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.     

Cauchy problem on the plane for the dispersionless Kadomtsev-Petviashvili equation

We construct the formal solution to the Cauchy problem for the dispersionless Kadomtsev-Petviashvili equation as an application of the inverse scattering transform for the vector field corresponding to a Newtonian particle in a time-dependent potential. This is in full analogy with the Cauchy problem for the Kadomtsev-Petviashvili equation, associated with the inverse scattering transform of the time-dependent Schrödinger operator for a quantum particle in a time-dependent potential.     

Wave Decoherence for the Random Schrödinger Equation with Long-Range Correlations

In this paper, we study the loss of coherence of a wave propagating according to the Schrödinger equation with a time-dependent random potential. The random potential is assumed to have slowly decaying correlations. The main tool to analyze the decoherence phenomena is a properly rescaled Wigner transform of the solution of the random Schrödinger equation. We exhibit anomalous wave decoherence effects at different propagation scales.     

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.     

Soliton solutions of the non linear Schrödinger equation with an external time-dependent field

A functional transformation between solutions of the one-dimensional nonlinear Schrödinger equation with time- and coordinate-dependent coefficients and solutions of the conventional nonlinear Schrödinger equation (NSE) is constructed. Exact solutions of the NSE with a homogeneous time-dependent external electric field and the NSE with oscillator potential are obtained.     

Nonlinear electrohydrodynamic Rayleigh-Taylor instability with mass and heat transfer subject to a vertical oscillating force and a horizontal electric field

Weakly nonlinear stability of interfacial waves propagating between two electrified inviscid fluids influenced by a vertical periodic forcing and a constant horizontal electric field is studied. Based on the method of multiple-scale expansion for a small-amplitude periodic force, two parametric nonlinear Schrödinger equations with complex coefficients are derived in the resonance cases. A standard nonlinear Schrödinger equation with complex coefficients is derived in the nonresonance case. A temporal solution is carried out for the parametric nonlinear Schrödinger equation. The stability analysis is discussed both analytically and numerically.     

Long Time Energy Transfer in the Random Schrödinger Equation

We consider the long time behavior of solutions of the d-dimensional linear Boltzmann equation that arises in the weak coupling limit for the Schrödinger equation with a time-dependent random potential. We show that the intermediate mesoscopic time limit satisfies a Fokker–Planck type equation with the wave vector performing a Brownian motion on the (d ? 1)-dimensional sphere of constant energy, as in the case of a time-independent Schrödinger equation. However, the long time limit of the solution with an isotropic initial data satisfies an equation corresponding to the energy being the square root of a Bessel process of dimension d/2.     

Dilaton Quantum Cosmology with a Schrödinger-like Equation

A quantum cosmological model with radiation and a dilaton scalar field is analyzed. The Wheeler–DeWitt equation in the minisuperspace induces a Schrödinger equation, which can be solved. An explicit wavepacket is constructed for a particular choice of the ordering factor. A consistent solution is possible only when the scalar field is a phantom field. Moreover, although the wavepacket is time-dependent, a Bohmian analysis allows to extract a bouncing behavior for the scale factor.     

Quantum superarrivals and information transfer through a time-varying boundary

The time-dependent Schrödinger equation is solved numerically for the case of a Gaussian wave packet incident on a time-varying potential barrier. The time evolving reflection and transmission probabilities of the wave packet are computed for several different time-dependent boundary conditions obtained by reducing or increasing the height of the potential barrier. We show that in the case when the barrier height is reduced to zero, a time interval is found during which the reflection probability is larger (superarrivals) compared to the unperturbed case. We further show that the transmission probability exhibits superarrivals when the barrier is raised from zero to a finite value of its height. Superarrivals could be understood by ascribing the features of a real physical field to the Schrödinger wave function which acts as a carrier through which a disturbance, resulting from the boundary condition being perturbed, prpagates from the barrier to the detectors measuring reflected and transmitted probabilities. The speed of propagation of this effect depends upon the rate of reducing or raising the barrier height, thus suggesting an application for secure information transfer using superarrivals.     

Form-preserving Transformations for the Time-dependent Schrödinger Equation in (n + 1) Dimensions

We define a form-preserving transformation (also called point canonical transformation) for the time-dependent Schrödinger equation (TDSE) in (n+1) dimensions. The form-preserving transformation is shown to be invertible and to preserve L ²-normalizability. We give a class of time-dependent TDSEs that can be mapped onto stationary Schrödinger equations by our form-preserving transformation. As an example, we generate a solvable, time-dependent potential of Coulombic ring-shaped type together with the corresponding exact solution of the TDSE in (3+1) dimensions. We further consider TDSEs with position-dependent (effective) masses and show that there is no form-preserving transformation between them and the conventional TDSEs, if the spatial dimension of the system is higher than one.     

High intensity of interband transitions in double-barrier structures with a high-frequency electric field

A solution of a two-band analogue of the time-dependent Schrödinger equation describing resonance coherent electron interaction with a high-frequency field was found for a symmetric double-barrier structure, and an analytic expression was obtained for the small-signal conductivity proportional to the electron transition intensity. It was found that the high-frequency conductivity of double-barrier structures in the case of interband transitions could be significantly higher than in the case of intersubband transitions.     

Formal exact solution for the Heisenberg spin system in a time-dependent magnetic field and Aharonov-Anandan phase

We study the time evolution of the Heisenberg spin system in a time-dependent magnetic field. By a unitary transformation we obtain the formal exact solutions of the Schrödinger equation for this system. Based on the formal exact solutions, the Aharonov-Anandan phases are worked out.     

The propagator for a time-dependent mass subject to a harmonic potential with a time-dependent frequency

We find the quantum propagator of the harmonic oscillator with a time-dependent mass solving directly the Schrödinger equation through a change of variable and a time reparametrization.     

Green's function for a three-dimensional oscillator in a variable uniform electromagnetic field

By means of the Feynman integral along trajectories, an exact expression is found for Green's function in the Cauchy problem involving the Schrödinger equation, which describes the behavior of a three-dimensional variable-frequency oscillator in a variable uniform electromagnetic field. The problem reduces to that of finding the fundamental solution to a secondorder ordinary linear differential equation with variable coefficients.     

On stationary solutions of the Schrödinger equation with a self-consistent potential satisfying boussinesq's equation

We derive an equation for a dense perturbation valid in a resonance region, M → 1, associated with the Schrödinger equation for a complex amplitude of the high-frequency Langmuir-oscillation electrical field. A solution of the above system in the linear approximation is obtained and discussed.     

Dynamic stabilization of atomic ionization in a high-frequency laser field with different initial angular momenta

We investigated the ionization of an atom with different orbital angular momenta in a high-frequency laser field by solving the time-dependent Schrödinger equation. The results showed that the ionization stabilization features changed with the relative direction between the angular momentum of the initial state and the vector field of the laser pulse. The ionization mechanism of the atom irradiated by a high frequency was explained by calculating the transition matrix and evolution of the time-dependent wave packet. This study can provide comprehensive understanding to improve atomic nonadiabatic ionization.     

Description of Pulse Propagation in Optical Strip Waveguides by the Generalized Nonlinear Schrödinger Equation

Taking into consideration the transverse confinement, the dispersion of the linear as well as of the nonlinear part of the refractive index and the nonlinear interaction of the electromagnetic field with the guiding material, the yielding equation for the envelope function of the fundamental mode propagating in a strip waveguide is the generalized nonlinear Schrödinger equation. In contrast to the nonlinear Schrödinger equation, solitons of the generalized nonlinear Schrödinger equation exist in the regimes of negative and positive group dispersion and are asymmetric.     

The symmetry group of the quantum harmonic oscillator in an electric field

In this paper we present two results. First, we derive the most general group of infinitesimal transformations for the Schrödinger Equation of the general time-dependent Harmonic Oscillator in an electric field. The infinitesimal generators and the commutation rules of this group are presented and the group structure is identified. From here it is easy to construct a set of unitary operators that transform the general Hamiltonian to a much simpler form. The relationship between squeezing and dynamical symmetries is also stressed. The second result concerns the application of these group transformations to obtain solutions of the Schrödinger equation in a time-dependent potential. These solutions are believed to be useful for describing particles confined in boxes with moving boundaries. The motion of the walls is indeed governed by the time-dependent frequency function. The applications of these results to non-rigid quantum dots and tunnelling through fluctuating barriers is also discussed, both in the presence and in the absence of a time-dependent electric field. The differences and similarities between both cases are pointed out.     

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.     

An alternative approximate solution to the time-dependent schrödinger equation

A second-order time-differential equation is proposed for the approximate solution of the time-dependent Schrödinger equation with a limited basis. The resulting nonunitary solutions allow for a flux of probability into the excluded space. Simple examples are used in demonstration.