Bound State Wave Functions Research Articles

A Schrödinger Potential Conditionally Integrable in Terms of the Hermite Functions

We introduce a biconfluent Heun potential well for the one-dimensional stationary Schrodinger equation which is composed of a confining fraction-power term and a repulsive centrifugal-barrier core. This is a conditionally integrable potential in that the strength of the centrifugal barrier is fixed to a constant. The potential supports a countable infinite number of bound states. We present the general solution of the Schrodinger equation, deduce the exact equation for the energy spectrumand derive a highly accurate approximation for energy levels. The bound state wave functions are written as irreducible linear combinations with constant coefficients of two Hermite functions of a scaled and shifted argument.

Bound states of Newton’s equivalent finite square well

In this paper, a one-parameter family of Newton’s equivalent Hamiltonians (NEH) for finite square well potential is analyzed in order to obtain bound state energy spectrum and wave functions. For a generic potential, each of the NEH is classically equivalent to one another and to the standard Hamiltonian yielding Newton’s equations. Quantum mechanically, however, they are expected to be different from each other. The Schrödinger’s equation coming from each NEH with finite square well potential is an infinite order differential equation. The matching conditions, therefore, demand the wave functions to be infinitely differentiable at the well boundaries. To handle this, we provide a way to consistently truncate these conditions. It turns out as expected that bound state energy spectrum and wave functions are dependent on the parameter [Formula: see text] which is used to characterize different NEH. As [Formula: see text], the energy spectrum coincides with that from the standard quantum finite square well.

Partonic quasidistributions in two-dimensional QCD

As a sequel of our preceding work [1] we carry out a comprehensive comparative study between the quasi parton distribution functions (PDFs), distribution amplitudes (DAs) and their light-cone counterparts for various flavor-neutral mesons, in the context of the 't Hooft model, that is, the two-dimensional QCD in the large $N$ limit. In contrast to the original derivation via diagrammatic techniques exemplified by Dyson-Schwinger and Bethe-Salpeter equations, here we employ the Hamiltonian operator approach to reconstruct the celebrated 't Hooft equation in light-front quantization, and Bars-Green equations in equal-time quantization. The novelty of our derivation is to employ the soft momentum cutoff as the IR regulator. As a virtue of this operator approach, the functional form of the quasi distributions can be transparently built out of the Bars-Green wave functions and the Bogoliubov angle with the aid of bosonization technique. Equipped with various bound-state wave functions numerically inferred in [1], we then investigate how rapidly the quasi distributions approach their light-cone counterparts with the increasing meson momentum. We observe that, light mesons' quasi distributions approach the light-cone distributions in a slower pace than the heavy quarkonia. Curiously, lattice simulations of quasi distributions in four-dimensional QCD also discover this feature. Furthermore, we also compute the partonic light-cone PDF and quasi-PDF to one-loop order in perturbation theory, again employing the momentum cutoff as the IR regulator. We explicitly verify one of the backbones underlying the large momentum effective field theory (LaMET), namely, both quasi-PDFs and light-cone PDFs in ${\rm QCD}_2$ indeed possess the same IR behavior at leading order in $1/P^z$

Neutron–proton scattering and singular potentials

We consider a Bargmann-type rational parametrization of the nucleon scattering phase shifts. Applying Marchenko’s method of quantum inverse scattering we show that the scattering data suggest a singular repulsive core of the potential of the form 2/r2 and 6/r2 in natural units, for the and channels, respectively. The simplest solution in the channel contains three parameters only but reproduces all features of the potential and bound state wave function within one percent error. We also consider the – coupled channel problem with the coupled channel Marchenko inversion method.

Atomic-orbital-dependent photoelectron momentum distributions for F− ions by orthogonal two-color laser fields

We theoretically investigate the two-dimensional photoelectron momentum distributions (PMDs) of ${\mathrm{F}}^{\ensuremath{-}}$ ions in an orthogonal two-color laser field with equal intensities. The PMDs for different atomic orbitals are simulated by an exact solution to the three-dimensional time-dependent Schr\odinger equation and the strong-field approximation method, respectively. Through the comparison of the calculations of these methods, we confirm that the asymptotic behavior of initial bound-state wave function plays a crucial role in forming the main shape of PMDs at large momenta. Based on the saddle-point method and the imaginary time theory, we show that the PMDs of ${\mathrm{F}}^{\ensuremath{-}}$ ions can be decoded to reveal the definite imprint of the photoelectron sub-barrier phase from the subcycle interference structures. We demonstrate the sub-barrier phases from different atomic orbitals have different impacts on the subcycle interference structures.

Analysis of N-dimensional Klein–Gordon equation for hydrogen molecule in the non-central potential field

We investigate a complete normalized relativistic solutions for N-dimensional Hartmann potential and relativistic effects of non-central potential field for hydrogen molecule within the framework of asymptotic iteration approach. We also discuss in detail the influence of dimensions on the angular probability distribution of hydrogen molecule. N-dimensional bound state wavefunction solutions including angular and radial solutions are expressed in terms of the Jacobi and Laguerre polynomials, respectively. N-dimensional angular wavefunctions of hydrogen molecule were visualized with Matlab software. In hyperspherical coordinates, the recurrence relations between diagonal matrix elements have been derived for N-dimensional Hartmann potential in relativistic theory. We also obtain analytical expressions of diagonal matrix elements and the recurrence formula between off-diagonal matrix elements in N-dimensions.

Direct fragmentation of $ \eta_{c}^{}$ η c (1S) and $ \eta_{b}^{}$ η b (1S) considering the transverse momentum effect of constituents

In this work, we calculate the fragmentation function (FF) of pseudoscalar S -wave heavy quarkonia from gluon splitting. We also study, for the first time, the Fermi motion effect of constituent quarks on the fragmentation production of the $ \eta_c(1S)$ and $ \eta_b(1S)$ states. In all previous studies of heavy quarkonium FFs, by ignoring the Fermi motion of constituents a Dirac-delta functional form was approximated for the wave function of bound state. Here, to study the Fermi motion of quarkonium constituents we consider a typical wave function which is different from the delta form and is the nonrelativistic limit of the solution of the Bethe-Salpeter equation with the QCD kernel. We shall present our results for the fragmentation of $ g\rightarrow (\eta_c, \eta_b)$ and show how the Fermi motion effect changes the results.

Analytical results for periodically-driven two-level models in relation to Heun functions

We introduce three different types of periodically-driven multiparametric two-level models whose analytical solutions are given in terms of Heun functions. These results are applied to obtain exact analytical results for certain types of periodic potentials and asymmetric double-well potentials. In particular, it is shown that under special parameter conditions, an experimentally realised periodic potential supports the exact in-gap solutions. In the asymmetric double-well potentials, some exact results of the bound-state wave functions and associated energies are found in explicit form.

Nonadiabatic Josephson current pumping by chiral microwave irradiation

Irradiating a Josephson junction with microwaves can operate not only on the amplitude but also on the phase of the Josephson current. This requires breaking time inversion symmetry, which is achieved by introducing a phase lapse between the microwave components acting on the two{\dag} sides of the junction. General symmetry arguments and the solution of a specific single level quantum dot model show that this induces chirality in the Cooper pair dynamics, due to the topology of the Andreev bound state wavefunction. Another essential condition is to break electron-hole symmetry within the junction. A shift of the current-phase relation is obtained, which is controllable in sign and amplitude with the microwave phase and an electrostatic gate, thus producing a "chiral" Josephson transistor. The dot model is solved in the infinite gap limit by Floquet theory and in the general case with Keldysh nonequilibrium Green's functions. The chiral current is nonadiabatic: it is extremal and changes sign close to resonant chiral transitions between the Andreev bound states.

A Study of Bound States of Systems of Helium and Lithium Using the Method of Discrete-Variable Representation

The systems of particles He2, 6Li–He, 7Li–He, He3, 6Li–He2, and 7Li–He2, the binding energies of which are small and the bound-state wave functions of which are widely distributed in space, are considered. Because the interaction potential is weak and rather localized compared to the characteristic sizes of wave functions of these systems, the problem of an accurate determination of binding energy and wave functions is complicated. Small changes in input parameters or an inaccuracy of calculations can lead to considerable deviations of calculated results from true values. An essential part of the study is the development and application of the discrete-variable representation method. This method is based on the determination of basis functions and the nodes and weights of a quadrature formula in such way that the values of a function are zero at all these nodes but one. With this representation the time required for calculating the Hamiltonianmatrix elements is reduced several times. The binding energies of several systems consisting of helium and lithium atoms were obtained using the method of discrete-variable representation. Thanks to the application of this approach, the calculation time was significantly reduced without loss in accuracy.

Low-energy outer-shell photo-detachment of the negative ion of aluminum

We investigate the photodetachment of the negative ion of aluminum, initially both in the ground state and the excited 1D state. A multiconfiguration Hartree–Fock method with nonorthogonal, term-dependent orbitals is used to generate accurate initial bound-state and final continuum-state wavefunctions. The collision of the ejected electron in the field of the residual ion is described by the B-spline R-matrix method for photon energies ranging from threshold to 12 eV. All principal scattering channels for photodetachment from both the and orbitals are included in the close-coupling expansion. We predict a number of prominent resonance features, whose verification presents new challenges in studies of this highly correlated process. Partial cross sections, together with the contributions of the principal scattering channels, are discussed in order to classify the resonance structure. The predicted cross sections are expected to be the most comprehensive and accurate currently available dataset for photodetachment of Al− at low energies.

Application of the Frobenius Method to the Schr？dinger Equation for a Spherically Symmetric Hyperbolic Potential

In this paper, an efficient technique for computing the bound state energies and wave functions of the Schrodinger Equation (SE) associated with a new class of spherically symmetric hyperbolic potentials is developed. This technique is based on a recent approximation scheme for the orbital centrifugal term and on the use of the Frobenius method (FM). The bound state eigenvalues are given as zeros of calculable functions. The corresponding eigenfunctions can be obtained by substituting the calculated energies into the recurrence relations for the expanding coefficients of the Frobenius series representing the solution. The excellent performance of this technique is illustrated through numerical results for some special cases like Poschl-Teller potential (PTP), Manning-Rosen potential (MRP) and Poschl-Teller polynomial potential (PTPP), with an application to the Gaussian potential well (GPW). Comparison with other methods is presented. Our results agree noticeably with the previously reported ones.

Light WIMP Searches Involving Electron Scattering

In the present work we examine the possibility of detecting electrons in light dark matter searches. These detectors are considered to be the most appropriate for detecting dark matter particles with a mass in the MeV region. We analyze theoretically some key issues involved in such detection. More specifically we consider a particle model involving WIMPs interacting with fermions via Z-exchange. We find that for WIMPs with mass in the electron mass range the cross section for WIMP-atomic electron scattering is affected by the electron binding. For WIMPs more than 20 times heavier than the electron, the binding affects the kinematics very little. As a result, many electrons can be ejected with energy which increases linearly with the WIMP mass, but the cross section is somewhat reduced depending on the bound state wave function employed. On the other hand for lighter WIMPs, the effect of binding is dramatic. More specifically at most 10 electrons, namely, those with binding energy below 10 eV, become available even in the case of WIMPs with a mass as large as 20 times the electron mass. Even fewer electrons contribute if the WIMPs are lighter. The cross section is, however, substantially enhanced by the Fermi function corrections, which become more important at low energies of the outgoing electrons. Thus events of 0.5–2.5 per kg-y become possible.

Frobenius Series Solutions of the Schrodinger Equation with Various Types of Symmetric Hyperbolic Potentials in One Dimension

The Schrodinger equation (SE) for a certain class of symmetric hyperbolic potentials is solved with the aid of the Frobenius method (FM). The bound state energies are given as zeros of a calculable function. The calculated bound state energies are successively substituted into the recurrence relations for the expanding coefficients of the Frobenius series representing even and odd solutions in order to obtain wave functions associated with even and odd bound states. As illustrative examples, we consider the hyperbolic Poschl-Teller potential (HPTP) which is an exactly solvable potential, the Manning potential (MP) and a model of the Gaussian potential well (GPW). In each example, the bound state energies obtained by means of the FM are presented and compared with the exact results or the literature ones. In the case of the HPTP, we also make a comparison between exact bound state wave functions and the eigenfunctions obtained by means of the present approach. We find that our results are in good agreement with those given by other methods considered in this work, and that our class of potentials can be a perfect candidate to model the GPW.

ON THE COMMON LIMIT OF THE PT-SYMMETRIC ROSEN–MORSE II AND FINITE SQUARE WELL POTENTIALS

Two <em>PT</em>-symmetric potentials are compared, which possess asymptotically finite imaginary components: the <em>PT</em>-symmetric Rosen-Morse II and the finite <em>PT</em>-symmetric square well potentials. Despite their different mathematical structure, their shape is rather similar, and this fact leads to similarities in their physical characteristics. Their bound-state energy spectrum was found to be purely real, an this finding was attributed to their <br />asymptotically non-vanishing imaginary potential components. Here the <em>V(x</em>)= <em>γδ</em>(<em>x</em>)+ i2Λ sgn(<em>x</em>) potential is discussed, which can be obtained as the common limit of the two other potentials. The energy spectrum, the bound-state wave functions and the transmission and reflection coefficients are studied in the respective limits, and the results are compared.

Plasma-screening effects on positronium formation**Project supported by the National Natural Science Foundation of China (Grant Nos. 11404223, 11447158, and 11604223) and the Doctoral Program Foundation of Shenyang Aerospace University, China (Grant No. 13YB26).

Plasma-screening effects on positronium (Ps) formation for positron–hydrogen collisions in a Debye plasma environment is further investigated using the screening approximation model with the inclusion of the modified structure of Ps. More accurate Ps formation cross sections (n = 1, 2) are obtained for various Debye lengths from the Ps formation thresholds to 50 eV. The influence of considering modified bound-state wave functions and eigenenergies for the Ps is found to result in the reduction of the Ps formation cross sections at low energies, whereas it cannot counteract the enhancement of the Ps formation by the Debye screening.

Contact formalism for coupled channels

The contact formalism, a useful tool for analyzing short-range correlations, is generalized here for systems with coupled channels, such as in nuclear physics. The relevant asymptotic form is presented and contact matrices are defined. Generally, for the case of two coupled channels, two two-body functions are included in the asymptotic form, resulting a 2 x 2 contact matrix. Nevertheless, it is shown that if the coupling terms of the potential are very weak or very strong, only a single two-body function is needed, resulting a single contact. This universal result is directly relevant to nuclear systems, and provides a theoretical explanation for the fact that proton-neutron short-range correlations can be described using the single bound-state deuteron wave function. It is achieved by applying an appropriate boundary condition on the two-body functions. This boundary condition can be interpreted as a mean field potential imposed on the correlated pair due to the residual system.

Conditional electron confinement in graphene via smooth magnetic fields

In this article we discuss confinement of electrons in graphene via smooth magnetic fields which are finite everywhere on the plane. We shall consider two types of magnetic fields leading to systems which are conditionally exactly solvable and quasi exactly solvable. The bound state energies and wavefunctions in both cases have been found exactly.

In-medium bound states and pairing gap

The propagator of two nucleons in infinite nuclear matter is evaluated by a diagonalization of the $pphh$ RPA Hamiltonian. This effective Hamiltonian is non-Hermitian and, for specific density domains and partial waves, yields pairs of complex conjugated eigenvalues representing in-medium bound states of two nucleons. The occurrence of these complex poles in the two-particle Greens function is tightly related to the well known BCS pairing approach. It is demonstrated that these complex eigenvalues and the corresponding bound state wavefunctions contain all information about the BCS gap function. This is illustrated by calculations for $^1S_0$ and $^3PF_2$ pairing gaps in neutron matter which essentially coincide with the corresponding gap functions extracted from conventional solutions of the gap equation. Differences between the bound states in the conventional BCS approach and the $pphh$ RPA are arising in the case of $^3SD_1$ channel in symmetric nuclear matter at low densities. These differences are discussed in the context of transition from BEC for quasi-deuterons to the formation of BCS pairing.

Modelling laser-matter interactions for simple molecules

SynopsisWe consider the interaction of a molecule with an ultrashort laser pulse within the mean field approximation. A model has been generalized to treat the interaction of atomic hydrogen with an electromagnetic pulse. The kernel of the Coulomb potential in the momentum space is replaced by a sum of products of separable potentials supporting a bound state of the atom and built from the corresponding bound state wave function. The molecular orbitals are expressed in terms of gaussians and generated by means of a quantum chemistry software. We analyze the ionization yield as a function of the size of the gaussian basis for H2 and H2O molecules and the results are compared to those already existing in the literature.