Refined comparison theorems for the Dirac equation in d dimensions

A single spin‐1/2 particle obeys the Dirac equation in spatial dimension and is bound by an attractive central monotone potential which vanishes at infinity (in one dimension the potential is even). This work refines the relativistic comparison theorems which were derived by Hall 1 . The new theorems allow the graphs of the two comparison potentials and to crossover in a controlled way and still imply the spectral ordering for the eigenvalues at the bottom of each angular momentum subspace. More specifically in a simplest case we have: in dimension , if , then ; and in dimensions, if , where and , then .

     

Casimir effect modified surface energy of a nanocavity in homogeneous media

We determine the regularized van der Waals contribution to pressure within a spherical cavity of vapor in a homogeneous, isotropic, infinite medium. The spherical Hamaker function, , has been defined, for the first time, in contrast to the conventional Hamaker function for planar surfaces, . For the materials under consideration, the pressure inside the cavity varies as , where a is the radius of the cavity. For radii below a transition radius, the surface energy (or surface tension) becomes size dependent and could have important implications for homogeneous nucleation of nanosized bubbles in liquids, as well as cavitation of bubbles.

     

Comparative analysis of electric field influence on the quantum wells with different boundary conditions.

Analytical solutions of the Schrödinger equation for the one‐dimensional quantum well with all possible permutations of the Dirichlet and Neumann boundary conditions (BCs) in perpendicular to the interfaces uniform electric field are used for the comparative investigation of their interaction and its influence on the properties of the system. Limiting cases of the weak and strong voltages allow an easy mathematical treatment and its clear physical explanation; in particular, for the small , the perturbation theory derives for all geometries a linear dependence of the polarization on the field with the BC‐dependent proportionality coefficient being positive (negative) for the ground (excited) states. Simple two‐level approximation elementary explains the negative polarizations as a result of the field‐induced destructive interference of the unperturbed modes and shows that in this case the admixture of only the neighboring states plays a dominant role. Different magnitudes of the polarization for different BCs in this regime are explained physically and confirmed numerically. Hellmann‐Feynman theorem reveals a fundamental relation between the polarization and the speed of the energy change with the field. It is proved that zero‐voltage position entropies are BC independent and for all states but the ground Neumann level (which has ) are equal to while the momentum entropies depend on the edge requirements and the level. Varying electric field changes position and momentum entropies in the opposite directions such that the entropic uncertainty relation is satisfied. Other physical quantities such as the BC‐dependent zero‐energy and zero‐polarization fields are also studied both numerically and analytically. Applications to different branches of physics, such as ocean fluid dynamics and atmospheric and metallic waveguide electrodynamics, are discussed.

     

Equivalence and hermiticity of Dirac Hamiltonians in the Kerr gravitational field

In the paper, for the Kerr field, we prove that Chandrasekhar's Dirac Hamiltonian and the self‐adjoint Hamiltonian with a flat scalar product of the wave functions are physically equivalent. Operators of transformation of Chandrasekhar's Hamiltonian and wave functions to the η representation with a flat scalar product are defined explicitly. If the domain of the wave functions of Dirac's equation in the Kerr field is bounded by two‐dimensional surfaces of revolution around the z axis, Chandrasekhar's Hamiltonian and the self‐adjoint Hamiltonian in the η representation are Hermitian with equality of the scalar products, .

     

Relic photon temperature versus redshift and the cosmic neutrino background

Presuming that CMB photons are described by the deconfining phase of an SU(2) Yang‐Mills theory with the critical temperature for the deconfining‐preconfining phase transition matching the present CMB temperature K (SU(2)_CMB), we investigate how CMB temperature T connects with the cosmological scale factor a in a Friedmann‐Lemaître‐Robertson‐Walker Universe. Owing to a violation of conformal scaling at late times, the tension between the (instantaneous) redshift of reionisation from CMB observation () and quasar spectra () is repealed. Also, we find that the redshift of CMB decoupling moves from to which questions ΛCDM cosmology at high redshifts. Adapting this model to the conventional physics of three flavours of massless cosmic neutrinos, we demonstrate inconsistency with the value N_eff ～ 3.36 extracted from Planck data. Interactions between cosmic neutrinos and the CMB implies a common temperature T of (no longer separately conserved) CMB and neutrino fluids. N_eff ～ 3.36 then entails a universal, temperature induced cosmic neutrino mass with . Our above results on z_re and z_dec, derived from SU(2)_CMB alone, are essentially unaffected when including such a neutrino sector.

     

Theory of the Robin quantum wall in a linear potential. I. Energy spectrum,polarization and quantum‐information measures

Information‐theoretical concepts are employed for the analysis of the interplay between a transverse electric field applied to a one‐dimensional surface and Robin boundary condition (BC), which with the help of the extrapolation length Λ zeroes at the interface a linear combination of the quantum mechanical wave function and its spatial derivative, and its influence on the properties of the structure. For doing this, exact analytical solutions of the corresponding Schrödinger equation are derived and used for calculating energies, dipole moments, position and momentum quantum information entropies and their Fisher information and and Onicescu information energies and counterparts. It is shown that the weak (strong) electric field changes the Robin wall into the Dirichlet, (Neumann, ), surface. This transformation of the energy spectrum and associated waveforms in the growing field defines an evolution of the quantum‐information measures; for example, it is proved that for the Dirichlet and Neumann BCs the position (momentum) quantum information entropy varies as a positive (negative) natural logarithm of the electric intensity what results in their field‐independent sum . Analogously, at and the position and momentum Fisher informations (Onicescu energies) depend on the applied voltage as () and its inverse, respectively, leading to the field‐independent product (). Peculiarities of their transformations at the finite nonzero Λ are discussed and similarities and differences between the three quantum‐information measures in the electric field are highlighted with the special attention being paid to the configuration with the negative extrapolation length.

     

Scalar Aharonov‐Bohm‐type effect induced by Lorentz symmetry breaking effects

A possible scenario of the Lorentz symmetry violation is discussed based on the arising of geometric quantum phases yielded by the effects of the Lorentz symmetry violation in the CPT‐even gauge sector of Standard Model Extension. Analogues of the Anandan quantum phase and the scalar Aharonov‐Bohm effect for a neutral particle [J. Anandan, Phys. Lett. A 138 , 347 (1989)] are obtained from the parity‐odd sector of the tensor . Moreover, we build quantum holonomies associated with the analogue of the Anandan quantum phase and discuss a possible analogy with the geometric quantum computation [A. Ekert et al., J. Mod. Opt. 47 , 2501 (2000)].

     

Nodal theorems for the Dirac equation in d ≥ 1 dimensions

A single particle obeys the Dirac equation in spatial dimensions and is bound by an attractive central monotone potential that vanishes at infinity. In one dimension, the potential is even, and monotone for The asymptotic behavior of the wave functions near the origin and at infinity are discussed. Nodal theorems are proven for the cases and , which specify the relationship between the numbers of nodes n₁ and n₂ in the upper and lower components of the Dirac spinor. For , whereas for if and if where and This work generalizes the classic results of Rose and Newton in 1951 for the case Specific examples are presented with graphs, including Dirac spinor orbits      

Quantum information entropies for an asymmetric trigonometric Rosen–Morse potential

Shannon entropy for the position and momentum eigenstates of an asymmetric trigonometric Rosen–Morse potential for the ground and first excited states is evaluated. The position and momentum information entropies and are calculated numerically. Also, it is found that is obtained analytically and increases with the potential depth and width. Some interesting features of the information entropy densities and are demonstrated graphically. The Bialynicki‐Birula–Mycielski inequality is also tested and found to hold good.     

Theory of the Robin quantum wall in a linear potential. II. Thermodynamic properties

A theoretical analysis of the thermodynamic properties of the Robin wall characterized by the extrapolation length Λ in the electric field that pushes the particle to the surface is presented both in the canonical and two grand canonical representations and in the whole range of the Robin distance with the emphasis on its negative values which for the voltage‐free configuration support negative‐energy bound state. For the canonical ensemble, the heat capacity at exhibits a nonmonotonic behavior as a function of the temperature T with its pronounced maximum unrestrictedly increasing for the decreasing fields as and its location being proportional to . For the Fermi‐Dirac distribution, the specific heat per particle is a nonmonotonic function of the temperature too with the conspicuous extremum being preceded on the T axis by the plateau whose magnitude at the vanishing is defined as , with N being a number of the particles. The maximum of is the largest for and, similar to the canonical ensemble, grows to infinity as the field goes to zero. For the Bose‐Einstein ensemble, a formation of the sharp asymmetric feature on the ‐T dependence with the increase of N is shown to be more prominent at the lower voltages. This cusp‐like dependence of the heat capacity on the temperature, which for the infinite number of bosons transforms into the discontinuity of , is an indication of the phase transition to the condensate state. Some other physical characteristics such as the critical temperature and ground‐level population of the Bose‐Einstein condensate are calculated and analyzed as a function of the field and extrapolation length. Qualitative and quantitative explanation of these physical phenomena is based on the variation of the energy spectrum by the electric field.     

Mechanical detection of ultraslow,Debye‐like Li‐ion motions in LiAlO single crystals

Single crystalline LiAlO is known as a very poor ion conductor. Thus, in its crystalline form it unequivocally disqualifies itself from being a powerful solid electrolyte in modern energy storage systems. On the other hand, its interesting crystal structure proves beneficial to sharpen our understanding of Li ion dynamics in solids which in return might influence application‐oriented research. LiAlO allows us to apply and test techniques that are sensitive to extremely slow Li ion dynamics. This helps us clarifying their diffusion behaviour from a fundamental point of view. Here, we combined two techniques to follow Li ion translational hopping in LiAlO that can be described by the same physical formalism: dynamic mechanical relaxation and electrical relaxation, i.e., ionic conductivity measurements. Via both methods we were able to track the same transport mechanism in LiAlO. Moreover, this enabled us to directly probe extremely slow Li exchange rates at temperatures slightly above 430 K. The results were compared with recent insights from nuclear magnetic resonance spectroscopy. Altogether, an Arrhenius‐type Li diffusion process with an activation energy of ca. 1.12 eV was revealed over a large dynamic range covering 10 orders of magnitude, i.e., spanning a dynamic range from the nano‐second time scale down to the second time scale.

     

An analogue of the quantum Hall conductivity for a magnetic quadrupole moment

The quantum dynamics of a moving particle with a magnetic quadrupole moment that interacts with electric and magnetic fields is introduced. Then, it is discussed which conditions the external fields must satisfy so that an analogue of the Landau quantization can be obtained. Finally, by dealing with the lowest Landau level associated with the magnetic quadrupole system, an analogue of the quantum Hall conductivity is obtained.

     

Finsler geometry of topological singularities for multi‐valued fields: Applications to continuum theory of defects

Topological singularity in a continuum theory of defects and a quantum field theory is studied from a viewpoint of differential geometry. The integrability conditions of singularity (Clairaut‐Schwarz‐Young theorem) are expressed by a torsion tensor and a curvature tensor when a Finslerian intrinsic parallelism holds for the multi‐valued function. In the context of the quantum field theory, the singularity called an extended object is expressed by the torsion when the intrinsic parallelism is related to the spontaneous breakdown of symmetry. In the continuum theory of defects, the path‐dependency of point and line defects within a crystal is interpreted by the non‐vanishing condition of torsion tensor in a non‐Riemannian space osculated from the Finsler space, and the domain is not simply connected. On the other hand, for the rotational singularity, an energy integral (J‐integral) around a disclination field is path‐independent when a nonlinear connection is single‐valued. This means that the topological expression for the sole defect (Gauss‐Bonnet theorem with genus ) is understood by the integrability of nonlinear connection.

     

Fascinating journeys into blue light (Nobel Lecture)

Isamu Akasaki is known for inventing the bright gallium nitride (GaN) p‐n junction blue LED in 1989 and subsequently the high‐brightness GaN blue LED. Together with Shuji Nakamura and Hiroshi Amano, he is one of the three recipients of the 2014 Nobel Prize in Physics. In his Nobel Lecture, he describes the historical progress that led to the invention of the first p‐n junction blue/UV LED and related optical devices. ***

     

Extra dimensions and phase experiments

In modern Kaluza‐Klein theories which successfully unify gravity, electromagnetism and a scalar field, null geodesics in five dimensions lead to simplified expressions for phase shifts in four‐dimensional spacetime. It might be possible to test for an extra dimension by experiments such as those where neutron interferometry is used to measure the Aharonov‐Bohm effect.

     

A generalized G‐function for the Quantum Rabi Model

The analytical solution of the quantum Rabi model is based on a transcendental function , the zeros of which determine the eigenenergies. is generalized here to a function , which allows a much better numerical control of the high‐energy part of the spectrum by an appropriate choice of the complex parameter z. Additionally, it is shown that all zeros of correspond to eigenvalues of the Hamiltonian as well as the zeros of for imaginary z.     

A particular solution of Heun equation for Hulthen and Woods‐Saxon potentials

In this article a particular solution of Heun equation is derived by making use of the Nikiforov‐Uvarov (NU) method which provides exact solutions for general hypergeometric equation and eigenvalues together with eigenfunctions of the Heun equation for this particular solution are obtained. One to one correspondence (isomorphism) of the aforesaid equation with the radial Schrödinger equation is emphasized and also physical counterparts of the parameters in this equation are put forward by introducing solutions for two different potential functions (Hulthen and Woods‐Saxon potentials).

     

X‐ray and neutron scattering on disordered nanosize clusters: a case study of lead‐zirconate‐titanate solid solutions

Defects and frequently used defect models of solids are reviewed. Signatures for identifying the disorder from x‐ray and neutron scattering data are given. To give illustrative examples how technologically important defects contribute to x‐ray and neutron scattering numerical method able to treat non‐periodical solids possessing several simultaneous defect types is given for simulating scattering in nanosize disordered clusters. The approach takes particle size, shape, and defects into account and isolates element specific signals. As a case study a statistical approximation model for lead‐zirconate titanate [Pb(Zr_xTi)O₃, PZT] is introduced. PZT is a material possessing several defect types, including substitutional, displacement and surface defects. Spatial composition variation is taken into account by introducing a model in which the edge lengths of each cell depend on the distribution of Zr and Ti ions in the cluster. Spatially varying edge lengths and angles is referred to as microstrain. The model is applied to compute the scattering from ellipsoid shaped PZT clusters and to simulate the structural changes as a function of average composition. Two‐phase co‐existence range, the so called morphotropic phase boundary composition is given correctly. The composition at which the rhombohedral and tetragonal cells are equally abundant was . Selected x‐ray and neutron Bragg reflection intensities and line shapes were simulated. Examples of the effect of size and shape of the scattering clusters on diffraction patterns are given and the particle dimensions, computed through Scherrer equation, are compared with the exact cluster dimensions. Scattering from two types of 180° domains in spherical particles, one type assigned to Ti‐rich PZT and the second to the MPB and Zr‐rich PZT, is computed. We show how the method can be used for modelling polarization reversal.

     

THz and above THz electron or hole oscillations in DNA dimers and trimers

A non conventional source or receiver of THz and above THz electromagnetic radiation is proposed. Specifically, electron or hole oscillations in DNA dimers (two interacting DNA base‐pairs or monomers) are predicted, with frequency in the range 0.25–100 THz (period 10–4000 fs) i.e. potentially absorbing or emitting electromagnetic radiation mainly in the mid‐ and far‐infrared with wavelengths ≈ 3–1200 μm. The efficiency of charge transfer between the two monomers which make up the dimer is described with the maximum transfer percentage p  and the pure maximum transfer rate . For dimers made of identical monomers , but for dimers made of different monomers . The investigation is extended to DNA trimers (three interacting DNA base‐pairs or monomers). For trimers made of identical monomers the carrier oscillates periodically with 0.5–33 THz ( 30–2000 fs); for 0 times crosswise purines , for 1 or 2 times crosswise purines . For trimers made of different monomers the carrier movement may be non periodic. Generally, increasing the number of monomers above three, the system becomes more complex and periodicity is lost; even for the simplest tetramer the carrier movement is not periodic.