The Dirac particle in a one-dimensional “hydrogen atom”

Specific features of the behavior of the spectrum of steady states of the Dirac particle in a regularized ??Coulomb?? potential V??(z) = ?q/(|z| + ??) as a function of the cutting parameter of ?? in 1 + 1 D are investigated. It is shown that in such a one-dimensional relativistic ??hydrogen atom?? at ?? ? 1, the discrete spectrum becomes a quasi-periodic function of ??; this effect depends on the bonding constant analytically and has no nonrelativistic analog. This property of the Dirac spectral problem clearly demonstrates the presence of a physically containable energy spectrum at arbitrary small ?? > 0 and simultaneously the absence of the regular limiting transition to ?? ?? 0. Thus, the necessity of extension of a definition for the Dirac Hamiltonian with irregularized potential in 1 + 1 D is confirmed at all nonzero values of the bonding constant q. It is also noted that the three-dimensional Coulomb problem possesses a similar property at q = Z?? > 1, i.e., when the selfconsistent extension is required for the Dirac Hamiltonian with an irregularized potential.     

Approximate path integral solution for a Dirac particle in a deformed Hulthén potential

The problem of a Dirac particle moving in a deformed Hulthén potential is solved in the framework of the path integral formalism. With the help of the Biedenharn transformation, the construction of a closed form for the Green’s function of the second-order Dirac equation is done by using a proper approximation to the centrifugal term and the Green’s function of the linear Dirac equation is calculated. The energy spectrum for the bound states is obtained from the poles of the Green’s function. A Dirac particle in the standard Hulthén potential (q = 1) and a Dirac hydrogen-like ion (q = 1 and a → ∞) are considered as particular cases.     

Reconstruction of the coordinate dependences of quadratic susceptibility tensor \hat \chi ^{(2)} (z, ω1 ± ω2; ω1, ±ω2) components for the one-dimensionally inhomogeneous absorbing medium

A method for the unambiguous reconstruction of the spatial profiles of all components (except for ?? _zzz ) of the quadratic susceptibility complex tensor {ie165-2} (z, ??₁ + ??₂; ??₁, ??₂), which is responsible for the sumfrequency generation in a one-dimensionally inhomogeneous plate is proposed and proven for the first time. Such reconstruction is possible if the symmetry of the medium provides the diagonal character of the linear permittivity tensor {ie165-3} (z, ??). The procedure involves the measurement of the complex amplitude of the new wave with the frequency ??₁ + ??₂ that is reflected from the plate for a certain interval of the angles of incidence of the wave with the frequency ??₂. The reflected wave results from the nonlinear interaction of the wave with frequency ??₂ and the wave with frequency ??₁ that exhibits the normal incidence. A similar approach can be used to determine the profiles of the components of the quadratic susceptibility tensor {ie165-4}(z, ??₁ ? ??₂; ??₁, ? ??₂), which is responsible for the difference-frequency generation.     

The formation of correlated states and the increase in barrier transparency at a low particle energy in nonstationary systems with damping and fluctuations

We consider peculiarities in the formation of a coherent correlated state (CCS) of a particle in a periodically modulated harmonic oscillator with damping for various types of stochastic perturbation. It is shown that in the absence of stochastic perturbation, an optimal relation exists between the damping parameter (damping coefficient) and the modulation depth, for which the ??extrinsic?? characteristics of the oscillator (amplitudes of ??classical?? oscillation and the momentum of a particle) remain unchanged, while the correlation coefficient rapidly increases from |r| = 0 to |r|_max ?? 1; this corresponds to a completely correlated coherent state. Under nonoptimal conditions, the formation of the CCS with a simultaneous increase in is accompanied by damping or excitation of the oscillator. It is shown that for a certain relation between the damping coefficient and the modulation depth, the presence of a stochastic external force acting on the nonstationary oscillator does not prevent the formation of a CCS with |r|_max ?? 1. A fundamentally different effect is observed under a stochastic influence on the nonstationary frequency of the oscillator; this effect always limits the value of |r| at a level |r|_max < 1; a CCR cannot be formed with an unlimited increase in its intensity, and |r|_max ?? 0. The influence of the CCS formation on the averaged probability ??D?? of the tunnel effect (transparency of the potential barrier) is considered for a particle in an oscillator with damping both in the absence and in the presence of a stochastic force. It is shown using a specific example that complete clearing of the potential barrier and the increase in the barrier transparency from the initial value ??D _r=0?? = 10^?80 to ??D?? ?? 1 can occur over a comparatively short time interval in both these cases. These effects can be used to obtain highly efficient nuclear fusion at a low energy of interacting particles.     

Hydrogen-like atom in a superstrong magnetic field: Photon emission and relativistic energy level shift

Following our previous work, additional arguments are presented that in superstrong magnetic fields B ? (Zα)² B ₀, B ₀ = m ² c ³/e? ≈ 4.41 × 10¹³ G, the Dirac equation and the Schrödinger equation for an electron in the nucleus field following from it become spatially one-dimensional with the only z coordinate along the magnetic field, “Dirac” spinors become two-component, while the 2 × 2 matrices operate in the {0; 3} subspace. Based on the obtained solution of the Dirac equation and the known solution of the “onedimensional” Schrödinger equation by ordinary QED methods extrapolated to the {0; 3} subspace, the probability of photon emission by a “one-dimensional” hydrogen-like atom is calculated, which, for example, for the Lyman-alpha line differs almost twice from the probability in the “three-dimensional” case. Similarly, despite the coincidence of nonrelativistic energy levels, the calculated relativistic corrections of the order of (Zα)⁴ substantially differ from corrections in the absence of a magnetic field. A conclusion is made that, by analyzing the hydrogen emission spectrum and emission spectra at all, we can judge in principle about the presence or absence of superstrong magnetic fields in the vicinity of magnetars (neutron stars and probably brown dwarfs). Possible prospects of applying the proposed method for calculations of multielectron atoms are pointed out and the possibility of a more reliable determination of the presence of superstrong magnetic fields in magnetars by this method is considered.     

Theoretical prediction of cosmological constant Λ in Veneziano ghost theory of QCD

Based on the Veneziano ghost theory of QCD, we predict the cosmological constant ??, which is related to energy density of cosmological vacuum by $ \Lambda = \frac{{8\pi G}} {3}\rho _\Lambda $ . In the Veneziano ghost theory, the vacuum energy density ?? _?? is expressed by absolute value of the product of quark vacuum condensate and quark current mass: $ \rho _\Lambda = \frac{{2N_f H}} {{m_{\eta '} }}c|m_q < 0|:\bar qq:|0 > | $ . We calculate the quark local vacuum condensates ??0|: $ \bar q $ q: |0?? by solving Dyson-Schwinger Equations for a fully dressed confining quark propagator S _f (p) with an effective gluon propagator G _???? ^ab (q). The quark current mass m _q is predicted by use of chiral perturbation theory. Our theoretical result of ??, with the resulting ??0|: 471-4 q: |0?? = ?(235 MeV)³ and light quark current mass m _q ? 3.29?C6.15 MeV, is in a good agreement with the observable of the ?? ?? 10^?52 m^?2 used widely in a great amount of literatures.     

The far-infrared spectrum of tetrahydrothiophene: The bend/twist vibrations and associated barriers

The far-infrared spectrum of tetrahydrothiophene is reinvestigated with a resolution of 0.12 cm^?1 in the region of 50–350 cm^?1. In addition to the bend transitions (ν_b) below 120 cm^?1 reported previously, a number of sequences revealed by the improved resolution are observed for the first time and assigned to transitions of 2ν_b, of the twist (ν_t), and of difference combinations (ν_t - ν_b). Simple one-dimensional modeling of the twist sequence, which is derived from a self-consistent bend-twist energy level diagram, with a quadratic-quartic Hamiltonian suggests a barrier to planarity on the order of 4250 cm^?1. A two-dimensional potential function in the dimensionless coordinates is found to be V(q_b, q_t) = ?249.6q_b² + 4.48q_b⁴ ? 215.5q_t² + 2.73q_t⁴ + 7.00q_b²q_t².     

Modes of natural vibration for magnetic domains

We analyze wall-vibration modes for the case of plane parallel stripe domains in a uniaxial film whose easy axis is normal to the film plane, using Landau-Lifshitz equations carried to the limit of vanishing wall thickness. We take into account long-range dipole interactions and wall-moment twist due to stray fields from magnetic charges on the film surfaces. The small-amplitude wall displacement q(k, z) depends on the position coordinate z normal to the film plane, and on a two dimensional wave vector k parallel to the film plane. Numerically computed natural frequencies v_n(k) depend on the number of nodes n(=0, 1, 2 …) in the dependence of q on z. Surface and bulk modes are distinguished by the z-dependence of computed eigenmodes q_n(k, z). The spectrum of computed natural frequencies compares favorably with available experimental data.     

Exclusive semi-leptonic decays of bottom mesons in the spectator quark model

We calculate the exclusive semileptonic bottom meson decays \(B \to D(D*) + l^ - + \bar v_l \) in the spectator quark model. The helicity structure of the mesonic current transitionsB→D(D ^*) is matched to the helicity structure of the free quark current transitionsb→c at minimum momentum transferq ²=0. The results are continued toq ²≠=0 by pole-dominated form factors. Our results are compared to recent calculations that use quark model dynamics at maximum momentum transferq _max ² = (M ₁ ?M ₂)². We find agreement atq _max ² . Atq ²=0 there are significant differences between the predictions of the two approaches leading to marked differences in the predictions for the shape of the lepton energy spectrum, the shape of theq ²-distribution, and the helicity composition of the transition measurable in the angular distributions of the decaysD ^*→Dπ and \(W_{virtual}^ - \to l^ - + \bar v_l \) .     

Polar Phase of Superfluid <Superscript>3</Superscript>He: Dirac Lines in the Parameter and Momentum Spaces

The time reversal symmetric polar phase of the spin-triplet superfluid ³He has two types of Dirac nodal lines. In addition to the Dirac loop in the spectrum of the fermionic Bogoliubov quasiparticles in the momentum space (p _x , p _y , p _z ), the spectrum of bosons (magnons) has Dirac loop in the 3D space of parameters—the components of magnetic field (H _x ,H _y ,H _z ). The bosonic Dirac system lives on the border between the type-I and type-II.     

Semiempirical determination of the atom-surface interaction

We discuss three methods of determining V(z), the lateral average (G = 0 Fourier component) of the atom-surface interaction, from the bound state spectrum found in beam scattering. One method uses the Rydberg-Klein-Rees technique, which yields the width of the potential (i.e., separation of classical turning points) as a function of energy. This method incorporates also the known asymptotic form V ～ ?C₃z^?3, whereC₃ is derived from the polarizability and dielectric function of atom and solid, respectively. The second method uses a hybrid potential, constructed from a Morse potential with shifted zero of energy connected to the asymptotic form,?C₃z^?3, requiring continuity of V and $\text{d}\text{V}\text{d}\text{z}$. The third potential is a Lennard-Jones 3–9 interaction. Results are presented for H and He scattering from LiF and NaF.     

Overdamped motion of interacting particles in general confining potentials: time-dependent and stationary-state analyses

By comparing numerical and analytical results, it is shown that a system of interacting particles under overdamped motion is very well described by a nonlinear Fokker-Planck equation, which can be associated with nonextensive statistical mechanics. The particle-particle interactions considered are repulsive, motivated by three different physical situations: (i) modified Bessel function, commonly used in vortex-vortex interactions, relevant for the flux-front penetration in disordered type-II superconductors; (ii) Yukawa-like forces, useful for charged particles in plasma, or colloidal suspensions; (iii) derived from a Gaussian potential, common in complex fluids, like polymer chains dispersed in a solvent. Moreover, the system is subjected to a general confining potential, ??(x)?=?(??|x| ^z )/z (???>?0, z?>?1), so that a stationary state is reached after a sufficiently long time. Recent numerical and analytical investigations, considering interactions of type (i) and a harmonic confining potential (z?=?2), have shown strong evidence that a q-Gaussian distribution, P(x,t), with q?=?0, describes appropriately the particle positions during their time evolution, as well as in their stationary state. Herein we reinforce further the connection with nonextensive statistical mechanics, by presenting numerical evidence showing that: (a) in the case z?=?2, different particle-particle interactions only modify the diffusion parameter D of the nonlinear Fokker-Planck equation; (b) for z????2, all cases investigated fit well the analytical stationary solution P _st(x), given in terms of a q-exponential (with the same index q?=?0) of the general external potential ??(x). In this later case, we propose an approximate time-dependent P(x,t) (not known analytically for z????2), which is in very good agreement with the simulations for a large range of times, including the approach to the stationary state. The present work suggests that a wide variety of physical phenomena, characterized by repulsive interacting particles under overdamped motion, present a universal behavior, in the sense that all of them are associated with the same entropic form and nonlinear Fokker-Planck equation.     

Internal structure for the pionization spectrum

Peripheral models for the inclusive q_τ² spectrum in the central plateau region are considered in a unified treatment. The highest energy ISR spectrum for p + p → π + X is fit for all q_τ.     

The particle in a box problem inq-quantum mechanics

Aq-deformed,q-Hermitian kinetic energy operator is realised and hence aq-Schrödinger equation (q-SE) is obtained. Theq-SE for a particle confined in an infinite potential box is solved and the energy spectrum is found to have an upper bound.     

The proton recoil spectrum of thermal neutrons decaying in vacuum

The decay of neutrons in an evacuated tangential reactor beam tube is considered. Due to the thermal motion of the decaying neutrons, the proton recoil spectrum in the laboratory system (z-axis parallel to the beam tube) differs from that in the c.m. system. This difference is calculated for a point at the beam tube exit, and expressed in terms of the average¯q _z and of the variance of thez-component of neutron momentum. An activation method for measuring the neutron current densityJ _z (which is closely related to¯q _z) is suggested. The distortion of the spectrum by the finite resolution of an electrostatic spectrometer is also given. The results are to be used in connection with a planned measurement of the proton recoil spectrum in neutron decay.     

Inverse Problem for the Discrete 1D Schrödinger Operator with Small Periodic Potentials

Consider the discrete 1D Schrödinger operator on ? with an odd 2k periodic potential q. For small potentials we show that the mapping: q→ heights of vertical slits on the quasi-momentum domain (similar to the Marchenko-Ostrovski maping for the Hill operator) is a local isomorphism and the isospectral set consists of 2 ^k distinct potentials. Finally, the asymptotics of the spectrum are determined as q→0.     

Quantum deformation of compactified Minkowski space

The quantum GrassmanianG(2|0; ? _q ^4|0 ) of “quantum 2-planes ? _q ^2|0 in the quantum 4-plane ? _q ^4|0 ”, which provides aq-deformation of compactified complexified Minkowski space, is proposed. A quantum analogue of classical Plücker embedding of the usual GrassmanianG(2; ?²) into a non-degenerate quadric in ??⁵ is found.     

Classical Kepler-Coulomb problem on SO(2, 2) hyperboloid

In the present work, the problem of the motion of the classical particle in the Kepler-Coulomb field in three-dimensional hyperbolic space H ₂ ² : z ₂ ⁰ + z ₂ ¹ ? z ₂ ² ? z ₂ ³ = R ² is solved in the framework of Hamilton-Jacobi equation. The requirements for the existence of bounded motion of particle are formulated. The equation of the trajectory of particle is obtained, and it is shown that all the finite trajectories are closed. It is also demonstrated that under the certain values (zero or negative) of the separation constant A the fall of the particle onto the center takes place.     

Surface states in a one-dimensional crystal

The present short paper considers the role of the shape of the surface potential, particularly its long range character, on the existence and energies of the electronic surface states. For that purpose, a one-dimensional crystal is being considered represented by a Kronig-Penney potential

$\text{V}{}_{\mathrm{I}}\text{(z)=?U}{}_0\text{+}\text{∑}\text{n}\text{h}{}^2\text{m}\text{P}\text{a}\text{(s+na)}$

, (P < 0) for z < ? terminated by an image potential of the form V_II (z) = ?Ce²/z(z >?). The physical values of U₀ and a given only two gaps between energies ?U₀ and O. It is found that for a step barrier surface potential at z = ? there is only one surface state in each gap. On the contrary, for an image type potential, the number of surface states depends on the value assumed for ? or C(U₀ = Ce²/?). Varying ? or C, one can modify the shape of the potential from almost a step barrier to an image potential (C = 1) and study the existence of surface states in every case. In particular for C ? 1 (? ? 1 Å) more than one surface state in the higher gap are obtained.     

Pseudospin symmetry hidden in Dirac equation with deformed potential

The pseudospin symmetry hidden in the Dirac equation is found for the case of the deformed potential both in the large and small components of the Dirac wave function. When the pseudospin approximation is good, the absolute values of the amplitudes of the pseudospin pair levels with the unnatural ? _z (unphysical components) almost agree each other both in the large and small components. In contrast to these pseudospin pair levels, the absolute values of the amplitudes with the natural ? _z (physical components) agree for the spin pair levels both in the large and small components. It is confirmed by the numerical analysis based on the relativistic mean-field theory for ¹⁵⁴Sm.