates narrow resonant states that cannot be produced by a local potential.
51(2010); http://dx.doi.org/10.1063/1.3526963View Description Hide Description
The subject of the paper is Schrödinger operators on tree graphs which are radial, having the branching number at all the vertices at the distance from the root. We consider a family of coupling conditions at the vertices characterized by real parameters. We prove that if the graph is sparse so that there is a subsequence of growing to infinity, in the absence of the potential the absolutely continuous spectrum is empty for a large subset of these vertex couplings, but on the the other hand, there are cases when the spectrum of such a Schrödinger operator can be purely absolutely continuous.
Madelung representation of damped parametric quantum oscillator and exactly solvable Schrödinger–Burgers equations51(2010); http://dx.doi.org/10.1063/1.3524505View Description Hide Description
We construct a Madelung fluid model with time variable parameters as a dissipative quantum fluid and linearize it in terms of Schrödinger equation with time-dependent parameters. It allows us to find exact solutions of the nonlinear Madelung system in terms of solutions of the Schrödinger equation and the corresponding classical linear ordinary differential equation with variable frequency and damping. For the complex velocity field, the Madelung system takes the form of a nonlinear complex Schrödinger–Burgers equation, for which we obtain exact solutions using complex Cole–Hopf transformation. In particular, we give exact results for nonlinear Madelung systems related with Caldirola–Kanai-type dissipative harmonic oscillator. Collapse of the wave function in dissipative models and possible implications for the quantum cosmology are discussed.
- Quantum Information and Computation
51(2010); http://dx.doi.org/10.1063/1.3511335View Description Hide Description
We introduce quantum versions of the -divergence, provide a detailed analysis of their properties, and apply them in the investigation of mixing times of quantum Markov processes. An approach similar to the one presented in the literature for classical Markov chains is taken to bound the trace-distance from the steady state of a quantum processes. A strict spectral bound to the convergence rate can be given for time-discrete as well as for time-continuous quantum Markov processes. Furthermore, the contractive behavior of the -divergence under the action of a completely positive map is investigated and contrasted to the contraction of the trace norm. In this context we analyze different versions of quantum detailed balance and, finally, give a geometric conductance bound to the convergence rate for unital quantum Markov processes.
51(2010); http://dx.doi.org/10.1063/1.3521499View Description Hide Description
We consider the scenario in which Alice transmits private classical messages to Bob via a classical-quantum channel, part of whose output is intercepted by an eavesdropper Eve. We prove the existence of a universal coding scheme under which Alice's messages can be inferred correctly by Bob, and yet Eve learns nothing about them. The code is universal in the sense that it does not depend on specific knowledge of the channel. Prior knowledge of the probability distribution on the input alphabet of the channel, and bounds on the corresponding Holevo quantities of the output ensembles at Bob's and Eve's end suffice.
51(2010); http://dx.doi.org/10.1063/1.3519379View Description Hide Description
There is an intriguing relation between quantum informationtheory and super gravity, discovered by M. J. Duff and S. Ferrara. It relates entanglement measures for qubits to black hole entropy, which in a certain case involves the quartic invariant on the 56-dimensional representation of the Lie group E7. In this paper we recall the relatively straightforward manner in which three-qubits lead to E7, or at least to the Weyl group of E7. We also show how the Fano plane emerges in this context.
51(2010); http://dx.doi.org/10.1063/1.3512994View Description Hide Description
Nonlocality without inequality is an elegant argument introduced by Hardy for two qubit systems, and later generalised to nqubits, to establish contradiction of quantum theory with local realism. Interestingly, for n = 2 this argument is actually a corollary of Bell-type inequalities, viz., the CH–Hardy inequality involving Bell correlations, but for n greater than two it involves n-particle probabilities more general than Bell-correlations. In this paper, we first derive a chain of completely new local realistic inequalities involving joint probabilities for nqubits and then associated with each such inequality, we provide a new Hardy-type local reality constraint without inequalities. Quantum mechanical maximal violations of the chain of inequalities and of the associated constraints are also studied by deriving appropriate Cirel'son-type theorems. These results involving joint probabilities more general than Bell correlations are expected to provide a new systematic tool to investigate entanglement.
51(2010); http://dx.doi.org/10.1063/1.3523478View Description Hide Description
Alternative partial Boolean structures, implicit in the discussion of classical representability of sets of quantum mechanical predictions, are characterized, with definite general conclusions on the equivalence of the approaches going back to Bell and Kochen–Specker. An algebraic approach is presented, allowing for a discussion of partial classical extension, amounting to reduction of the “number of contexts,” classical representability arising as a special case. As a result, known techniques are generalized and some of the associated computational difficulties overcome. The implications on the discussion of Boole–Bell inequalities are indicated.
- Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)
51(2010); http://dx.doi.org/10.1063/1.3506403View Description Hide Description
We construct the time-evolution for the second-quantized Dirac equation subject to a smooth, compactly supported, time dependent electromagnetic potential and identify the degrees of freedom involved. Earlier works on this (e.g., Ruijsenaars) observed the Shale–Stinespring condition and showed that the one-particle time-evolution can be lifted to Fock space if and only if the external field had zero magnetic components. We scrutinize the idea, observed earlier by Fierz and Scharf, that the time-evolution can be implemented between time varying Fock spaces. In order to define these Fock spaces we are led to consider classes of reference vacua and polarizations. We show that this implementation is up to a phase independent of the chosen reference vacuum or polarization and that all induced transition probabilities are well-defined and unique.
51(2010); http://dx.doi.org/10.1063/1.3515844View Description Hide Description
This paper concerns standard supersymmetryalgebras in diverse dimensions, involving bosonic translational generators and fermionic supersymmetry generators. A cohomology related to these supersymmetryalgebras, termed supersymmetryalgebra cohomology, and corresponding “primitive elements” are defined by means of a BRST (Becchi-Rouet-Stora-Tyutin)-type coboundary operator. A method to systematically compute this cohomology is outlined and illustrated by simple examples.
51(2010); http://dx.doi.org/10.1063/1.3503773View Description Hide Description
We investigate the geometric interpretation of quantized Nambu–Poisson structures in terms of noncommutative geometries. We describe an extension of the usual axioms of quantization in which classical Nambu–Poisson structures are translated to n-Lie algebras at quantum level. We demonstrate that this generalized procedure matches an extension of Berezin–Toeplitz quantization yielding quantized spheres, hyperboloids, and superspheres. The extended Berezin quantization of spheres is closely related to a deformation quantization of n-Lie algebras as well as the approach based on harmonic analysis. We find an interpretation of Nambu–Heisenberg n-Lie algebras in terms of foliations of by fuzzy spheres, fuzzy hyperboloids, and noncommutative hyperplanes. Some applications to the quantum geometry of branes in M-theory are also briefly discussed.
51(2010); http://dx.doi.org/10.1063/1.3526964View Description Hide Description
We follow and modify the Feshbach–Villars formalism by separating the Klein–Gordon equation into two coupled time-dependent Schrödinger equations for particle and antiparticlewave function components with positive probability densities. We find that the equation of motion for the probability densities is in the form of relativistic hydrodynamics where various forces have their classical counterparts, with the additional element of the quantum stress tensor that depends on the derivatives of the amplitude of the wave function. We derive the equation of motion for the Wigner function and we find that its approximate classical weak-field limit coincides with the equation of motion for the distribution function in the collisionless kinetic theory.
51(2010); http://dx.doi.org/10.1063/1.3525805View Description Hide Description
The Nicole model is a conformal field theory in a three-dimensional space. It has topological soliton solutions classified by the integer-valued Hopf charge, and all currently known solitons are axially symmetric. A volume-preserving flow is used to construct soliton solutions numerically for all Hopf charges from 1 to 8. It is found that the known axially symmetric solutions are unstable for Hopf charges greater than 2 and new lower energy solutions are obtained that include knots and links. A comparison with the Skyrme–Faddeev model suggests many universal features, though there are some differences in the link types obtained in the two theories.
51(2010); http://dx.doi.org/10.1063/1.3520529View Description Hide Description
In Bohmian mechanics, a version of quantum mechanics that ascribes world lines to electrons, we can meaningfully ask about an electron's instantaneous speed relative to a given inertial frame. Interestingly, according to the relativistic version of Bohmian mechanics using the Dirac equation, a massive particle's speed is less than or equal to the speed of light, but not necessarily less. That is, there are situations in which the particle actually reaches the speed of light—a very nonclassical behavior. That leads us to the question of whether such situations can be arranged experimentally. We prove a theorem,Theorem 5, implying that for generic initial wave functions the probability that the particle ever reaches the speed of light, even if at only one point in time, is zero. We conclude that the answer to the question is no. Since a trajectory reaches the speed of light whenever the quantum probability current is a lightlike 4-vector, our analysis concerns the current vector field of a generic wave function and may thus be of interest also independently of Bohmian mechanics. The fact that the current is never spacelike has been used to argue against the possibility of faster-than-light tunneling through a barrier, a somewhat similar question. Theorem 5, as well as a more general version provided by Theorem 6, are also interesting in their own right. They concern a certain property of a function that is crucial to the question of reaching the speed of light, namely being transverse to a certain submanifold of along a given compact subset of space-time. While it follows from the known transversality theorem of differential topology that this property is generic among smooth functions , Theorem 5 asserts that it is also generic among smooth solutions of the Dirac equation.
51(2010); http://dx.doi.org/10.1063/1.3521553View Description Hide Description
A classic theorem in the theory of connections on principal fiber bundles states that the evaluation of all holonomy functions gives enough information to characterize the bundle structure (among those sharing the same structure group and base manifold) and the connection up to a bundle equivalence map. This result and other important properties of holonomy functions have encouraged their use as the primary ingredient for the construction of families of quantum gauge theories. However, in these applications often the set of holonomy functions used is a discrete proper subset of the set of holonomy functions needed for the characterization theorem to hold. We show that the evaluation of a discrete set of holonomy functions does not characterize the bundle and does not constrain the connection modulo gauge appropriately. We exhibit a discrete set of functions of the connection and prove that in the abelian case their evaluation characterizes the bundle structure (up to equivalence), and constrains the connection modulo gauge up to “local details” ignored when working at a given scale. The main ingredient is the Lie algebra valued curvature function defined below. It covers the holonomy function in the sense that .