Three–dimensional effective mass Schrödinger equation: harmonic and Morse-type potential solutions

In this work, a scheme to generate exact wave functions and eigenvalues for the spherically symmetric three-dimensional position-dependent effective mass Schrödinger equation is presented. The methodology is implemented by means of separation of variables and point canonical transformations that allow to recognize a radial dependent equation with important differences as compared with the one-dimensional position dependent mass problem, which has been widely studied. This situation deserves to consider the boundary conditions of the emergent problem. To obtain specific exact solutions, the methodology requires known solutions of ordinary one-dimensional Schrödinger equations. We have preferred those applications that use the harmonic oscillator and the Morse oscillator solutions.     

High Algebraic Order Methods for the Numerical Solution of the Schrödinger Equation

Abstract

A family of new hybrid four-step tenth algebraic order methods with phase-lag of order 16(2)22 is developed for the numerical solution of the Schrödinger equation. Based on the new methods a variable-step procedure is introduced. Numerical illustrations obtained for the approximation of the phase shift problem for the well known case of the Lenard-Jones potential and for the numerical solution of the coupled equations arising from the Schrödinger equation show that these new methods are better than other finite difference methods.     

A P-stable exponentially-fitted method for the numerical integration of the Schrödinger equation

In this paper we present a P-stable exponentially-fitted four-step method for the numerical integration of the radial Schrödinger equation. More specifically we present a method than satisfies the property of P-stability and also integrates exactly any linear combination of the functions {1, x, exp( ± w x), x exp( ± w x), x ²exp( ± w x)}. We tested the efficiency of our newly developed scheme against well known methods, with excellent results. The numerical illustration showed that our method is considerably more efficient compared to well‐known methods used for the numerical integration of resonance problem of the radial Schrödinger equation.     

Bound state solutions of Schrödinger equation with modified Mobius square potential (MMSP) and its thermodynamic properties

We solved the Schrödinger equation with the modified Mobius square potential model using the modified factorization method. Within the framework of the Greene–Aldrich approximation for the centrifugal term and using a suitable transformation scheme, we obtained the energy eigenvalues equation and the corresponding eigenfunction in terms of the hypergeometric function. Using the resulting eigenvalues equation, we calculated the vibrational partition function and other relevant thermodynamic properties. We also showed that the modified Mobius square potential can be reduced to the Hua potential model using appropriate potential constant values.     

Rovibrational energy and spectroscopic constant calculations of complexes pairing via dihydrogen bonds

In the present investigation, we performed a thorough study of potential energy curves, rovibrational spectra, and spectroscopic constants for complexes pairing via dihydrogen bonds. In particular, we dealt with LiH???HX (X = F, CN, CCH, CCF, CCCl) complexes by employing accurate electronic energy calculations at the MP2/aug-cc-pVDZ level of theory. Following this, the Numerov method was applied to solve the nuclear Schrödinger equation, thus obtaining spectroscopic constants and rovibrational spectra. Good linear correlation between the magnitudes of the interaction energies for interaction of HX with LiH, and the most positive electrostatic potentials of hydrogen in HX, was established.     

A compact model to predict quantized sub-band energy levels and inversion layer centroids of MOSFETs with a parabolic potential well approximation

A compact model to predict sub-band energy levels and inversion charge centroids in the MOSFET surface inversion layer has been presented in this paper for parabolic potential well approximation. Based on a coupled solution of the Schrödinger equation and the Poisson equation following the WKB method, one transcendental equation of the sub-band energy level has been rigorously derived and then the approximate analytical solutions for the sub-band energy levels and the inversion charge centroids have been obtained. The analytical results are compared with the numerical data and a good agreement between the analytical and numerical is found.     

Three-Dimensional Structures of the Spatiotemporal Nonlinear Schr?dinger Equation with Power-Law Nonlinearity in PT-Symmetric Potentials

The spatiotemporal nonlinear Schrödinger equation with power-law nonlinearity in -symmetric potentials is investigated, and two families of analytical three-dimensional spatiotemporal structure solutions are obtained. The stability of these solutions is tested by the linear stability analysis and the direct numerical simulation. Results indicate that solutions are stable below some thresholds for the imaginary part of -symmetric potentials in the self-focusing medium, while they are always unstable for all parameters in the self-defocusing medium. Moreover, some dynamical properties of these solutions are discussed, such as the phase switch, power and transverse power-flow density. The span of phase switch gradually enlarges with the decrease of the competing parameter k in -symmetric potentials. The power and power-flow density are all positive, which implies that the power flow and exchange from the gain toward the loss domains in the cell.     

Energy transport mechanism in the form of proton soliton in a one-dimensional hydrogen-bonded polypeptide chain

The dynamics of protons in a one-dimensional hydrogen-bonded (HB) polypeptide chain (PC) is investigated theoretically. A new Hamiltonian is formulated with the inclusion of higher-order molecular interactions between peptide groups (PGs). The wave function of the excitation state of a single particle is replaced by a new wave function of a two-quanta quasi-coherent state. The dynamics is governed by a higher-order nonlinear Schrödinger equation and the energy transport is performed by the proton soliton. A nonlinear multiple-scale perturbation analysis has been performed and the evolution of soliton parameters such as velocity and amplitude is explored numerically. The proton soliton is thermally stable and very robust against these perturbations. The energy transport by the proton soliton is more appropriate to understand the mechanism of energy transfer in biological processes such as muscle contraction, DNA replication, and neuro-electric pulse transfer on biomembranes.     

Higher Dimensional Gaussian-Type Solitons of Nonlinear Schr?dinger Equation with Cubic and Power-Law Nonlinearities in PT-Symmetric Potentials

Two families of Gaussian-type soliton solutions of the (n+1)-dimensional Schrödinger equation with cubic and power-law nonlinearities in -symmetric potentials are analytically derived. As an example, we discuss some dynamical behaviors of two dimensional soliton solutions. Their phase switches, powers and transverse power-flow densities are discussed. Results imply that the powers flow and exchange from the gain toward the loss regions in the cell. Moreover, the linear stability analysis and the direct numerical simulation are carried out, which indicates that spatial Gaussian-type soliton solutions are stable below some thresholds for the imaginary part of -symmetric potentials in the defocusing cubic and focusing power-law nonlinear medium, while they are always unstable for all parameters in other media.     

Models of Darwinian processes and evolutionary principles

Evolutionary processes are described as stochastic motions in a genotype space (set of sequences with a Hamming distance) and a phenotype space (vector space of phenotypic properties). Real value functions are introduced which form a landscape over these spaces; smoothness postulates are formulated. Evolution is considered as a kind of hill climbing on these adaptive landscapes. A rather simple diffusion approximation for the phenotypic processes is proposed which leads to similar mathematical problems as the Schrödinger equation for disordered potential distributions.     

Modulational Collapsing and Adjusting with External Potential in a Complex Nonlinear Plasma

Plasma Physics Reports - Modulational collapse of a wave pulse in a complex nonlinear plasma described by a complex nonlinear Schrödinger equation (CNSE) including the effects of viscous...     

Confining annealed branched polymers inside spherical capsids

The Lifshitz equation for the confinement of a linear polymer in a spherical cavity of radius R has the form of the Schrödinger equation for a quantum particle trapped in a potential well with flat bottom and infinite walls at radius R. We show that the Lifshitz equation of a confined annealed branched polymer has the form of the Schrödinger equation for a quantum harmonic oscillator. The resulting confinement energy has a 1/R⁴ dependence on the confinement radius R, in contrast to the case of confined linear polymers, which have a 1/R² dependence. We discuss the application of this result to the problem of the confinement of single-stranded RNA molecules inside spherical capsids.     

Stability of dust acoustic wavepackets suffering from polarization force due to the presence of trapped ions

The combined effects of the polarization force, free and trapped ions, and dust charge variation are incorporated in a rigorous study of the nonlinear dust acoustic waves (DAWs) propagating in an unmagnetized dusty plasma. Owing to the departure from the Boltzmann ion distribution, it is found that the nonlinear DAWs are governed by a modified Korteweg?de Vries (mKdV) equation. The association between the mKdV solitary wave and the DAW envelope in the system under consideration is discussed. A modified nonlinear Schrödinger equation appropriate for describing the modulated DAWs is derived. The modulation instability (MI) and the dependence of the system physical parameters on the polarization force, trapped ions, and dust charge variation have been analyzed. It is found that the critical curve separating the stable/unstable regions is strongly influenced by both of the polarization and the ion trapping parameters. Moreover, increasing the polarization leads to an increase of the critical wave number, while increasing the trapping parameter yields the opposite effect. The MI maximum growth rate decreases (increases) as the polarization (trapped ion) increases. The obtained results may be helpful in better understanding of space observations of the solar energetic particle flows in interplanetary space and the energetic particle events in the Earth’s magnetosphere.     

Extracellular potentials of a single myelinated nerve fiber in an unbounded volume conductor

The extracellular potentials of a single myelinated nerve fiber in an unbounded volume conductor were studied. The spatial distribution of the transmembrane potential was obtained by integrating the system of partial differential equations characterizing the electric processes in the active myelinated nerve fiber. The spatial distribution of the extracellular potentials at various radial distances in the volume conductor were calculated using the line source model. Up to a certain radial distance (500 m) the discontinuity of the action potential propagation is reflected in the extracellular potentials, while further in the volume conductor the potentials are smooth. The effect of the fiber diameter and the internodal distance on the volume conductor potentials as well as the changes in the magnitude of the extracellular potential (in the time domain) between two adjacent nodes at various radial distances were studied. The radial decline of the peak-to-peak amplitude of the extracellular potential depends on the radial coordinater of the field point and increases with the increase ofr.     

Equilibrium electropotentials in electrolyte systems

The determinationof electric potentials in finite regions of symmetrical electrolyte in one-dimensional equilibrium situations requires the solution of the one-dimensional Poisson-Boltzmann equation in which the dependent variable is linearly related to the electric potential and contains unknown parameters. These require evaluation as part of the solution to a given boundary value problem. The general solution of the equation is presented. This involves elliptic functions and integrals and is sectionally isomorphic with respect to an integration parameter. The application to problems posed in terms of both initial values and two-point boundary values is discussed. The solution is used to determine the potential and concentration distributions between two flat-faced charged particles immersed in an electrolyte liquid and having interacting double layers.     

Can contact potentials reliably predict stability of proteins?

The simplest approximation of interaction potential between amino acid residues in proteins is the contact potential, which defines the effective free energy of a protein conformation by a set of amino acid contacts formed in this conformation. Finding a contact potential capable of predicting free energies of protein states across a variety of protein families will aid protein folding and engineering in silico on a computationally tractable time-scale. We test the ability of contact potentials to accurately and transferably (across various protein families) predict stability changes of proteins upon mutations. We develop a new methodology to determine the contact potentials in proteins from experimental measurements of changes in protein's thermodynamic stabilities (DeltaDeltaG) upon mutations. We apply our methodology to derive sets of contact interaction parameters for a hierarchy of interaction models including solvation and multi-body contact parameters. We test how well our models reproduce experimental measurements by statistical tests. We evaluate the maximum accuracy of predictions obtained by using contact potentials and the correlation between parameters derived from different data-sets of experimental (DeltaDeltaG) values. We argue that it is impossible to reach experimental accuracy and derive fully transferable contact parameters using the contact models of potentials. However, contact parameters may yield reliable predictions of DeltaDeltaG for datasets of mutations confined to the same amino acid positions in the sequence of a single protein.     

Potentials of mean force for the interaction of blocked alanine dipeptide molecules in water and gas phase from MD simulations

We calculate potentials of mean force (PMFs) for the intermolecular interaction of two blocked alanine dipeptide (AcAlaNHMe) molecules in water and gas phase at two temperatures, 278 and 300 K, from all-atom molecular dynamics simulations. Simple models based on buried solvent accessible surface and one-dimensional potentials derived from distance-based radial distribution functions are not capable of expressing the short- and long-range complexity of the solute-solute interactions in water. Instead, radial and angular variations in the PMFs are observed with the two-dimensional potentials. The strength of the interactions for specific relative orientations of the molecules in the two-dimensional PMFs is more than double that observed in the one-dimensional PMFs. The populations of specific blocked alanine dipeptide conformations in water, such as alpha(R) and PPII, vary with temperature, and most significantly, with the distance between the centers of mass. A preference for helical conformations is observed at close encounter between molecules.     

Using DL_POLY to study the sensitivity of liquid structure to potential parameters

Two case studies are presented showing the local structure in liquids and how it responds to changes in the intermolecular potential. The idea is to use realistic and unrealistic potentials in order to determine the sensitivity of local liquid structure to potential parameters. The first case study concerns two families of modified water models. In the “hybrid” family, the hydrogen bond strength is reduced, but the geometry kept constant; in the second family, the molecular geometry is changed by reducing the bond angle, keeping a constant molecular dipole moment. The local structure is measured by radial distribution functions, three-dimensional probability distribution functions and three-body angular correlations. The second case study concerns the ionic liquid dimethylimidazolium chloride ([C₁mim]Cl). The effect of reducing the hydrogen bonding potential of the cations while maintaining their charge is examined.     

DNA conformation and its changes upon binding transcription factors

Pyrimidine-purine steps are flexible and can roll around the major groove. This feature is used to follow the protein surface and thereby create a particular superstructure of DNA. The interaction of the two molecules can be understood in terms of a close fitting of the two surfaces, in particular, how the DNA surface adapts to the protein surface. For analysing the fitting the widths of two DNA grooves are useful parameters.Much progress has been made since Schrödinger predicted “a gene to be an aperiodic solid (or a crystal)” (29) and we hope that our review may contribute in a small way.     

Compact modeling of the threshold voltage in silicon nanowire MOSFET including 2D-quantum confinement effects

A quantum-mechanical compact model of the threshold voltage (V _T) for quantum nanowire MOSFETs has been developed. This approach is based on analytical solutions for the decoupled 2D Schrödinger and 1D Poisson equations solved in the silicon channel. A quantum correction based on the perturbation theory has been also introduced to improve the model accuracy. Finally, the validity of the model has been verified by comparison with data obtained with a 2D/3D Poisson-Schrödinger drift-diffusion simulation code.