Relativistic theory of surface states

A Green's function method is used to investigate the role of relativistic effects in determining the electronic states in a semi-infinite Kronig-Penney model. Various boundary conditions are considered and it is found that relativistic effects do not introduce new surface states.     

Relativistic electronic states in disordered systems

The energy spectrum of relativistic electrons in one-dimensional disordered systems is investigated. Using an approximate solution to the one-dimensional Dirac equation as given by Steslicka and Davison and following a method reported earlier by the author for nonrelativistic cases, an equation connecting the wave functions at three consecutive atomic sites is derived. This “connection equation” is then analysed with the help of (1) Worpitzky's theorem and (2) a perturbational approach due to Phariseau. With the help of these approaches, explicit equations are derived for electronic energies in disordered systems with small deviations from periodicity. These explicit equation indicate that, in general, a band structure of electronic energies exists in each region of the system; they reveal further that there occur energies in addition to those characterising the local bands and these extra energies constitute a disturbance to the (local) band structures.     

Relativistic analysis of four-body final states

The constraints of unitarity and analyticity on four-body final states are studied. It is shown that unitarity alone forces the amplitudes to be coherent and have singular behaviour. The implementation of unitarity with total energy analyticity yields a set of relativistic linear integral equations for the four-body amplitude. This is the minimal set consistent with quantum mechanics and also is the full dynamical set of equations with two-body separable interactions. These equations will provide important ingredients for the phenomenological analysis of four-body final states using the isobar model.     

Relativistic wave functions for single-proton resonant states

With the relativistic boundary condition, single-proton resonant states in spherical nuclei are studied by an analytic continuation in the coupling constant (ACCC) method within the framework of the self-consistent relativistic mean-field (RMF) theory. In this scheme, we investigate the wave functions for l≠ 0 proton resonant states close to the continuum threshold in the stable nuclide ¹²⁰Sn for the first time. Some hints for pseudospin symmetry in the resonant states of nuclei are mentioned as well.     

Borromean Bound States

The Hall-Post inequalities relating N-body to (N − 1)-body energies of quantum bound states are applied to delimit, in the space of coupling constants, the domain of Borromean binding where a composite system is bound while the smaller subsystems are unbound.     

Relativistic quantum correlations in bipartite fermionic states

The influences of relative motion, the size of the wave packet and the average momentum of the particles on different types of correlations present in bipartite quantum states are investigated. In particular, the dynamics of the quantum mutual information, the classical correlation and the quantum discord on the spin correlations of entangled fermions are studied. In the limit of small average momentum, regardless of the size of the wave packet and the rapidity, the classical and the quantum correlations are equally weighted. On the other hand, in the limit of large average momentum, the only correlations that exist in the system are the quantum correlations. For every value of the average momentum, the quantum correlations maximize at an optimal size of the wave packet. It is shown that after reaching a minimum value, the revival of quantum discord occurs with increasing rapidity.     

Exact Three-Dimensional Relativistic Equation for qq Bound states

Starting with the QCD generating functional, a new derivation and a new form of the Dirac-Schriidinger (D-S) equation,catisfied by the equal-time Bethe-Salpeter amplitude of quark-antiquark bound states have been given. In this equation, the effective interaction kernel has a compact expression and can be directly calculated by means of the conventional QCD Feynman rules. F'urthermore, an equivalent reduction of the above equation to a Pauli-Schriidinger equation has also been achieved and a closed form of the effective interaction Hamiltonian appearing in the latter equation has been explicitly written out.     

Relativistic resonances as non-orthogonal states in Hilbert space

We analyze the energy-momentum properties of relativistic short-lived particles with the result that they are characterized by two 4-vectors: in addition to the familiar energy-momentum vector (timelike) there is an energy-momentum `spread vector' (spacelike). The wave functions in space and time for unstable particles are constructed. For the relativistic properties of unstable states we refer to Wigner's method of Poincaré group representations that are induced by representations of the space-time translation and rotation groups. If stable particles, unstable particles and resonances are treated as elementary objects that are not fundamentally different one has to take into account that they will not generally be orthogonal to each other in their state space. The scalar product between a stable and an unstable state with otherwise identical properties is calculated in a particular Lorentz frame. The spin of an unstable particle is not infinitely sharp but has a `spin spread' giving rise to `spin neighbors'. This opens the possibility of a non-zero scalar product between states with unequal spin. - A first practical application of non-orthogonal states is seen in diffraction dissociation reactions whose large cross-sections are attributed to interference of states that are `partially identical'. Received: 28 June 2002 / Revised version: 13 November 2002 / Published online: 14 March 2003 RID="a" ID="a" e-mail: walter.blum@cern.ch RID="b" ID="b" e-mail: hns@mppmu.mpg.de     

Borromean rings and linkings

For links of 3 components, such as Borromean rings, which escape the detection of Gauss linking, we define and compute combinatorically and explicitly the higher linking. And as in perturbative quantum field theory, this higher linking is presented as a sum of Chern–Simons–Witten configuration space integrals.     

Relativistic constraints on the distinguishability of orthogonal quantum states

The constraints imposed by special relativity on the distinguishability of quantum states are discussed. An explicit expression relating the probability of an error in distinguishing two orthogonal single-photon states to their structure, the time t at which a measurement starts, and the interval of time T elapsed from the start of the measurement until the time at which the outcome is obtained by an observer is given as an example.     

Borromean three-body heteroatomic resonances

We present an approach to analyze recent experimental evidences of Efimov resonant states in mixtures of ultracold gases, by considering two-species three-body atomic systems bound in a Borromean configuration, where all the two-body interactions are unbound. For such Borromean three-body systems, it is shown that a continuum three-body s-wave resonance emerges from an Efimov state as a scattering length or a three-body scale is moved. The energy and width of the resonant state are determined from a scaling function with arguments given by dimension-less energy ratios relating the two-body virtual state subsystem energies with the shallowest three-body bound state. The peculiar behavior of such resonances is that their peaks are expected to move to lower values of the scattering length, with increasing width, as one raises the temperature. For Borromean systems, two resonant peaks are expected in ultralow-temperature regimes, which will disappear at higher energies. It is shown how a Borromean-Efimov excited bound state turns out to a resonant state by tuning the virtual two-body subsystem energies or scattering lengths, with all energies written in units of the next deeper shallowest Efimov state energy. The resonance position and width for the decay into the continuum are obtained as universal scaling functions (limit cycle) of the dimensionless ratios of the two and three-body scales, which are calculated numerically within a zero-range renormalized three-body model.     

Quantum mechanics of Klein-Gordon fields II: Relativistic coherent states

We use the formulation of the quantum mechanics of first-quantized Klein-Gordon fields given in the first of this series of papers to study relativistic coherent states. In particular, we offer an explicit construction of coherent states for both charged and neutral (real) free Klein-Gordon fields as well as for charged fields interacting with a constant magnetic field. Our construction is free from the problems associated with charge-superselection rule that complicated the previous studies. We compute various physical quantities associated with our coherent states and present a detailed investigation of their classical (nonquantum) and nonrelativistic limits.     

Relativistic particle scattering states with tensor potential and spatially-dependent mass

We investigate the relativistic equation for particles with spin 1/2 in the q-parameter modified Pöschl-Teller potential, including Coulomb-like tensor interaction with spatially-dependent mass for the D-dimension. We present approximate solutions of the Dirac equation with these potentials for any spin-orbit quantum number κ under spin symmetry. The normalized wave functions are expressed in terms of the hyper-geometric series of the scattering states on the k/2π scale. We also give the formula for the phase shifts, and use the Nikiforov-Uvarov method to obtain the energy eigen-values equation.     

Relativistic calculations of ground states of single-electron diatomic molecular ions

We have performed relativistic calculations of ground-state energies for a series of single-electron homonuclear dimers A ₂ ^(2Z?1)+ with nucleus charge Z = 1, 2, 10, 20, 30, 40, 50, 60, 70, 80, 90, 92, and 100 and internuclear distances R = 2/Z. The work involves the Born-Oppenheimer approximation and the single-electron two-center Dirac Hamiltonian, which describes the interaction between an electron and two immovable point charges. Analysis of the convergence process and comparison with data presented in other works for H ₂ ⁺ and Th ₂ ¹⁷⁹⁺ dimers shows that the relative error of the obtained results is on the order of 10^?11–10^?12. High-accuracy values of ground-state energies for some dimers other than Z = 1 and 90 have been obtained in this work for the first time.