Klein tunneling of a quasirelativistic Bose-Einstein condensate in an optical lattice

A proof-of-principle experiment simulating effects predicted by relativistic wave equations with ultracold atoms in a bichromatic optical lattice that allows for a tailoring of the dispersion relation is reported. We observe the analog of Klein tunneling, the penetration of relativistic particles through a potential barrier without the exponential damping that is characteristic for nonrelativistic quantum tunneling. Both linear (relativistic) and quadratic (nonrelativistic) dispersion relations are investigated, and significant barrier transmission is observed only for the relativistic case.     

Theory of complete orthonormal sets of relativistic tensor wave functions and Slater tensor orbitals of particles with arbitrary spin in position, momentum and four-dimensional spaces

Using the complete orthonormal basis sets of nonrelativistic and quasirelativistic orbitals introduced by the author in previous papers for particles with arbitrary spin the new analytical relations for the 2(2s+1)-component relativistic tensor wave functions and Slater tensor orbitals in position, momentum and four-dimensional spaces are derived, where s=1/2,1,3/2,2,… . The relativistic tensor function sets are expressed through the corresponding nonrelativistic and quasirelativistic orbitals. The analytical formulas for overlap integrals over relativistic Slater tensor orbitals in position space are also derived.     

Cubic dispersion relation for a relativistic backward-waveoscillator

The cubic approximation to the dispersion relation for a relativistic backward-wave oscillator is obtained, and the utility and limits of the approximation are presented. The approximation is obtained by Taylor series expansion of the wave admittance in the dispersion relation for the transverse-magnetic and free-streaming modes of a relativistic, thin, hollow, cylindrical electron beam moving along the axis of a disc-loaded waveguide in a strong axial magnetic field. The resulting cubic dispersion relation yields instability growth rates and frequencies which fall off beyond their maximum more sharply with increasing wavenumber than for the complete dispersion relation. The approximation is found to be quite good near the operating points of contemporary high-power relativistic backward-wave oscillators, namely, for relatively long wavelength and small ratio of Budker's parameter to the relativistic gamma factor of the beam     

Bethe-salpeter equation with instantaneous harmonic oscillator exchange

The Bethe-Salpeter equation in the form due to Cung et al. (Ann. Phys. (N.Y.)98 (1976), 516) is investigated for the special case of instantaneous harmonic oseillator exchange, An exact reduction to a pair of coupled ordinary differential equations for the radial excitations of the ³(J ± 1)_J modes is achieved. The equations in the mass zero case are brought to a form which is quite close to Whittaker's equation. This similarity to Whittaker's equation is exploited in a computer study of the level structure as a function of the quark mass. This study covers the region from a highly relativistic spectrum depending only upon J to the nonrelativistic regime where the spectrum depends only upon L. An expression for the leptonic width of a ³S₁ state in terms of the Bethe-Salpeter wave function is derived and applied to the ψ-family. The effect of relativistic corrections is to reduce the predicted value of the leptonic width compared to the value calculated by assuming nonrelativistic kinematics. It is also shown that the relativistic treatment allows a ³D₁ state to couple directly to a virtual photon.     

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导出了无磁化无碰撞各向同性相对论正负电子对等离子体纵波的色散关系。使用逐次近似的标准技术,对两种极限情况下的纵波色散方程进行了解析研究。将完全色散方程进行变换,使它们完全适用于在需要完全相对论处理的温度范围内进行数值计算。用数值模拟方法得到了用解析方法不能得到的完全的色散曲线。     

Semi-phenomenological neutron density distributions

The simple algebraic form for the nuclear densities designed to incorporate correctly, the two physical requirements, namely, the asymptotic behaviour and the behaviour near the centre, is used to calculate the neutron distributions in several nuclei. Sample neutron densities for 58,64 Ni, ¹¹⁶Sn and ²⁰⁸Pb are presented. The calculation reveals an excellent agreement with the experiment as well as with the relativistic and nonrelativistic microscopic mean field calculations. The use of this algebraic form of the densities in the analytic studies is strongly advocated.     

Relativistic Linear Theory in the Absence of External Fields

The dielectric tensor for a multi-component, homogeneous, field-free relativistic plasma is derived in manifestly covariant form. From the dielectric tensor, linear dispersion relations are obtained explicitly when each component of the plasma is isotropic in its rest frame. If the components are relativistic Maxwellians, these dispersion relations are expressible in terms of the relativistic plasma dispersion function. Special attention is given to the Weible and two-stream instabilities and to the normal modes of a quiescent, hot electron gas. For the last case the dispersion relations are solved numerically and compared against computer simulation data. An appendix applies the formalism to cold plasmas.     

Application of modulus-phase dispersion relations to πN backward scattering

We derive modulus-phase dispersion relations for the πN backward scattering amplitudes. The convenience of these dispersion relations in treating πN backward scattering is pointed out. A numerical evaluation of the π^?p backward scattering phase is also given.     

Phonon dispersion of carbon nanotubes

We present the phonon dispersion relations of single-wall carbon nanotubes calculated within a force-constants approach. By using the full symmetry group of the tubes, we are able to calculate the dispersion relations for any chirality starting from one single carbon atom. We find an overbending in the highest optical branch between 6 and 12 cm⁻¹ independent of the tube diameter. The order of the high-energy modes at the Γ-point differs from the results derived from simple zone folding. The splitting between the two Raman active optical modes with A₁ symmetry at the Γ-point of chiral tubes is ≈4 cm⁻¹ for typical diameters; it increases with decreasing tube diameter.     

Spatial-dispersion and relativistic effects in the optical sum rules

We describe a procedure to take into account the spatial dispersion of the optical excitations in the susceptibility sum rules. We show that this implies that relativistic corrections of the same order must be considered. The final result is a decrease of the total oscillator strength equal to the ratio of the average electron kinetic energy with mc². We propose experiments with synchrotron radiation sources on crystals of heavy elements to observe the described effect. Received 5 June 2001     

Ion cyclotron instabilities in a mildly relativistic hydrogen-deuterium fusion plasma

A dispersion relation for the perpendicular propagation of the electromagnetic ion cyclotron wave around the second harmonic of the deuterium ion gyrofrequency in a mildly relativistic, anisotropic Maxwellian plasma with hydrogen as the majority species and deuterium as the minority component has been derived. The work has been carried out in the frame of reference of the majority hydrogen ions; to these ions the waves at 2Θ_D would be at its own gyrofrequency. Using a small quantityɛ to order all relevant parameters of the plasma, it was possible to derive the dispersion relations in a simple form. To the lowest order the relativistic factors do not enter the dispersion relation. The plasma can now support two modes—one above and the other below the hydrogen gyrofrequency in agreement with the assumptions. This was also verified numerically using a standard root solver thereby justifying the correctness of the ordering scheme. In the next higher order, the dispersion relation is a quartic equation and is sensitively dependent on the relativistic factors. The plasma can now support four modes, both above and below the hydrogen gyrofrequency and consistent with the ordering scheme used. However the modes can now coalesce resulting in complex conjugate roots to the dispersion relation thereby indicating an instability. The advantage of such a scheme is that two dispersion relations — one of which is independent of the relativistic factors and the other which is sensitively dependent on them can be separated out.     

Relativistic interactions without fields or potentials

Employing Poincaré degrees of freedomM ^jk=(¯K,¯J) andP ^k=(E,¯p) transforming linearly (but inhomogeneously) under the action of the Poincaré group we define a number of quantities which we later identify with physical observables. The identifications are consistent with the nonrelativistic limit and with other requirements following from the Poincaré covariance. Next, we treat a free relativistic particle as composed of two interacting parts. Relativistic quantum commutation relations for their Poincaré algebras and a kind of (inverse) relativistic correspondence principle are used to generate (quasi-) classical equations of their relative motion. A simple example based on these ideas is explicitly solved.I am indebted to Prof. B. Laurent, Dr. S. Flodmark, and Prof. I. Fischer-Hjalmars for pointing this out to me.     

A relativistic quark model for mesons based on numerical solutions of the Bethe-Salpeter equation

A study is made to determine if the results of the nonrelativistic quark model can be reproduced by a fully relativistic model of deeply bound $\text{spin}\text{-}\text{1}\text{2}$ quarks. It is found that the relativistic model does not reproduce the nonrelativistic results, even when the quarks have nonrelativistic momenta. However, the model is rather successful in accounting for the known properties of mesons.Numerical solutions to the Bethe-Salpeter equation are obtained for pseudoscalar and vector bound states of equal mass quark-antiquark pairs, with either a scalar, pseudoscalar, or neutral vector exchange interaction. The interaction function corresponds to single particle exchange, with the addition of either one or two regulating terms. It is found that the second regulator allows the internal quark momentum to be nonrelativistic, but that the spinor structure of the wave function remains highly relativistic.Only the scalar interaction can account for the observed spectrum of states. The pseudoscalar interaction yields a vector state of lower mass than the pseudoscalar state, and the vector interaction leads to a vector state which lies approximately one quark mass above the pseudoscalar state. The λ quark is taken as slightly heavier than the p and n, and the perturbation treatment of the mass difference leads to a quadratic mass formula.The decay amplitudes for π, K → μν are calculated, and it is found, independent of parameters, that $\text{?}{}_{\mathrm{\pi }}\text{≈ ?}{}_{\mathrm{K}}$ for either a scalar or vector interaction, in agreement with experiment and in contrast with the nonrelativistic model. The amplitudes for ?^o, ω, φ → e⁺e^?, μ⁺μ^? are also calculated, but in this case the ratios (again parameter independent) are in minor discrepancy with experiment.The question of the additivity of quark amplitudes is examined by calculating (with significant restrictions) the magnetic moments of the vector mesons and the amplitudes for magnetic transitions such as ω → π^oγ. The magnetic moments of the vector mesons have the same (trivial) ratios to each other as in the nonrelativistic model, but they are strongly enhanced over the sum of the quark magnetic moments. The amplitude for magnetic transitions, however, is related to the quark magnetic moments in approximately the same ratio as in the nonrelativistic model.The model is also used to obtain parameter dependent predictions for the masses and decay amplitudes. These predictions are not experimentally correct, but are generally well within an order of magnitude for a wide range of the parameters.The most significant defect discovered of the model is the presence of ghost states (the daughters of the vector mesons, with J^PC = 0^+?) with masses of about 2 BeV.     

Relativistic shell models and meson-exchange currents

We discuss the connection between nonrelativistic pair exchange currents and relativistic mean field theories. Heavy mesons, especially the σ-meson, are shown to yield strong enhancements to the matrix element of “odd” Dirac operators, such as γ and γ₄γ₅, in complete analogy to the strong enhancements obtained in relativistic mean field theories. For isoscalar operators, there is a cancelling contribution from ω-pair diagrams, called “backflow” in relativistic theories, but for isovector operators the equivalent ρ-pair diagram is very weak. Agreement between traditional nonrelativistic calculations and experiment is worsened by the introduction of heavy-meson pair diagrams. The situation can be improved on adding vertex form factors and short-range correlation functions that reduce the heavy-meson pair contribution by about a factor of two. The differences between the two approaches are emphasized and experimental tests are discussed.     

Conformal symmetry of relativistic and nonrelativistic systems and AdS/CFT correspondence

The nonlinear realization of conformal so(2,d) symmetry for relativistic systems and the dynamical conformal so(2,1) symmetry of nonrelativistic systems are investigated in the context of AdS/CFT correspondence. We show that the massless particle in d-dimensional Minkowski space can be treated as the system confined to the border of the AdS_d+1 of infinite radius, while various nonrelativistic systems may be canonically related to a relativistic (massless, massive, or tachyon) particle on the AdS₂ × S^d−1. The list of nonrelativistic systems “unified” by such a correspondence comprises the conformal mechanics model, the planar charge-vortex and three-dimensional charge-monopole systems, the particle in a planar gravitational field of a point massive source, and the conformal model associated with the charged particle propagating near the horizon of the extreme Reissner-Nordström black hole.     

Polariton dispersion of periodic quantum well structures

We studied the polariton dispersion relations of a periodic quantum-well structure with a period in the vicinity of half the exciton resonance wavelength, i.e., the Bragg structure. We classified polariton modes using an approximation of a large number of quantum wells. The polariton effective masses are found to be very small and equal to 10^?3?10^?4 of the free-electron mass.     

A relativistic quark model for baryons

The Bethe-Salpeter equation for the relativistic three quark bound state is solved for an instantaneous interaction in the ladder approximation. The particular solution obtained is valid for simple potentials in both the weak and strong binding situations. The general method for calculating matrix elements for the interaction of the bound state with an external electromagnetic field is presented. Particular attention is paid to the emergence of the nonrelativistic quark model interaction as the lowest order approximation in a perturbative expansion in the inverse quark mass. Relativistic corrections to this approximation are investigated, and their importance is seen to depend on the quark mass. For light quarks these corrections can be large, and to reproduce the proton magnetic moment, for instance, a substantial anomalous moment is necessary. The model has several encouraging features. The form factors with a harmonic potential have an asymptotic k^?2 behavior, and the relativistic corrections to the SU(6) results for the form factors are of the correct order of magnitude.