Standard Schrodinger Equation Research Articles

From Generalized Hamilton Principle to Generalized Schrodinger Equation

The Hamilton principle is a variation principle describing the isolated and conservative systems, its Lagrange function is the difference between kinetic energy and potential energy. By Feynman path integration, we can obtain the standard Schrodinger equation. In this paper, we have given the generalized Hamilton principle, which can describe the heat exchange system, and the nonconservative force system. On this basis, we have further given their generalized Lagrange functions and Hamilton functions. With the Feynman path integration, we have given the generalized Schrodinger equation of nonconservative force system and the heat exchange system.

Quantum Foundations and Energy Distribution Processes of Inner Oscillations

We consider the problem of computing energy distribution of inner harmonic oscillations of a nanoparticle in its phase space, when the particle moves in a medium under certain temperature. It is assumed that the particle obeys the Brownian motion under the action of the medium and the force field given by a potential function. In the present paper we provide and study an equation describing the problem, generalizing the Klein-Kramers equation. It is shown that for large value of medium resistance, the process of energy distribution of inner harmonic oscillations of the nanoparticle is represented as the composition of a rapid transition process and a slow process. After the rapid transition process, the system goes to a quasi-stationary state. The slow process is approximately described by the standard Schrodinger equation used for description of quantum processes. Thus, the process can serve as models of quantum processes.

Quantum mechanics with a nonzero quantum correlation time

We propose an extension of Quantum Mechanics based on the idea that the underlying "quantum noise" has a non-zero, albeit very small, correlation time $\tau_c$. The standard (non-relativistic) Schrodinger equation is recovered to zeroth order in $\tau_c$, and the first correction to energy levels is computed. Some consequences are discussed, in particular the violation of Heisenberg's uncertainty principle and the restoration of locality at short times.

Dynamics of finite energy Airy beams modeled by the fractional Schrödinger equation with a linear potential

The dynamics of finite energy Airy beams modeled by the fractional Schrodinger equation (FSE) with a linear potential are numerically investigated. Different from the propagation properties of Airy beams described by the standard Schrodinger equation, the splitting phenomenon, which is presented under the FSE without potential, is influenced by the quadratic chirp and the Levy index. As the linear potential is considered, the periodic evolution of Airy beams is shown, and the period is inversely proportional to the linear potential coefficient. The beam width can undergo an abrupt decrease or increase depending on the sign of potential coefficient. The beam deviation distance increases with the beam width greatly changed if the Levy index increases. Moreover, the quadratic chirp does not influence the evolution period but the intensity distribution. In addition, the intriguing properties are analytically clarified. All these features confirm the promising applications of Airy beams in optical manipulation and optical switch.

Exact Solution of T-Difference Radial Schrödinger Equation

In this paper, we get an exact discrete analog of the radial Schrodinger equation. We propose a difference equation that exactly corresponds to the standard radial Schrodinger equation. The suggested equation contains \({{{\mathcal {T}}}}\)-differences that are represented by infinite series. From a physical point of view, this discrete equation describes a lattice with long-range interactions of power-law type. From a mathematical point of view, it is a uniquely selected difference equation that exactly corresponds to continuous radial Schrodinger equation. Solution of the suggested difference radial Schrodinger equation is obtained.

Effective Potential from the Generalized Time-Dependent Schrödinger Equation

We analyze the generalized time-dependent Schrödinger equation for the force free case, as a generalization, for example, of the standard time-dependent Schrödinger equation, time fractional Schrödinger equation, distributed order time fractional Schrödinger equation, and tempered in time Schrödinger equation. We relate it to the corresponding standard Schrödinger equation with effective potential. The general form of the effective potential that leads to a standard time-dependent Schrodinger equation with the same solution as the generalized one is derived explicitly. Further, effective potentials for several special cases, such as Dirac delta, power-law, Mittag-Leffler and truncated power-law memory kernels, are expressed in terms of the Mittag-Leffler functions. Such complex potentials have been used in the transport simulations in quantum dots, and in simulation of resonant tunneling diode.

Transformations and Soliton Solutions for a Variable-coefficient Nonlinear Schrödinger Equation in the Dispersion Decreasing Fiber with Symbolic Computation

Describing the dispersion decreasing fiber, a variable-coe fficient nonlinear Schrodinger equation is hereby under investigation. Three transformations have been obtained from such a equation to the known standard and cylindrical nonlinear Schrodinger equations with the relevant constraints on the variable coefficients presented, which t urn out to be more general than those previously published in the literature. Meanwhile, several families of exact dark-soliton-like and bright-soliton-like solutions are constructed. Also, we o btain some similarity solutions, which can be illustrated in terms of the elliptic and the second Painle ve transcendent equations.

Effective mass Schrödinger equation with Thomas-Fermi potential

The exactly-solvable position-dependent mass Schrodinger equation (PDMSE) for the Thomas-Fermi potential is presented. The PDMSE is transformed into a standard Schrodinger equation (SSE) with constant mass by means of a point canonical transformation scheme. By proposing an exponential type potential of the SSE it is possible to determine a PDMSE with the Thomas-Fermi potential. The resulting PDMSE is carried to the Sturm- Liouville form and the corresponding theory is developed for the particular resulting problem in order to obtain closed solutions in terms of Bessel functions of the first kind. Beyond the case considered in this work, the approach is general and can be useful in the study of electronic properties of non-uniform materials in which the carrier effective mass depends on the position as well as in the search of new solvable potentials suitable for quantum systems.

Fractal Transport Phenomena through the Scale Relativity Model

A correspondence between Nottale’s scale relativity model and Cresson’s mathematical procedures is analyzed. It results that the “synchronization” of the movements at different scales (fractal scale, differential scale etc.) gives conductive type properties to the fractal fluid, while the absence of “synchronization” is inducing properties of convective type. The behavior of a conductive fractal fluid is illustrated through the numerical simulation of plasma diffusion that is generated by laser ablation. Rotational and irrotational convective behaviors of a fractal fluid are established. Particularly, at Compton spatial and temporal scales, the irrotational behavior implies the standard Schrodinger equation.

Pointwise convergence of solutions to the nonelliptic Schr\"odinger equation

It is conjectured that the solution to the Schrodinger equation in R n+1 converges almost everywhere to its initial datum f, for all f ∈ H s (R n ), if and only if s ≥ 1 4. It is known that there is an s < 1 2 for which the solution converges for all f ∈ H s (R 2 ). We show that the solution to the nonelliptic Schrodinger equation, i∂ t u + (∂ 2 x - ∂ 2 y )u = 0, converges to its initial datum f, for all f ∈ H s (R 2 ), if and only if s ≥ 1 2. Thus the pointwise behaviour is worse than that of the standard Schrodinger equation. In higher dimensions, we have similar results with the loss of the endpoint.

Multitaper Techniques and Filter Diagonalization Methods—A Comparison

In the present contribution we compare the new Multitaper Filtering technique with the very popular Filter Diagonalization Method. The substitution of a time-independent problem, like the standard Schrodinger equation, by a time-dependent one from the Filter Diagonalization Method allows the employment of and comparison with standard signal processing filtration machinery. The use of zero-order prolate spheroidal tapers as filtering functions is here extended and exactly formulated using techniques originating from general investigations of prolate spheroidal wave functions. We investigate the modifications presented with respect to accuracy and general effectiveness. The approach may be useful in various branches of physics and engineering sciences including signal processing applications as well as possibly also in general time-dependent processes.

An integrable Hamiltonian hierarchy and its constrained flows with generalized Hamiltonian regular representations, as well as its expanding integrable system

A new subalgebra of loop algebra A ̃ 2 is first constructed. It follows that an isospectral problem is established. Using Tu-pattern gives rise to a new integrable hierarchy, which possesses bi-Hamiltonian structure. As its reduction cases, the well-known standard Schrodinger equation and MKdV equation are presented, respectively. Furthermore, by making use of bi-symmetry constraints, generalized Hamiltonian regular representations for the hierarchy are obtained. At last, we obtain an expanding integrable system of this hierarchy by applying a scalar transformation between two isospectral problems and constructing a five-dimensional loop algebra G̃. In particular, the expanding integrable models of Schrodinger equation and MKdV equation are presented, respectively.