Spectrum Of Bound States Research Articles

Energy-level structure of the hydrogen atom confined by a penetrable cylindrical cavity

The bound-state energy spectrum and its evolution for a hydrogen atom located along the axis of a standard cylindrical confining cavity with either impenetrable or penetrable confining boundaries are obtained by solving the stationary Schrödinger equation using a finite differences approach. New results are first presented for a nuclear-centered position for the penetrable case as the barrier height and cavity size change. Special attention is then given to the energy-level dependence on the nuclear position along the cylinder axis, where proper identification for the evolution of states from the nuclear-centered position (centered states) up to the cylinder cap (cap states) is proposed, while the corresponding state evolution for intermediate nuclear positions (intermediate states) remains consistent with node conservation and symmetry. It is found that in general the energy levels evolve with an increasing value as the nuclear position is shifted from the central position up to a cylinder cap. As the barrier height (and cavity size) are reduced, a progressive extinction of bound states appears in the order cap states, intermediate states and centered states. Finally, a predominance of cavity-size over barrier-height effects on the energy level shift is found.

OPE convergence in non-relativistic conformal field theories

Motivated by applications to the study of ultracold atomic gases near the unitarity limit, we investigate the structure of the operator product expansion (OPE) in non-relativistic conformal field theories (NRCFTs). The main tool used in our analysis is the representation theory of charged (i.e. non-zero particle number) operators in the NRCFT, in particular the mapping between operators and states in a non-relativistic "radial quantization" Hilbert space. Our results include: a determination of the OPE coefficients of descendant operators in terms of those of the underlying primary state, a demonstration of convergence of the (imaginary time) OPE in certain kinematic limits, and an estimate of the decay rate of the OPE tail inside matrix elements which, as in relativistic CFTs, depends exponentially on operator dimensions. To illustrate our results we consider several examples, including a strongly interacting field theory of bosons tuned to the unitarity limit, as well as a class of holographic models. Given the similarity with known statements about the OPE in SO(2,d) invariant field theories, our results suggest the existence of a bootstrap approach to constraining NRCFTs, with applications to bound state spectra and interactions. We briefly comment on a possible implementation of this non-relativistic conformal bootstrap program.

The SUSY Yang–Mills plasma in a T-matrix approach

In this paper, the thermodynamic properties of [Formula: see text] supersymmetric Yang–Mills theory with an arbitrary gauge group are investigated. In the confined range, we show that identifying the bound state spectrum with a Hagedorn one coming from noncritical closed superstring theory leads to a prediction for the value of the deconfining temperature [Formula: see text] that agrees with recent lattice data. The deconfined phase is studied by resorting to a [Formula: see text]-matrix formulation of statistical mechanics in which the medium under study is seen as a gas of quasigluons and quasigluinos interacting nonperturbatively. Emphasis is put on the temperature range (1–5) [Formula: see text], where the interactions are expected to be strong enough to generate bound states. Binary bound states of gluons and gluinos are indeed found to be bound up to 1.4 [Formula: see text] for any gauge group. The equation of state is then computed numerically for [Formula: see text] and [Formula: see text], and discussed in the case of an arbitrary gauge group. It is found to be nearly independent of the gauge group and very close to that of nonsupersymmetric Yang–Mills when normalized to the Stefan–Boltzmann pressure and expressed as a function of [Formula: see text].

Group theoretic approach to rationally extended shape invariant potentials

The exact bound state spectrum of rationally extended shape invariant real as well as PT symmetric complex potentials is obtained by using potential group approach. The generators of the potential groups are modified by introducing a new operator U(x,J3±12)to express the Hamiltonian corresponding to these extended potentials in terms of Casimir operators. Connection between the potential algebra and the shape invariance is elucidated.

Mass-profile quantum dots in graphene and artificial periodic structures

We analyze the bound-state spectra of mass-profile quantum dots in graphene, a system at current experimental reach. Homogeneous perpendicular magnetic fields are also considered which result in breaking the valley degeneracy. The spectra show rich features, arising from the chiral band structure of graphene and its Landau levels and we identify three different regimes depending on the ratio between the radius of the dot and the magnetic length. We further carry out a comparison with potential-well quantum dots discussed in [Recher et al, Phys. Rev. B 79, 085407 (2009)] and conclude that mass confinement may offer significant advantages for optical applications in the THz and infrared regime. Also due to experimental advances, we additionally analyze the band structure of a linear chain of mass-profile quantum dots, where overlap-assisted hopping processes play a main role for closely packed arrays. Here, the inclusion of Coulomb interactions shows that Frenkel excitons can be hosted by the system for certain values of the lattice constant. A Bose-Hubbard model can be mimicked under such a choice of the distance between the dots.

Spectrum of three-body bound States in a finite volume.

The spectrum of a bound state of three identical particles with a mass m in a finite cubic box is studied. It is shown that in the unitary limit, the energy shift of a shallow bound state is given by ΔE=c(κ^{2}/m)(κL)^{-3/2}|A|^{2}exp(-2κL/sqrt[3]), where κ is the bound-state momentum, L is the box size, |A|^{2} denotes the three-body analog of the asymptotic normalization coefficient of the bound state wave function, and c is a numerical constant. The formula is valid for κL≫1.

Approximate Solutions of the Dirac Equation for the Hua Plus Modified Eckart Potential

The relativistic symmetries of the Dirac equation with Hua plus modified Eckart potential including the generalized tensor interaction are presented. We obtain the energy eigenvalues and the corresponding eigenfunction using parametric Nikiforov–Uvarov method. We compute the bound-state energy spectrum to show the effects of the tensor interaction, and by using an approximate scheme to the centrifugal term, the arbitrary-state solutions are reported and the scattering states are studied. Our results show that the presence of the generalized tensor interaction removes the degeneracies between spin and pseudospin doublets. Some special cases of this potential are also discussed.

Efimov States in Li–Cs Mixtures within a Minimal Model

We use a minimal zero-range model for describing the bound state spectrum of three-body states consisting of two Cesium and one Lithium atom. Using a broad Feshbach resonance model for the two-body interactions, we show that recent experimental data can be described surprisingly well for particular values of the three-body parameter that governs the short-range behavior of the atomic potentials and is outside the scope of the zero-range model. Studying the spectrum as a function of the three-body parameter suggests that the lowest state seen in experiment could be influenced by finite range corrections. We also consider the question of Fermi degeneracy and corresponding Pauli blocking of the Lithium atoms on the Efimov states.

Kratzer’s molecular potential in quantum mechanics with a generalized uncertainty principle

The Kratzer’s potential V(r)=g1/r2−g2/r is studied in quantum mechanics with a generalized uncertainty principle, which includes a minimal length (ΔX)min=ħ5β. In momentum representation, the Schrödinger equation is a generalized Heun’s differential equation, which reduces to a hypergeometric and to a Heun’s equations in special cases. We explicitly show that the presence of this finite length regularizes the potential in the range of the coupling constant g1 where the corresponding Hamiltonian is not self-adjoint. In coordinate space, we perturbatively derive an analytical expression for the bound states spectrum in the first order of the deformation parameter β. We qualitatively discuss the effect of the minimal length on the vibration–rotation energy levels of diatomic molecules, through the Kratzer interaction. By comparison with an experimental result of the hydrogen molecule, an upper bound for the minimal length is found to be of about 0.01 Å. We argue that the minimal length would have some physical importance in studying the spectra of such systems.

The spinless relativistic Hulthén problem

The spinless Salpeter equation can be regarded as the eigenvalue equation of a Hamiltonian that involves the relativistic kinetic energy and therefore is, in general, a nonlocal operator. Accordingly, it is hard to find solutions of this bound-state equation by exclusively analytic means. Nevertheless, a lot of tools enables us to constrain the resulting bound-state spectra rigorously. We illustrate some of these techniques for the example of the Hulthén potential.

Repeated output coupling of ultracold Feshbach molecules from a Cs BEC

We investigate magnetoassociation of ultracold Feshbach molecules from a Bose-Einstein condensate of Cs atoms and explore the spectrum of weakly bound molecular states close to the atomic threshold. By exploiting the variation of magnetic field experienced by a molecular cloud falling in the presence of a magnetic field gradient, we demonstrate the repeated output coupling of molecules from a single atomic cloud using a Feshbach resonance at 19.89 G. Using this method we are able to produce up to 24 separate pulses of molecules from a single atomic condensate, with a molecular pulse created every 7.2 ms. Furthermore, by careful control of the magnetic bias field and gradient we are able to utilise an avoided crossing in the bound state spectrum at 13.3 G to demonstrate exquisite control over the dynamics of the molecular clouds.

Bound state spectra and properties of the doublet states in three-electron atomic systems

The bound state spectra of the doublet states in three-electron atomic systems are investigated. By using different variational expansions we determine various bound state properties in these systems. Such properties include the electron–nucleus and electron–electron delta-functions and cusp values. The general structure of the bound state spectra in several three-electron atomic systems (Li, Be+, C3+ and F6+) is investigated with the use of the Hylleraas-Configuration Interaction and the Configuration Interaction wave functions. The advantage of our Configuration Interaction based procedure is that it provides high numerical accuracy for all rotationally excited states, including the bound states with L⩾7.

Studies on bound-state spectra of Manning–Rosen potential

Accurate ro-vibrational energies, eigenfunctions, radial densities, expectation values are presented for the exponential-type Manning–Rosen (MR) potential. Bound states, accurate up to ten significant figure are obtained by employing a simple, reliable generalized pseudospectral (GPS) method. All 55 eigenstates with n ≤10 are treated for arbitrary values of potential parameters, covering a wide range of interaction, through a non-uniform, optimal spatial radial discretization. A detailed investigation has been made on energy changes with respect to screening and other potential parameters. A systematic estimation of critical screening parameters is given for these eigenstates. Special emphasis has been given to higher states and in the vicinity of critical screening region. A thorough comparison with literature results is made wherever possible. This surpasses the accuracy of all other existing methods currently available. Several new states are reported for the first time. In short, a simple, efficient scheme for accurate calculation of this and other molecular potentials is offered.

Path integral of a relativistic spinning particle in (1+1) dimension with vector and scalar linear potentials in the presence of a minimal length

Using the path integral formalism in (1+1) dimension of the energy–momentum space representation, we calculate the Green function of relativistic spinning particle subjected to the action of combined vector and scalar linear potentials in the framework of deformed Heisenberg algebra which distinguish to the appearance of nonzero minimum position uncertainty given by [Formula: see text] where β is the deformation parameter of the modified Heisenberg uncertainty relation [Formula: see text]. We adapt the space–time transformation method to evaluate quantum corrections and this latter are dependent on the α-point discretization interval. The exact bound states spectrum and the corresponding momentum space wave function are obtained.

Parity Measurement in Topological Josephson Junctions

We study the properties of a topological Josephson junction made of both edges of a two-dimensional topological insulator. We show that, due to fermion parity pumping across the bulk, the global parity of the junction has a clear signature in the periodicity and critical value of the Josephson current. In particular, we find that the periodicity with the flux changes from 4π in a junction with an even number of quasiparticles to 2π in the odd sector. In the case of long junctions, we exhibit a rigorous mathematical connection between the spectrum of Andreev bound states and the fermion parity anomaly, through bosonization. Additionally, we discuss the rather quantitative effects of Coulomb interactions on the Josephson current.

Bound state spectrum of the triplet states in the Be atom

The bound state spectrum of the low-lying triplet states in the Be atom is investigated. In particular, we perform accurate computations of the bound triplet S,P,D,F,G, H and I states in the Be atom. The results of these calculations are employed to draw the spectral diagram which contains the energy levels of the triplet states. Based on our computational results we can observe transitions from the low-lying bound states to the weakly-bound Rydberg states. For the 23S,33S and 43S states in the Be atom we also determine a number of bound states properties.

Energy Levels of Spin-1/2 Particles with Yukawa Interaction

In this article, Yukawa interaction is used to study the relativistic spin-1/2 particles and obtain their energy levels. The role of Yukawa potential on the spin and pseudospin symmetry solution is investigated systematically by solving the Dirac equation with attractive scalar S(r) and repulsive vector V(r) potentials. Bound state spectrum and wave functions of Yukawa potential are obtained. It is found that the energy eigenvalues strongly depend on the potential parameters.

Studies on the Bound-State Spectrum of Hyperbolic Potential

Bound states of hyperbolic potential is investigated by means of a generalized pseudospectral method. Significantly improved eigenvalues, eigenfunctions are obtained efficiently for arbitrary n, ℓ quantum states by solving the relevant non-relativistic Schrödinger equation allowing a non-uniform, optimal spatial discretization. Eigenvalues accurate up to tenth decimal place are reported for a large range of potential parameters; thus covering a wide range of interaction. Excellent agreement with available literature results is observed in all occasions. Special attention is paid for higher states. Some new states are given. Energy variations with respect to parameters in the potential are studied in considerable detail for the first time.

Impurity states and cooperative magnetic order in Fe-based superconductors

We study impurity bound states and impurity-induced order in the superconducting state of LiFeAs within a realistic five-band model based on the band structure and impurity potentials obtained from density functional theory (DFT). In agreement with recent experiments, we find that Co impurities are too weak produce sub-gap bound states, whereas stronger impurities like Cu do. We also obtain the bound state spectrum for magnetic impurities, such as Mn, and show how spin-resolved tunnelling may determine the nature of the various defect sites in iron pnictides, a prerequisite for using impurity bound states as a probe of the ground state pairing symmetry. Lastly we show how impurities pin both orbital and magnetic order, providing an explanation for a growing set of experimental evidence for unusual magnetic phases in doped iron pnictides.

Ro-vibrational studies of diatomic molecules in a shifted Deng-Fan oscillator potential

Bound-state spectra of shifted Deng-Fan oscillator potential are studied by means of a generalized pseudospectral method. Very accurate results are obtained for both low as well as high states by a nonuniform optimal discretization of the radial Schrodinger equation. Excellent agreement with literature data is observed in both s-wave and rotational states. Detailed variation of energies with respect to potential parameters is discussed. Application is made to the ro-vibrational levels of representative diatomic molecules (H2, LiH, HCl, CO). Nine states having are calculated with good accuracy along with 15 other higher states for each of these molecules. Variation of energies with respect to state indices n, show behavior similar to that in the Morse potential. Many new states are reported here for the first time. In short, a simple, accurate, and efficient method is presented for this and other similar potentials in molecular physics. © 2013 Wiley Periodicals, Inc.