Scattering states of modified Pöschlben Teller potential in D-dimension

We present a new approximation scheme for the centrifugal term, and apply this new approach to the Schrödinger equation with the modified Pöschl-Teller potential in the D-dimension for arbitrary angular momentum states. The approximate analytical solutions of the scattering states are derived. The normalized wave functions expressed in terms of the hypergeometric functions of the scattering states on the k/2π scale and the calculation formula of the phase shifts are given. The numerical results show that our results are in good agreement with those obtained by using the amplitude-phase method (APM).     

A Schrödinger Potential Conditionally Integrable in Terms of the Hermite Functions

We introduce a biconfluent Heun potential well for the one-dimensional stationary Schrödinger equation which is composed of a confining fraction-power term and a repulsive centrifugal-barrier core. This is a conditionally integrable potential in that the strength of the centrifugal barrier is fixed to a constant. The potential supports a countable infinite number of bound states. We present the general solution of the Schrödinger equation, deduce the exact equation for the energy spectrumand derive a highly accurate approximation for energy levels. The bound state wave functions are written as irreducible linear combinations with constant coefficients of two Hermite functions of a scaled and shifted argument.

     

Eigenstate expansions in a soluble model of identical particle scattering

Solutions to the Schrödinger equation and the inhomogeneous equation for the case of two identical particles interacting with a center of force are studied. Eigenstate expansions for solving each equation are explicitly introduced and their properties discussed. The case when the interparticle interaction v₁₂ is zero is then examined; this is a completely soluble problem. The eigenstate expansion solutions for the Schrödinger and inhomogeneous equations are used to explore the means by which the correct solution is obtained. Finally, approximate solutions, obtained by truncating the eigenfunction expansions, are introduced. It is seen that both methods lead to the correct amplitude when τ₁₂ = 0, even though the approximate solution to the inhomogeneous equation does not lead, in the end, to an antisymmetric solution.     

A New Approach to the Exact Solutions of the Effective Mass Schrödinger Equation

Effective mass Schrödinger equation is solved exactly for a given potential. Nikiforov-Uvarov method is used to obtain energy eigenvalues and the corresponding wave functions. A free parameter is used in the transformation of the wave function. The effective mass Schrödinger equation is also solved for the Morse potential transforming to the constant mass Schrödinger equation for a potential. One can also get solution of the effective mass Schrödinger equation starting from the constant mass Schrödinger equation.     

Optical potential study of Σ nuclear states

Single-particle energies and widths of Σ hypernuclear states are calculated in light systems (A ≤ 40) as energy eigenvalues of the Schrödinger equation for a complex optical potential that fits level shifts and widths of Σ^? atoms. The interpretation and significance of Σ (normalizable) bound states embedded in the Λ hypernuclear (as well as, sometimes, in the Σ hypernuclear) continuum are discussed and their properties are studied, primarily in order to identify relatively narrow (Γ ? 10 MeV) states. The connection between these calculations and the recently observed Σ hypernuclear states suggests that bound states embedded in the Σ continuum, rather than (nonnormalizable) Gamow resonant states, are produced in (K^?, π) nuclear reactions.     

New optical soliton solutions for nonlinear complex fractional Schrödinger equation via new auxiliary equation method and novel $$({G'}/{G})$$-expansion method

In this research, we apply two different techniques on nonlinear complex fractional nonlinear Schrödinger equation which is a very important model in fractional quantum mechanics. Nonlinear Schrödinger equation is one of the basic models in fibre optics and many other branches of science. We use the conformable fractional derivative to transfer the nonlinear real integer-order nonlinear Schrödinger equation to nonlinear complex fractional nonlinear Schrödinger equation. We apply new auxiliary equation method and novel \(\left( {G'}/{G}\right) \)-expansion method on nonlinear complex fractional Schrödinger equation to obtain new optical forms of solitary travelling wave solutions. We find many new optical solitary travelling wave solutions for this model. These solutions are obtained precisely and efficiency of the method can be demonstrated.     

Approximate analytical solution of continuous states for the l-wave Schrödinger equation with a diatomic molecule potential

The continuous states of the l-wave Schrödinger equation for the diatomic molecule represented by the hyperbolical function potential are carried out by a proper approximation scheme to the centrifugal term. The normalized analytical radial wave functions of the l-wave Schrödinger equation for the hyperbolical function potential are presented and the corresponding calculation formula of phase shifts is derived. Also, we interestingly obtain the corresponding bound state energy levels by analyzing analytical properties of scattering amplitude.     

Exact and approximate solutions of Schrödinger’s equation for a class of trigonometric potentials

The asymptotic iteration method is used to find exact and approximate solutions of Schrödinger’s equation for a number of one-dimensional trigonometric potentials (sine-squared, double-cosine, tangent-squared, and complex cotangent). Analytic and approximate solutions are obtained by first using a coordinate transformation to reduce the Schrödinger equation to a second-order differential equation with an appropriate form. The asymptotic iteration method is also employed indirectly to obtain the terms in perturbation expansions, both for the energies and for the corresponding eigenfunctions.     

A generalized Schrödinger equation via a complex Lagrangian of electrodynamics

In this paper we give a generalized form of the Schrödinger equation in the relativistic case, which contains a generalization of the Klein-Gordon equation. By complex Legendre transformation, the complex Lagrangian of electrodynamics produces a complex relativistic Hamiltonian H of electrodynamics, on the holomorphic cotangent bundle T′* M. By a special quantization process, a relativistic time dependent Schrödinger equation, in the adapted frames of (T′* M, H) is obtained. This generalized Schrödinger equation can be expressed with respect to the Laplace operator of the complex Hamilton space (T′*M, H). Finally, under some additional conditions on the proper time s of the complex space-time M and the time parameter t along the quantum state, by the method of separation of variables, we obtain two classes of solutions for the Schrödinger equation, one for the weakly gravitational complex curved space M, and the second in the complex space-time with Schwarzschild metric.     

Periodic Solutions for a Class of Nonlinear Partial Differential Equations in Higher Dimension

We prove the existence of periodic solutions in a class of nonlinear partial differential equations, including the nonlinear Schrödinger equation, the nonlinear wave equation, and the nonlinear beam equation, in higher dimension. Our result covers cases of completely resonant equations, where the bifurcation equation is infinite-dimensional, such as the nonlinear Schrödinger equation with zero mass, for which solutions which at leading order are wave packets are shown to exist.     

Cooperative state and resonance fluorescence

Using the exact solution of the Schrödinger equation for a system of N two-level atoms driven by a strong resonant field, the spectrum of cooperative resonance fluorescence consisting of the (ω₀ + mΩ)-harornics has been found.     

Theory of Stochastic Schrödinger Equation in Complex Vector Space

A generalized Schrödinger equation containing correction terms to classical kinetic energy, has been derived in the complex vector space by considering an extended particle structure in stochastic electrodynamics with spin. The correction terms are obtained by considering the internal complex structure of the particle which is a consequence of stochastic average of particle oscillations in the zeropoint field. Hence, the generalised Schrödinger equation may be called stochastic Schrödinger equation. It is found that the second order correction terms are similar to corresponding relativistic corrections. When higher order correction terms are neglected, the stochastic Schrödinger equation reduces to normal Schrödinger equation. It is found that the Schrödinger equation contains an internal structure in disguise and that can be revealed in the form of internal kinetic energy. The internal kinetic energy is found to be equal to the quantum potential obtained in the Madelung fluid theory or Bohm statistical theory. In the rest frame of the particle, the stochastic Schrödinger equation reduces to a Dirac type equation and its Lorentz boost gives the Dirac equation. Finally, the relativistic Klein–Gordon equation is derived by squaring the stochastic Schrödinger equation. The theory elucidates a logical understanding of classical approach to quantum mechanical foundations.     

A Fractional Schrödinger Equation and Its Solution

This paper presents a fractional Schrödinger equation and its solution. The fractional Schrödinger equation may be obtained using a fractional variational principle and a fractional Klein-Gordon equation; both methods are considered here. We extend the variational formulations for fractional discrete systems to fractional field systems defined in terms of Caputo derivatives to obtain the fractional Euler-Lagrange equations of motion. We present the Lagrangian for the fractional Schrödinger equation of order α. We also use a fractional Klein-Gordon equation to obtain the fractional Schrödinger equation which is the same as that obtained using the fractional variational principle. As an example, we consider the eigensolutions of a particle in an infinite potential well. The solutions are obtained in terms of the sines of the Mittag-Leffler function.     

Energy levels of a particle confined in an ellipsoidal potential well

The Schrödinger equation is solved for a particle confined within the ellipsoidal potential well using the perturbation theory and the Hamiltonian diagonalization method. The explicit expressions are obtained for the energy levels that are size and shape dependent and appropriate wave functions. The calculated energy levels are in a good qualitative and quantitative agreement with the result obtained by numerical solution of the Schrödinger equation. It is revealed that for the lowest states of a given symmetry the region of validity of the perturbation approximation is much larger than it follows from the usual condition of applicability of the perturbation theory. The optical properties of nanoparticles of a prolate and oblate ellipsoidal shape are discussed.     

A simple and efficient numerical scheme to integrate non-local potentials

As nuclear wave functions have to obey the Pauli principle, potentials issued from reaction theory or Hartree-Fock formalism using finite-range interactions contain a non-local part. Written in coordinate space representation, the Schrödinger equation becomes integro-differential, which is difficult to solve, contrary to the case of local potentials, where it is an ordinary differential equation. A simple and powerful method was proposed several years ago, with the trivially equivalent potential method, where the non-local potential is replaced by an equivalent local potential, which is state dependent and has to be determined iteratively. Its main disadvantage, however, is the appearance of divergences in potentials if the wave functions have nodes, which is generally the case. We will show that divergences can be removed by a slight modification of the trivially equivalent potential method, leading to a very simple, stable and precise numerical technique to deal with non-local potentials. Examples will be provided with the calculation of the Hartree-Fock potential and associated wave functions of ¹⁶O using the finite-range N³LO realistic interaction.     

Eigenfunction expansions in the imaginary Lobachevsky space

Eigenfunctions of the Schrödinger equation with the Coulomb potential in the imaginary Lobachevsky space are studied in two coordinate systems admitting solutions in terms of hypergeometric functions. Normalization and coefficients of mutual expansions for some sets of solutions are found.     

Asymptotic behavior of weakly collapsing solutions of the nonlinear Schrödinger equation

The generic asymptotic behavior of a three-parameter weakly collapsing solution of a nonlinear Schrödinger equation is examined. A discrete set of zero-energy states is shown to exist. In the (A, C ₁) parameter space, there are two close lines along which the amplitude of oscillating terms is exponentially small in the parameter C ₁.     

On a level shift occurring in the treatment of a discrete state coupled to two interacting continua

It is shown how the partitioning method of Löwdin may be used to obtain approximate solutions to the Dirac equation. By using a novel separation of the partitioned wave equation perturbation theory may be employed with the solutions of the Schrödinger equation as the zeroth order functions. The method is demonstrated for the 1s, 2s and 2p states of the hydrogen atom and in particular the energies correct to order mc ² α⁶ are obtained. The first and second order contributions to the energy are both finite so the problem of divergences is avoided.     

Time dependence of resonant tunneling in heterostructures with a three-trough energy spectrum

The time dependence of resonant electron tunneling inGaAs andAlAs quantum heterostructures is studied for a three-trough model. From an analysis of transmitted and reflected wave phases, the spectra of tunneling and reflection times are obtained. The tunneling of a wave packet through a two-barrier heterostructure is modeled by numerical solution of the nonstationary Schrödinger equation.