Maxwell perturbations in a cavity with Robin boundary conditions: two branches of modes with spectrum bifurcation on Schwarzschild black holes

We perform a systematic study of the Maxwell quasinormal spectrum in a mirror-like cavity following the generic Robin type vanishing energy flux principle, by starting with the Schwarzschild black holes in this paper. It is shown that, for black holes in a cavity, the vanishing energy flux principle leads to two different sets of boundary conditions. By solving the Maxwell equations with these two boundary conditions both analytically and numerically, we observe two distinct sets of modes. This indicates that the vanishing energy flux principle may be applied not only to asymptotically anti-de Sitter (AdS) black holes but also to black holes in a cavity. In the analytic calculations, the imaginary part of the Maxwell quasinormal modes are derived analytically for both boundary conditions, which match well with the numeric results. While in the numeric calculations, we complete a thorough study on the two sets of the Maxwell spectrum by varying the mirror radius $$r_m$$ r m , the angular momentum quantum number $$\ell $$ ℓ , and the overtone number N. In particular, we proclaim that the Maxwell spectrum may bifurcate for both modes when the mirror is placed around the black hole event horizon, which is analogous to the spectrum bifurcation effects found for the Maxwell fields on asymptotically AdS black holes. This observation provides another example to exhibit the similarity between black holes in a cavity and the AdS black holes.

The Magnetic Hooke-Newton Transmutation in Momentum Space

The magnetic Hooke-Newton transmutation, which emerges from the transformation design of the quadratic conformal mapping for the system of charged particles moving in the uniform magnetic field, is investigated in the momentum space. It is shown that there are two ways to turn the linear interaction force of the system into the inverse square interaction. The first one, which involves simply applying the mapping to the system, has the spectrum with the Landau levels of even angular momentum quantum number. The second one considers the geometric structure of the mapping as an effective potential which leads us to the transmuted Coulomb system with the novel quantum spectrum. The wave functions of momentum for the bound and scattering states of the transmutation system are given. It is also shown that the effective potential due to the geometric structure can be generalized to a general 2D surface, and the Schrödinger equation of a particle moving on the surface while under the action of the potential can be solved by the form-invariant Schrödinger equation of the free particle. The solution of a particle moving on the hyperbolic surface under the potential is given with the conclusion. The presentation manifests the transformation design of the quantum state, depending mainly on the geometric structure of the representation space, not on the action of the specific potential field. This characteristic makes it possible for us to use the geometric structure of different representation spaces to explore some novel behaviors of quantum particles.

Hydrogen-like Atomic Wavefunctions: A First Sketch

This chapter reiterates the quantum numbers for atomic orbitals, known from general chemistry, and places them into the context developed so far. It is sketched how the Schrodinger equation (SE) for the hydrogen atom hydrogen-like systems (one electron plus a nucleus of charge Z) is set up. When the nucleus is treated as a fixed point charge, the SE is only for the electron. The solutions of the SE can be obtained by switching to spherical polar coordinates, such that the variables are separable in terms of the electron distance from the nucleus, r, and two angles. The kinetic energy of the electron then has a radial component, and an angular component. The latter is associated with the angular momentum quantum number, which is codified by the letters s, p, d, f, and so forth. A step by step solution of the SE is provided later, in chapter 19.

Scattering of charged fermion to two-dimensional wormhole with constant axial magnetic flux

Scattering of charged fermion with $$(1+2)$$ ( 1 + 2 ) -dimensional wormhole in the presence of constant axial magnetic flux is explored. By extending the class of fermionic solutions of the Dirac equation in the curved space of wormhole surface to include normal modes with real energy and momentum, we found a quantum selection rule for the scattering of fermion waves to the wormhole. The newly found momentum–angular momentum relation implies that only fermion with the quantized momentum $$k=m'/a\sqrt{q}$$ k = m ′ / a q can be transmitted through the hole. The allowed momentum is proportional to an effective angular momentum quantum number $$m'$$ m ′ and inversely proportional to the radius of the throat of the wormhole a. Flux dependence of the effective angular momentum quantum number permits us to select fermions that can pass through according to their momenta. A conservation law is also naturally enforced in terms of the unitarity condition among the incident, reflected, and transmitted waves. The scattering involving quasinormal modes (QNMs) of fermionic states in the wormhole is subsequently explored. It is found that the transmitted waves through the wormhole for all scenarios involving QNMs are mostly suppressed and decaying in time. In the case of QNMs scattering, the unitarity condition is violated but a more generic relation of the scattering coefficients is established. When the magnetic flux $$\phi =mhc/e$$ ϕ = m h c / e , i.e., quantized in units of the magnetic flux quantum hc/e, the fermion will tunnel through the wormhole with zero reflection.

Determining dominant partial waves in photoproduction via moment analysis

Important insights into the excitation spectra of baryons are provided by measurements of polarization observables in reactions that involve particles with spin. The photoproduction of a single pseudoscalar meson constitutes an example-reaction that has been under intense investigation recently. We present the basic method of moment-analysis for pseudoscalar meson photoproduction, in which just the angular distributions are analyzed. Using this method, the total angular momentum quantum number of the dominant partial waves contributing in the data can be extracted quickly. Furthermore, the Legendre-coeffcients extracted from the angular distributions show interesting composition-patterns in terms of multipoles and allow for instructive comparisons to models. In this contribution, recent results for moment analyses of polarization data for the photoproduction of pions and eta-mesons are shown.

The dynamics of the Br + HgBr (v = 0, j = 0) → Br2 + Hg reaction based on quasi-classical trajectory calculations

To investigate the dynamics mechanism of the Br + HgBr → Br2 + Hg reaction, the quasi-classical trajectory calculations are performed on Balabanov’s potential energy surface (PES) of ground electronic state. Both the scalar and vector properties are investigated to recognize the dynamics of the title reaction. Reaction probability for the total angular momentum quantum number J = 0 is determined at the collision energies (denoted as Ec) in a range of 1–25 kcal/mol, and the product vibrational distributions are given and compared between Ec = 20 and 40 kcal/mol. Other calculation values characterizing product polarizations including polarization-dependent differential cross sections (PDDCSs), distributions of P(θr), P([Formula: see text]), and P(θr, [Formula: see text]), are all discussed and compared between the two different collision energies in detail to analyze the alignment and orientation characteristics. It is revealed that the products prefer forward scattering and the PDDCSs are anisotropic in the whole range of the scattering angle. The product rotational angular momentum j′ shows a tendency to align perpendicular to the reagent relative velocity k. In fact, the product polarization of the title reaction is weak at both collision energies. In terms of horizontal comparison, the alignment is slightly stronger but the orientation is even less remarkable at higher collision energy.

New Bound States for Modified Vibrational-Rotational Structure of Supersingular Plus Coulomb Potential of the Schrödinger Equation in One-Electron Atoms

In this study, three-dimensional modified time-independent Schrödinger equation of modified vibrational-rotational structure of supersingular plus Coulomb (v.r.s.c) potential was solved using Boopp’s shift method instead to apply star product, in the framework of both noncommutativity three dimensional real space and phase (NC: 3D-RSP). We have obtained the explicit energy eigenvalues for ground state and first excited state for interactions in one-electron atoms. Furthermore, the obtained corrections of energies are depended on infinitesimal parameters (Θ,χ) and (θ,σ) which are induced by position-position and momentum-momentum noncommutativity, respectively, in addition to the discreet atomic quantum numbers: j=l±1/2,s=±1/2,l and the angular momentum quantum number m. We have also shown that, the usual states in ordinary three dimensional spaces for ordinary vibrational-rotational structure of supersingular plus Coulomb potential are canceled and has been replaced by new degenerated 2(2l+1) sub-states in the extended new quantum symmetries of (NC: 3D-RSP).

Approximations for Brillouin and its reverse function

Purpose The mathematical complexity of the BJ(x) Brillouin function makes it unsuitable for most calculations and its application difficult for computer programming in magnetism. Here, its approximation with the tanh function is proposed to ease the mathematical operations for most cases. The approximation works with good accuracy, acceptable in practical calculations. This approximation has already formed the foundation of the “hyperbolic model” in magnetism for the study of hysteretic phenomena. The reversal of the Brillouin function is an important but difficult mathematical problem for practical purposes. Here, a proposal has been put forward for an easy approximation using an analytical expression. This provides a good workable solution for the BJ(x)−1 function dependent on J, the angular momentum quantum number of the material used. The proposed approximation is applicable within the working range of practical applications. The paper aims to discuss these issues. Design/methodology/approach The multi-variant Brillouin function is closely approximated by the tanh function to ease calculations. Its mathematically unsolved reversed function is approximated by a simple analytical expression with a good working accuracy. Findings The Brillouin function and its reversal can be approximated for practical users mostly for professionals working in Magnetism. Research limitations/implications Most if not all practical problems in Magnetism can be solved within the limitations of the two approximations. Practical implications Both proposed functions can ease the mathematical problems faced by researchers and other users in Magnetism. Social implications Ease the frustration of most users working in the field of Magnetism. Originality/value The application of the tanh function for replacing the Brillouin function led to the creation of the hyperbolic model of hysteresis. To the author's knowledge, the reverse function was mathematically only solved in 2015 with a vastly complicated mathematics, and is hardly suitable for practical calculations in Magnetism. The proposed simple expression can be very useful for theorists and experimental scientists.

A study of shell model neutron states in 207,209Pb using the generalized Woods–Saxon plus spin-orbit potential

The experimental binding energies of single-particle and single-hole neutron states belonging to neutron shells that extend from [Formula: see text] = 126–184 and 82–126, respectively, have been reproduced by solving the Schrödinger equation with a potential that has two components: the generalized Woods–Saxon (GWS) potential and the spin-orbit (SO) coupling term. The GWS potential contains the traditional WS potential plus a term (SU) whose intensity reaches a maximum in the nuclear surface. Our results indicate the existence of an explicit relationship between the strength of the SU potential and the orbital angular momentum quantum number [Formula: see text] of the state. This dependence has been used to make reasonable predictions for the excitation energy centroids of states located inside and outside the neutron shells investigated. Comparisons are made with results reported in previous investigations.