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.     

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.     

Entanglement in the Schrödinger experiment

The mixed and entanglement states have been analyzed in the Schrödinger experiment. It is known that, in an open system, the “Schrödinger cat” paradox is explained by the decoherence phenomenon, but, in a closed system, it is explained by the Everett-Wheeler many-world interpretation of quantum mechanics. The quantum real world can be presented as a complex multispatial geometric figure and the classical world is one of the faces of this figure. In this paper it is shown that this figure is the simplex that is well known in the functional analysis. Such an interpretation of quantum mechanics enables one to obtain the nonuniform wave equation, and the Schrödinger equation is the uniform equation of this one. Perhaps this equation is the equation of the subquantum world about which Einstein has written.     

Canonical averaging of the Schrödinger equation

The representation of the Schrödinger equation in the form of a classical Hamiltonian system makes it possible to construct a unified perturbation theory that is based on the theory of canonical transformations and covers both classical and quantum mechanics. Also, the closeness of the exact and approximate solutions of the Schrödinger equation can be approximately estimated with such a representation.     

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.     

Inverse scattering method and vector higher order non-linear Schrödinger equation

A generalized inverse scattering method has been developed for arbitrary n-dimensional Lax equations. Subsequently, the method has been used to obtain N-soliton solutions of a vector higher order non-linear Schrödinger equation, proposed by us. It has been shown that under a suitable reduction, the vector higher order non-linear Schrödinger equation reduces to the higher order non-linear Schrödinger equation. An infinite number of conserved quantities have been obtained by solving a set of coupled Riccati equations. Gauge equivalence is shown between the vector higher order non-linear Schrödinger equation and the generalized Landau–Lifshitz equation and the Lax pair for the latter equation has also been constructed in terms of the spin field, establishing direct integrability of the spin system.     

A generalized Schrödinger formalism as a Hilbert space representation of a generalized Liouville equation

A generalized Schrödinger formalism has been presented which is obtained as a Hilbert space representation of a Liouville equation generalized to include the action as a dynamical variable, in addition to the positions and the momenta. This formalism applied to a classical mechanical system had been shown to yield a similar set of Schrödinger like equations for the classical dynamical system of charged particles in a magnetic field. The novel quantum-like predictions for this classical mechanical system have been experimentally demonstrated and the results are presented.     

Prequantum Classical Statistical Field Theory: Complex Representation, Hamilton-Schrödinger Equation, and Interpretation of Stationary States

We develop a prequantum classical statistical model in that the role of hidden variables is played by classical (vector) fields. We call this model Prequantum Classical Statistical Field Theory (PCSFT). The correspondence between classical and quantum quantities is asymptotic, so we call our approach asymptotic dequantization. We construct the complex representation of PCSFT. In particular, the conventional Schrödinger equation is obtained as the complex representation of the system of Hamilton equations on the infinite-dimensional phase space. In this note we pay the main attention to interpretation of so called pure quantum states (wave functions) in PCSFT, especially stationary states. We show, see Theorem 2, that pure states of QM can be considered as labels for Gaussian measures concentrated on one dimensional complex subspaces of phase space that are invariant with respect to the Schrödinger dynamics. “A quantum system in a stationary state ψ” in PCSFT is nothing else than a Gaussian ensemble of classical fields (fluctuations of the vacuum field of a very small magnitude) which is not changed in the process of Schrödinger's evolution. We interpret in this way the problem of stability of hydrogen atom. One of unexpected consequences of PCSFT is the infinite dimension of physical space on the prequantum scale.     

Localization of Bose-Einstein condensates in optical lattices

The dynamics of repulsive bosons condensed in an optical lattice is effectively described by the Bose-Hubbard model. The classical limit of this model, reproduces the dynamics of Bose-Einstein condensates, in a periodic potential, and in the superfluid regime. Such dynamics is governed by a discrete nonlinear Schrödinger equation. Several papers, addressing the study of the discrete nonlinear Schrödinger dynamics, have predicted the spontaneous generation of (classical) breathers in coupled condensates. In the present contribute, we shall focus on localized solutions (quantum breathers) of the full Bose-Hubbard model. We will show that solutions exponentially localized in space and periodic in time exist also in absence of randomness. Thus, this kind of states, reproduce a novel quantum localization phenomenon due to the interplay between bounded energy spectrum and non-linearity.     

Extreme waves statistics for the Ablowitz-Ladik system

We examine statistics of waves for the problem of modulation instability development in the framework of discrete integrable Ablowitz-Ladik (AL) system. Modulation instability depends on one free parameter h that has the meaning of the coupling between the nodes on the lattice. For strong coupling h ? 1, the probability density functions (PDFs) for waves amplitudes coincide with that for the continuous classical nonlinear Schrödinger equation; the PDFs for both systems are very close to Rayleigh ones. When the coupling is weak h ～ 1, there appear highly localized waves with very large amplitudes, that drastically change the PDFs to significantly non-Rayleigh ones, with so-called “fat tails” when the probability of a large wave occurrence is by several orders of magnitude higher than that predicted by the linear theory. Evolution of amplitudes for such rogue waves with time is similar to that of the Peregrine solution for the classical nonlinear Schrödinger equation.     

Continuum spin system as an exactly solvable dynamical system

The continuum limit of a one-dimensional classical spins with nearest neighbour Heisenberg interaction is shown to be an exactly solvable system and that its dynamics describable by the nonlinear Schrödinger equation. N-soliton solutions for the energy density exist.     

Solitons and spectral renormalization methods in nonlinear optics

Localized wave solutions, often referred to as solitary waves or solitons, are important classes of solutions in nonlinear optics. In optical communications, weakly nonlinear, quasi-monochromatic waves satisfy the “classical” and the “dispersion-managed” nonlocal nonlinear Schrödinger equations, both of which have localized pulses as special solutions. Recent research has shown that mode-locked lasers are also described by similar equations. These systems are variants of the classical nonlinear Schrödinger equation, appropriately modified to include terms which model gain, loss and spectral filtering that are present in the laser cavity. To study their remarkable properties, a computational method is introduced to find localized waves in nonlinear optical systems governed by these equations.     

Bifurcations and the stability of the surface envelope solitons for a finite-depth fluid

The dynamics of the quasi-monochromatic surface gravitational waves in a finite-depth fluid is studied for the case where the product of the wavenumber by the depth of the fluid is close to the critical value k _cr h ≈ 1.363. Within the framework of the Hamiltonian formalism, the general nonlinear Schrödinger equation is derived. In contrast to the classical nonlinear Schrödinger equation, this equation involves the gradient terms to the four-wave interaction, as well as the six-wave interaction. This equation is used to analyze the modulation instability of the monochromatic waves, as well as the bifurcations of the soliton solutions and their stability. It is shown that the solitons are stable and unstable to finite perturbations for focusing and defocusing nonlinearities, respectively.     

On local symmetries of the Schrödinger equation

The multi-symplectic approach to the Schrödinger equation with a potential V = V(t,x^k) is given. The condition for a vector field X in the multi-symplectic space to be a symmetry field is found. For a spherically symmetrical potential all such symmetry fields are effectively found.The one-to-one correspondence between solutions of the free Schrödinger equation and solutions of the oscillator problem is given. This enables us to give a new geometric interpretation of the non-typical, given by A.O. Barut, symmetry of the Schrödinger equation.     

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.     

Growth of Sobolev Norms in the Cubic Nonlinear Schrödinger Equation with a Convolution Potential

Fix s > 1. Colliander et al. proved in (Invent Math 181:39–113, 2010) the existence of solutions of the cubic defocusing nonlinear Schrödinger equation in the two torus whose s-Sobolev norm undergoes arbitrarily large growth as time evolves. In this paper we generalize their result to the cubic defocusing nonlinear Schrödinger equation with a convolution potential. Moreover, we show that the speed of growth is the same as the one obtained for the cubic defocusing nonlinear Schrödinger equation in Guardia and Kaloshin (Growth of Sobolev norms in the cubic defocusing Nonlinear Schrödinger Equation. To appear in the Journal of the European Mathematical Society, 2012).     

Generalized Schrödinger cat states in cavity QED

A method of generating generalized Schrödinger cat states by considering the resonant interaction of a high-quality cavity with N two-level atoms driven by a strong classical field is proposed.