Boundary Conditions Of Wave Functions Research Articles

Pseudospin and spin symmetry in the relativistic generalized Woods-Saxon potential including Coulomb-like tensor potential

The Dirac Equation is solved approximately for relativistic generalized Woods-Saxon potential including Coulomb-like tensor potential in exact pseudospin and spin symmetry limits. The bound states energy eigenvalues are found by using wavefunction boundary conditions, and corresponding radial wavefunctions are obtained in terms of hypergeometric function. Some numerical examples are given for the dependence of bound states energy eigenvalues on quantum numbers and potential parameters.

5d Dirac fermion on quantum graph

In this paper, we investigate a five-dimensional Dirac fermion on a quantum graph that consists of a single vertex and N loops. We find that the model possesses a rich structure of boundary conditions for wavefunctions on the quantum graph and they can be classified into distinct categories. It is then shown that there appear degenerate four-dimensional chiral massless fermions in the four-dimensional mass spectrum. We briefly discuss how our model could naturally solve the problems of the fermion generation, the fermion mass hierarchy and the origin of the CP-violating phase.

Reexamining cluster radioactivity in trans-lead nuclei with consideration of specific density distributions in daughter nuclei and clusters

We further investigate the cluster emission from heavy nuclei beyond the lead region in the framework of the preformed cluster model. The refined cluster-core potential is constructed by the double-folding integral of the density distributions of the daughter nucleus and the emitted cluster, where the radius or the diffuseness parameter in the Fermi density distribution formula is determined according to the available experimental data on the charge radii and the neutron skin thickness. The Schr\odinger equation of the cluster-daughter relative motion is then solved within the outgoing Coulomb wave-function boundary conditions to obtain the decay width. It is found that the present decay width of cluster emitters is clearly enhanced as compared to that in the previous case, which involved the fixed parametrization for the density distributions of daughter nuclei and clusters. Among the whole procedure, the nuclear deformation of clusters is also introduced into the calculations, and the degree of its influence on the final decay half-life is checked to some extent. Moreover, the effect from the bubble density distribution of clusters on the final decay width is carefully discussed by using the central depressed distribution.

An artificial boundary approach for short-ranged interactions

Real physical systems are only understood, experimentally or theoretically, to a finite resolution so in their analysis there is generally an ignorance of possible short-range phenomena. It is also well-known that the boundary conditions of wavefunctions and fields can be used to model short-range interactions when those interactions are expected, a priori. Here, a real-space approach is described wherein an artificial boundary of ignorance is imposed to explicitly exclude from analysis the region of a system wherein short-distance effects may be obscure. The (artificial) boundary conditions encode those short-distance effects by parameterizing the possible UV completions of the wavefunction. Since measurable quantities, such as spectra and cross sections, must be independent of the position of the artificial boundary, the boundary conditions must evolve with the radius of the boundary in a particular way. As examples of this approach, an analysis is performed of the non-relativistic free particle, harmonic oscillator, and Coulomb potential, and some known results for point-like (contact) interactions are recovered, however from a novel perspective. Generically, observables differ from their canonical values and symmetries are anomalously broken compared to those of idealized models. Connections are made to well-studied physical systems, such as the binding of light nuclei and cold atomic systems. This method is arguably more physically transparent and mathematically easier to use than other techniques that require the regularization and renormalization of delta-function potentials, and may offer further generalizations of practical use.

Novel Understanding of Electron States Architecture and Its Dimensionality in Semiconductors

Some important insights into the electron-states-architecture (ESA) and its dimensionality (from 3 to 0) in a semiconductor (or generally crystalline) material are obtained. The self-consistency of the set of density of states (DOS) expressions with different dimensionalities is remediated through the clarification and rearrangement of the wave-function boundary conditions for working out the eigenvalues in the wave vector space. The actually too roughly observed and theoretically unpredicted critical points for the dimensionality transitions referring to the integer ones are revealed upon an unusual assumption of the intrinsic energy-level dispersion (ELD). The ELD based quantitative physical model had been established on an immediate instinct at the very beginning and has been properly modified afterwards. The uncertainty regarding the relationship between the de Broglie wavelength of electrons and the dimensionality transitions, seeming somewhat mysterious before, is consequentially eliminated. The effect of the material dimensions on the ELD width is also predicted and has been included in the model. The continuous evolution of the ESA dimensionality is convincingly and comprehensively interpreted and thus the area of the fractional ESA dimensionalities is opened. Another new assumption of the spatial extension shrinkage (SES) closely related to the ELD has also been made and thus the understanding of the behavior of an electron or, in a general sense, a particle has become more comprehensive. This work would manifest itself a new basis for further development of nanoheterostructures (or low dimensional heterostructures including the quantum wells, quantum wires, quantum dots and especially the hetero-dimensional structures). Expected should also be the possible inventions of some novel electronic and optoelectronic devices. More basically, it leads to a new quantum mechanical picture, the essential modifications of Schrödinger equation and Newtonian equation that give rise to a full cosmic-scope picture, and a super-low-speed relativity assumption.

Linear dynamic polarizability and the absorption spectrum of an exciton in a quantum ring in a magnetic field

The problem of an electron–hole system interacting through a contact potential and moving in a one-dimensional quantum ring threaded by an Aharonov–Bohm flux is considered, with respect to the system's energetics as well as its optical properties. An exact analytical expression for the energy spectrum is derived using a straightforward method based on boundary conditions for wavefunctions and their derivatives along the ring. The optical properties of this exciton system, namely linear dynamic polarizability and the absorption spectrum, are investigated and certain unusual features are demonstrated. It is shown, for example, that for special values of the magnetic flux there are energies in the spectrum that correspond to the dark excitonic states.

Calculation of Energy and Other Properties of Muonic Helium Atom Using Boundary Conditions of Wave Function

The properties of muonic helium atom (4He+2μ−e−) in ground state are considered. In this work, the energy and average distance between particles have been obtained using a wave function, which satisfies boundary conditions. It is shown that the obtained energy are very close to the values calculated by others. But the small differences of the expectation values of r2n are due to the incorporated boundary conditions in proposed wave function and are expected.

Neutral Möbius Aromatics: Derivatives of the Pyrrole Congener Aza[11]annulene as Promising Synthetic Targets

AbstractAmino‐ and methyl‐disubstituted derivatives of an aromatic 12π electron pyrrole homologue Möbius 1H‐aza[11]annulene are predicted computationally at the B3LYP/6‐31G* level of theory to be distinctly aromatic and a neutral ground state on its potential surface. The parent annulene has a nonaromatic conformation, which is 6.0 kcal mol–1 lower in energy at the CCSD(T)/D95*//B3LYP/6‐31G* level of theory than a structurally very similar Möbius conformation. After derivatization, however, geometry optimizations of the two lowest‐energy – Möbius and non‐Möbius – heteroannulene conformations both converged two a single Möbius minimum, whereas two (Möbius and non‐Möbius) distinct difluoro‐ substituted species could be located 1.8 kcal mol–1 apart in energy at the B3LYP level. The Möbius minimum is 7.5 kcal mol–1 lower in energy at the coupled cluster level than a “Hückel topology” form derived from a known bridged 1H‐[4,9]methano[11]annulene. Several possible reactive modes are investigated in order to assess the relative stability of the Möbius parent annulene. In addition, we report formal novel examples for charged and neutral Möbius triplet aromatic annulenic and heteroannulenic systems, some of which also exhibit distinct aromatic properties. Further fundamental insights into the phenomenon of “Möbius aromaticity” are suggested. The aromaticity of Möbius annulenes is for the first time linked to wavefunction boundary conditions. (© Wiley‐VCH Verlag GmbH & Co. KGaA, 69451 Weinheim, Germany, 2008)

Analytical solutions for the E ⊗ e dynamic Jahn–Teller problem in the strong coupling limit

Correct boundary conditions for wavefunctions in the E ⊗ e dynamic Jahn–Teller problem explicitly is considered to obtain approximate analytical solutions in the strong coupling limit. We demonstrate that the pseudorotational energy j 2/(2 g 2), where g is the dimensionless vibronic coupling constant, and j total angular momentum: j = 1/2, 3/2, …, in the conventional strong coupling expression for the vibronic levels of the lower sheet is exact. However, it is found that in the conventional treatment a term of the order of 1/ g 2 is missing. The solutions for the upper sheet with correct boundary conditions gives those which are entirely different from the corresponding Slonczewski’s solutions.

Nonstandard Boundary Conditions in Quantum Mechanics

The class of boundary conditions for wave functions which follow from the quantum mechanical continuity equation for the probability density and the probability current is considered.

Calculation of the ground‐state energy and studies of other properties of exotic helium atoms with the use of boundary conditions of wave function

AbstractThe minimum energy and average distance between particles of doubly muonic helium atoms Heμμ (He2+ + 2μ), are calculated with the use of a wave function that satisfies boundary conditions such as the behavior of the wave function when two particles are close to each other or far away from each other. In this wave function, the muon–muon correlation in doubly muonic helium atoms is described to arrive at the correct behavior for r12 tending to zero and infinity. It is shown that the obtained results are very close to the values calculated by others. Finally, to confirm the method and results, calculated values are compared with a similar electronic system, and it is shown that the small differences in the energies of Heμμ and He are due to the reduced masses, as expected. In addition to being very simple, the proposed wave function provides relatively accurate values for the energy and expectation values of r2n, emphasizing the importance of the local properties of the wave functions. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

Stratified reduction of many-body kinetic energy operators

The center-of-mass system of many bodies admits a natural action of the rotation group SO(3). According to the orbit types for the SO(3) action, the center-of-mass system is stratified into three types of strata. The principal stratum consists of nonsingular configurations for which the isotropy subgroup is trivial, and the other two types of strata consist of singular configurations for which the isotropy subgroup is isomorphic with either SO(2) or SO(3). Depending on whether the isotropy subgroup is isomorphic with SO(2) or SO(3), the stratum in question consists of collinear configurations or of a single configuration of the multiple collision. It is shown that the kinetic energy operator is expressed as the sum of rotational and vibrational energy operators on each stratum except for the stratum of multiple collision. The energy operator for nonsingular configurations has singularity at singular configurations. However, the singularity is not essential in the sense that both of the rotational and vibrational energy integrals have a finite value. This can be proved by using the boundary conditions of wave functions at singular configurations for three-body systems, for simplicity. It is shown, in addition, that the energy operator for collinear configurations has also singularity at the multiple collision, but the singularity is not essential either in the sense that the kinetic energy integral is not divergent at the multiple collision. Reduction procedure is applied to the respective energy operators for the nonsingular and the collinear configurations to obtain respective reduced operators, both of which are expressed in terms of internal coordinates.

Transfer matrix formalism applied to above barrier energy states in strained multiple quantum wells

We have used the effective mass approximation in a transfer matrix formalism to describe both confined and unconfined (above barrier) states in strained InGaAs/GaAs multiple quantum well (MQW) structures. We deduce the transfer matrices for a pseudomorphic MQW structure from the envelope wavefunction boundary conditions, modified to account for the discontinuities in the lattice constant. In the calculations we also take into account the finite width of the cap-layer, and the finite height of the vacuum potential. The potential profiles of the InGaAs/GaAs MQWs are derived using the calculated positions of the band-edges, taken from the model-solid theory. Photoluminescence excitation (PLE) spectroscopy has been performed on a series of InGaAs/GaAs MQWs (4-6 periods) at 11 K. We observed excitonic peaks at energies both below and above the GaAs free exciton. The values of the calculated transition energies are in good agreement with the experimental peaks.

A transfer matrix approach to conductance in quantum waveguides

A transfer matrix approach is presented for the study of electron conduction in an arbitrarily shaped cavity structure embedded in a quantum wire. Using the boundary conditions for wave functions, the transfer matrix at an interface with a discontinuous potential boundary is obtained for the first time. The total transfer matrix is calculated by multiplication of the transfer matrix for each segment of the structure as well as numerical integration of coupled second-order differential equations. The proposed method is applied to the evaluation of the conductance and the electron probability density in several typical cavity structures. The effect of the geometrical features on the electron transmission is discussed in detail. In the numerical calculations, the method is found to be more efficient than most of the other methods in the literature and the results are found to be in excellent agreement with those obtained by the recursive Green's function method.

Self-consistent calculation of electron density and muon hyperfine field in transition metals and their compounds

The use of finite cluster models to represent the electronic structure of solids in the framework of self-consistent local density theory is reviewed. The embedding problem is discussed, and practical variational approaches to treating crystalline potentials and wavefunction boundary conditions are presented. Hyperfine fields atμ+ sites in Fe, Ni, and Co are found to result from a delicate balance between negative contributions of deep-lying (bound, paired) states and positive contributions due to polarization of band levels near the Fermi energy. The nature of muon screening, and the potential forμ+ in Cu and Al are briefly considered. Finally, we explore the likely binding sites and bonding mechanisms forμ+ in the defect compound VO x .