Stationary Schr Research Articles

Stationary Schr\xf6dinger equation in the semi-classical limit: numerical coupling of oscillatory and evanescent regions

This paper is concerned with a 1D Schrödinger scattering problem involving both oscillatory and evanescent regimes, separated by jump discontinuities in the potential function, to avoid “turning points”. We derive a non-overlapping domain decomposition method to split the original problem into sub-problems on these regions, both for the continuous and afterwards for the discrete problem. Further, a hybrid WKB-based numerical method is designed for its efficient and accurate solution in the semi-classical limit: a WKB-marching method for the oscillatory regions and a FEM with WKB-basis functions in the evanescent regions. We provide a complete error analysis of this hybrid method and illustrate our convergence results by numerical tests.

SYMMETRY OF COMPONENTS FOR RADIAL SOLUTIONS OF γ-LAPLACIAN SYSTEMS

Abstract. In this paper, we give several sufficient conditions ensuringthat any positive radial solution (u,v) of the following γ-Laplacian sys-tems in the whole space R n has the components symmetry property u ≡ vˆ−div(|∇u| γ−2 ∇u) = f(u,v) in R n ,−div(|∇v| γ−2 ∇v) = g(u,v) in R n .Here n > γ, γ > 1.Thus, the systems will be reduced to a single γ-Laplacian equation:−div(|∇u| γ−2 ∇u) = f(u) in R n .Our proofs are based on suitable comparation principle arguments, com-bined with properties of radial solutions. 1. IntroductionIn 2008, Li and Ma [10] studied the stationary Schr¨odinger system(1.1)ˆ−∆u = u p v q in R n ,−∆v = u q v p in R n ,and obtained a components symmetry result:Proposition 1.1. Assume n > γ, 1 ≤ p,q ≤ n+2n−2 and p + q = n−2 . Thenany (L n2−n2 (R n )) 2 -positive solution pair (u,v) to (1.1) is radial symmetric, andhence u ≡ v = a(b 2 +|x −x 0 | 2 ) (2−n)/2 with a,b > 0 and x 0 ∈ R n .The proof was achieved by the classification result in [4] and the methodof moving planes based on the conformal invariant property. Afterwards, Leiand Li ([8]) studied the asymptotic radial symmetry and decay estimates ofpositive integrable solutions of(1.2)ˆ−div(|∇u|

Fast magnetohydrodynamic oscillation of longitudinally inhomogeneous prominence threads: an analogue with quantum harmonic oscillator

Previous works indicate that the frequency ratio of second and first harmonics of kink oscillations has tendency towards 3 in the case of prominence threads. We aim to study the magnetohydrodynamic oscillations of longitudinally inhomogeneous prominence threads and to shed light on the problem of frequency ratio. Classical Sturm--Liouville problem is used for the threads with longitudinally inhomogeneous plasma density. We show that the spatial variation of total pressure perturbations along the thread is governed by the stationary Schr\"{o}dinger equation, where the longitudinal inhomogeneity of plasma density stands for the potential energy. Consequently, the equation has bounded solutions in terms of Hermite polynomials. Boundary conditions at the thread surface lead to transcendental dispersion equation with Bessel functions. Thin flux tube approximation of the dispersion equation shows that the frequency of kink waves is proportional to the expression \alpha(2n+1), where \alpha is the density inhomogeneity parameter and n is the longitudinal mode number. Consequently, the ratio of the frequencies of second and first harmonics tends to 3 in prominence threads. Numerical solution of the dispersion equation shows that the ratio only slightly decreases for thicker tubes in the case of smaller longitudinal inhomogeneity of external density, therefore the thin flux tube limit is a good approximation for prominence oscillations. However, stronger longitudinal inhomogeneity of external density may lead to the significant shift of frequency ratio for wider tubes and therefore the thin tube approximation may fail. The tendency of frequency ratio of second and first harmonics towards 3 in prominence threads is explained by the analogy of the oscillations with quantum harmonic oscillator, where the density inhomogeneity of the threads plays a role of potential energy.

Quantum-mechanical analogy and supersymmetry of electromagnetic wave modes in planar waveguides

We study an analogue of the equations describing TE and TM modes in a planar waveguide with an arbitrary continuous dependence of the electric permittivity and magnetic permeability on coordinates with the stationary Schr\"odinger equation. The effective potential energies involved in the Schr\"odinger equation for TE and TM modes are found. In general, the effective potential energies for TE and TM modes are different but in the limit of a weak dependence of the permittivity and permeability on coordinates they coincide. In the case when the product of a position-dependent permittivity and permeability is constant, it means that the refractive index is constant, we find that the TE and TM modes are described by the supersymmtric quantum mechanics.

Transmission of Correlated Gaussian Packets Through a Delta-Potential

We study the evolution of the most general initial Gaussian packet with nonzero correlation coefficient between the coordinate and momentum operators in the presence of a repulsive delta potential barrier, using the known exact propagator of the time-dependent Schr\"odinger equation. For the initial packet localized far enough from the barrier, we define the transmission coefficient as the probability of discovering the particle in the whole semi-axis on the other side of the barrier. It appears that the asymptotical transmission coefficient (calculated in the large time limit) depends on two dimensionless parameters: the normalized ratio of the potential strength to the initial mean value of momentum and the ratio of the initial momentum dispersion to the initial mean value of momentum. For small values of the second parameter the result is reduced to the well known formula for the transparency of the delta barrier, obtained in the plane wave approximation by solving the stationary Schr\"odinger equation. For big values of the second parameter, the transmission coefficient can be much bigger than that calculated in the plane wave approximation. For a fixed initial spread of the packet in the coordinate space, the initial correlation coefficient influences the transparency of the barrier only indirectly, through the increase of the initial momentum dispersion.

A sharper energy method for the localization of the support to some stationary Schrödinger equations with a singular nonlinearity

We prove the compactness of the support of the solution of some stationary Schr{\"o}dinger equations with a singular nonlinear order term. We present here a sharper version of some energy methods previously used in the literature and, in particular, by the authors.

State-dependent potentials in a nanofiber-based two-color trap for cold atoms

We analyze the ac Stark shift of a cesium atom interacting with far-off-resonance guided light fields in the nanofiber-based two-color optical dipole trap realized by Vetsch \textit{et al.} [Phys. Rev. Lett. \textbf{104}, 203603, (2010)]. Particular emphasis is given to the fictitious magnetic field produced by the vector polarizability of the atom in conjunction with the ellipticity of the polarization of the trapping fields. Taking into account the ac Stark shift, the atomic hyperfine interaction, and a magnetic interaction, we solve the stationary Schr\"odinger equation at a fixed point in space and find Zeeman-state-dependent trapping potentials. In analogy to the dynamics in magnetic traps, a local degeneracy of these state-dependent trapping potentials can cause spin flips and should thus be avoided. We show that this is possible using an external magnetic field. Depending on the direction of this external magnetic field, the resulting trapping configuration can still exhibit state-dependent displacement of the potential minima. In this case, we find nonzero Franck-Condon factors between motional states of the potentials for different hyperfine-structure levels and propose the possibility of microwave cooling in a nanofiber-based two-color trap.

Exactly solvable two-dimensional stationary Schrödinger operators obtained by the nonlocal Darboux transformation

The Fokker-Planck equation associated with the two - dimensional stationary Schr\"odinger equation has the conservation low form that yields a pair of potential equations. The special form of Darboux transformation of the potential equations system is considered. As the potential variable is a nonlocal variable for the Schr\"odinger equation that provides the nonlocal Darboux transformation for the Schr\"odinger equation. This nonlocal transformation is applied for obtaining of the exactly solvable two - dimensional stationary Schr\"odinger equations. The examples of exactly solvable two - dimensional stationary Schr\"odinger operators with smooth potentials decaying at infinity are obtained.

Quantum Corrections to the Dynamics of the Expanding Universe

The dynamics of the expanding universe is analyzed in terms of the quantum geometrodynamical model. It is shown that the equations of quantum theory in the form of the eigenvalues equation similar to the stationary Schr\"{o}dinger equation complemented by the equations of motion for the momentum operator and its time derivative in Heisenberg's form reduce to the Einstein equations with an additional source of the gravitational field of quantum nature. The spatially closed universe with cosmological constant, originally filled with a uniform scalar field and radiation, is considered as quantum cosmological system. The perfect fluid in the form of radiation defines the material reference frame. The properties of the averaged scalar field which acts like ordinary matter are investigated. After averaging over its quantum states, the free scalar field turns into the Weyssenhoff fluid characterized by the energy density, pressure, and spin of constituent particles. The cases when the contribution of the quantum effects into the gravitational interaction becomes significant on macroscopic scale are analyzed. It is demonstrated that, unless the whole, at least a part of such matter-energy constituents as dark matter and dark energy may have a quantum origin.

Molecular Siegert states in an electric field

The Siegert states of atoms and molecules in a static electric field are the solutions of the stationary Schr\"odinger equation satisfying the regularity and outgoing-wave boundary conditions. Recently, an efficient method for calculating Siegert states in the single-active-electron approximation based on the adiabatic expansion in parabolic coordinates was proposed [P. A Batishchev et al., Phys. Rev. A 82, 023416 (2010); O. I. Tolstikhin et al., Phys. Rev. A 84, 053423 (2011)]. So far, this method has been implemented only for axially symmetric potentials, which corresponds to atoms and linear molecules aligned along the field. In the present work, we extend its implementation to a general class of soft-core molecular potentials. This makes it possible to calculate the Siegert eigenvalue $E=\mathcal{E}\ensuremath{-}i\ensuremath{\Gamma}/2$ defining the energy $\mathcal{E}$ and ionization rate $\ensuremath{\Gamma}$ of the corresponding state as functions of the electric field for arbitrarily oriented polyatomic molecules. The method is illustrated by calculations for the $1s\ensuremath{\sigma}$ and $2p\ensuremath{\pi}$ states of H${}_{2}^{+}$. Comparison of the results with the predictions of perturbation theory for $\mathcal{E}$ and weak-field asymptotic theory for $\ensuremath{\Gamma}$ is discussed.

On the foundations of quantum physics

Some aspects of the interpretation of microscopic physics in terms of quantum theory are discussed. It is first emphasized that quantum theory is formulated in a Cartesian coordinate system; in other coordinates the result obtained with the help of the Hamiltonian formalism and commutation relations between ''canonically conjugated'' coordinate and momentum operators leads to a wrong version of quantum mechanics. In this connection the Feynman integral formalism is also discussed. In this formalism the measure is not well defined, and there is no idea how to distinguish between the true version of quantum mechanics and an incorrect one. In this respect, the Feynman approach consists of a mnemonic rule to generate perturbation series from an undefined zero-order term. The origin of time in the quantum framework is then analyzed in detail and illustrated by the example of atomic collisions. It is shown that the time-dependent Schr ¨ odinger equation for the closed three-body (two nuclei + electron) system has no physical meaning because in the high-impact energy limit it transforms into an equation with two independent time-like variables; time naturally appears in the stationary Schr ¨ odinger equation as a result of extraction of a classical subsystem (two nuclei) from a closed three-body system. Finally, following the well-known Einstein-Rosen-Podolsky experiment and Bell's inequality, we reiterate that the wave function must be interpreted as an actual field of information, in a form as elementary as the usual material particles and electromagnetic fields. In fact, experimental measurements transfer this quantum information field into the classical world, which is directly discernable. In my conclusion, the relation between physical reality and its mathematical formulation is discussed. 2012 Physics Essays Publication. (DOI: 10.4006/0836-1398-25.1.27) R ´

On Darboux's Approach to R-Separability of Variables

We discuss the problem of $R$-separability (separability of variables with a factor $R$) in the stationary Schr\"odinger equation on $n$-dimensional Riemann space. We follow the approach of Gaston Darboux who was the first to give the first general treatment of $R$-separability in PDE (Laplace equation on ${\mathbb E}^3$). According to Darboux $R$-separability amounts to two conditions: metric is isothermic (all its parametric surfaces are isothermic in the sense of both classical differential geometry and modern theory of solitons) and moreover when an isothermic metric is given their Lam\'e coefficients satisfy a single constraint which is either functional (when $R$ is harmonic) or differential (in the opposite case). These two conditions are generalized to $n$-dimensional case. In particular we define $n$-dimensional isothermic metrics and distinguish an important subclass of isothermic metrics which we call binary metrics. The approach is illustrated by two standard examples and two less standard examples. In all cases the approach offers alternative and much simplified proofs or derivations. We formulate a systematic procedure to isolate $R$-separable metrics. This procedure is implemented in the case of 3-dimensional Laplace equation. Finally we discuss the class of Dupin-cyclidic metrics which are non-regularly $R$-separable in the Laplace equation on ${\mathbb E}^3$.

Microwave realization of quasi-one-dimensional systems with correlated disorder

A microwave setup for mode-resolved transport measurement in quasi-one-dimensional (quasi-1D) structures is presented. We will demonstrate a technique for direct measurement of the Green's function of the system. With its help we will investigate quasi-1D structures with various types of disorder. We will focus on stratified structures, i.e., structures that are homogeneous perpendicular to the direction of wave propagation. In this case the interaction between different channels is absent, so wave propagation occurs individually in each open channel. We will apply analytical results developed in the theory of one-dimensional (1D) disordered models in order to explain main features of the quasi-1D transport. The main focus will be selective transport due to long-range correlations in the disorder. In our setup, we can intentionally introduce correlations by changing the positions of periodically spaced brass bars of finite thickness. Because of the equivalence of the stationary Schr\"odinger equation and the Helmholtz equation, the result can be directly applied to selective electron transport in nanowires, nanostripes, and superlattices.

Atomic Siegert states in an electric field: Transverse momentum distribution of the ionized electrons

The Siegert states of atoms in a static uniform electric field, defined as the solutions to the stationary Schr\"odinger equation satisfying the regularity and outgoing-wave boundary conditions, are discussed. An efficient method to calculate not only the complex energy eigenvalue, but also the eigenfunction for a general class of one-electron atomic potentials is introduced. An exact expression for the transverse momentum distribution of the ionized electrons in terms of the Siegert eigenfunction in the asymptotic region is derived. The method is illustrated by calculations of the energy, ionization width, and transverse momentum distribution as functions of the electric field for several lowest states of H, outer $p$ shells of Ne, Ar, Kr, and Xe, and the active electron in ${\mathrm{H}}^{\ensuremath{-}}$. We also discuss the ionization of Ar by the pulse of a unidirectional time-dependent electric field, which illustrates the role of the Siegert states in the recently developed adiabatic theory of ionization of atoms by intense laser pulses [O. I. Tolstikhin et al., Phys. Rev. A 81, 033415 (2010)].

Explicit time-propagation method to treat the dynamics of driven complex systems

We describe the efficient implementation of an explicit method to solve systems of stiff differential equations either on a grid or within a spectral approach. This method is based on an ansatz that approximates the solution. This ansatz depends on stiffness parameters that are shown to be related to the eigenfrequencies of the system. The accuracy and the performance of the method are tested in three different cases. First, we treat a highly stiff single differential equation, where explicit schemes converge rather slowly. Then, we solve the stationary Schr\"odinger equation associated to the quantum reflection of an ultracold atom by a surface. Finally, we consider the interaction of atomic hydrogen with a strong low-frequency laser pulse whose duration is of the order of 25 fs. We focus on the calculation of the above-threshold ionization electron spectrum, a problem which, under such realistic physical conditions, is computationally very demanding.

Multiquantum eigenstates of a linear chain of coupled qubits

We present a technique to identify exact analytic expressions for the multiquantum eigenstates of a linear chain of coupled qubits. A choice of Hilbert subspaces is described that allows an exact solution of the stationary Schr\"odinger equation without imposing periodic boundary conditions and without neglecting end effects, fully including the dipole-dipole nearest-neighbor interaction between the atoms. The treatment is valid for an arbitrary coherent excitation in the atomic system, any number of atoms, any size of the chain relative to the resonant wavelength and arbitrary initial conditions of the atomic system. The procedure we develop is general enough to be adopted for the study of excitation in an arbitrary array of atoms including spin chains and one-dimensional Bose-Einstein condensates.

Quantum-mechanical and semiclassical study of the collinear three-body Coulomb problem: Inelastic collisions below the three-body disintegration threshold

A quantum-mechanical (QM) and semiclassical (SC) study of inelastic collisions in collinear three-body Coulomb systems below the three-body disintegration threshold is presented. The QM results are obtained by solving the stationary Schr\"odinger equation in hyperspherical coordinates using the slow/smooth variable discretization method. After appropriate rescaling of the hyperspherical coordinates, an asymptotic parameter $0\ensuremath{\leqslant}h\ensuremath{\leqslant}1$ that depends only on the masses of particles and has the meaning of an effective Planck's constant for the motion in hyperradius emerges. The SC results are obtained in the leading order approximation of the asymptotic expansion in $h$. The main attention is paid to investigating how the SC and QM results converge as $h\ensuremath{\rightarrow}0$. It is shown that the overall agreement for a wide spectrum of systems and processes is surprisingly good even for $h\ensuremath{\sim}1$. However, because of interference effects the convergence is not monotonic, and the SC results may be grossly in error in the situations where a destructive interference occurs. The analysis of hidden crossings clarifies mechanisms of the nonadiabatic transitions. It is shown that if the oppositely charged particle is located between the two others, the nonadiabatic transitions occur near the top of the potential barrier via the well-known $T$ series of hidden crossings. If it is located on one end of the system, then there is no potential barrier for real values of the angular variable, but there still exists an extremum in the complex plane; the mechanism of nonadiabatic transitions in this case is called the complex $T$ series of hidden crossings.

Nonlinear quantization of a degenerate charged Bose gas in an external Coulomb trap

We consider a degenerate charged Bose-Coulomb gas populating several discrete stationary boson bound states that are located in a spherical-symmetrical central Coulombian potential. Each such state is defined, through appropriate boundary conditions and normalization, by a so-called ``nonlinear eigenstate'' that is actually a solution of the coupled (linear) stationary Schr\"odinger-like Gross-Pitaevskii differential equation and the (nonlinear) Poisson equation. The corresponding eigenvalues allow us to define the energies of these degenerate boson states, much like the Koopmans orbital energy in atomic physics. This theory applies surprisingly well (compared with the corresponding Hartree-Fock results) to spherical-symmetrical $s$ orbital states in atomic physics (i.e., bosonlike restricted orbital states where the additional spin degree of freedom is already integrated out). Finally the superposition of two such stationary nonlinear eigenstates is investigated and given a semiclassical physical significance similar to a Thomas-Fermi approach. The resulting concepts apply particularly well (namely within an average 1% error bar with respect to spectroscopic data) to the $1{s}^{2}\text{\penalty1000-\hskip0pt}2{s}^{2}$ orbital states of the $3\ensuremath{\leqslant}Z\ensuremath{\leqslant}9$ atomic subsystems.

Hyperspherical time-dependent method with semiclassical outgoing waves for double photoionization of helium

The hyperspherical time-dependent method with semiclassical outgoing waves for study of double photoionization of helium is presented. It is closely related to the hyperspherical R-matrix method with semiclassical outgoing waves [Phys. Rev. A 65, 032711 (2002)]: both split configuration space into two regions to solve the stationary inhomogeneous Schr\"odinger equation associated with the one-photon ionization problem, and both apply the same treatment to the outer region. However, the two methods differ radically in their treatments of the problem in the inner region: the most recent one applies a time-dependent approach for calculating the stationary wave function, while the previous one uses a R-matrix treatment. The excellent agreement observed between the triple differential cross sections obtained from these two basically different methods provides very strong support for both of them. Importantly, the very different numerical structures of both methods might make the most recent one a better candidate for investigating the near-threshold region.

Solution of the Holstein equation of radiation trapping by the geometrical quantization technique. III. Partial frequency redistribution with Doppler broadening

We introduce an analytical method to investigate radiation trapping problems with Doppler frequency redistribution. The problem is formulated within the framework of the Holstein-Biberman-Payne equation. We interpret the basic integro-differential trapping equation as a generalized wave equation for a four-dimensional (4D) classical system (an associated quasiparticle). We then construct its analytical solution by a semiclassical approach, called the geometrical quantization technique (GQT). Within the GQT, it is shown that the spatial and frequency variables can be separated and that the frequency part of the excited atom distribution function obeys a stationary Schr\"odinger equation for a perturbed oscillator. We demonstrate that there is a noticeable deviation of the actual spectral emission profile from the Doppler line in the region of small opacities. The problem of calculating the spatial mode structure and the effective radiation trapping factors is reduced to the evaluation of wave functions and quantized energy values of the quasiparticle confined in the vapor cell. We formulate the quantization rules and derive the phase factors, which allow us to obtain analytically the complete spectrum of the trapping factors in 1D geometries (layer, cylinder, sphere) and other (2D and 3D) geometries when the separation of space variables is possible. Finally, we outline a possible extension of our method to treat radiation trapping effects for more general experimental situations including, for instance, a system of cold atoms.