On the motion of a quantum particle in the spinning cosmic string space–time

We analyze the energy spectrum and the wave function of a particle subjected to magnetic field in the spinning cosmic string space–time and investigate the influence of the spinning reference frame and topological defect on the system. To do this we solve Schrödinger equation in the spinning cosmic string background. In our work, instead of using an approximation in the calculations, we use the quasi-exact ansatz approach which gives the exact solutions for some primary levels.     

The dynamic natures of microscopic particles described by nonlinear Schrödinger equation

There are a lot of difficulties and troubles in quantum mechanics, when the linear Schrödinger equation is used to describe microscopic particles. Thus, we here replace it by a nonlinear Schrödinger equation to investigate the properties and rule of microscopic particles. In such a case we find that the motion of microscopic particle satisfies classical rule and obeys the Hamiltonian principle, Lagrangian and Hamilton equations. We verify further the correctness of these conclusions by the results of nonlinear Schrödinger equation under actions of different externally applied potential. From these studies, we see clearly that rules and features of motion of microscopic particle described by nonlinear Schrödinger equation are greatly different from those in the linear Schrödinger equation, they have many classical properties, which are consistent with concept of corpuscles. Thus, we should use the nonlinear Schrödinger equation to describe microscopic particles.     

Spinor with Schrödinger symmetry and non-relativistic supersymmetry

We construct the d-dimensional “half” Schrödinger equation, which is a kind of the root of the Schrödinger equation, from the (d+1)-dimensional free Dirac equation. The solution of the “half” Schrödinger equation also satisfies the usual free Schrödinger equation. We also find that the explicit transformation laws of the Schrödinger and the half Schrödinger fields under the Schrödinger symmetry transformation are derived by starting from the Klein-Gordon equation and the Dirac equation in d+1 dimensions. We derive the 3- and 4-dimensional super-Schrödinger algebra from the superconformal algebra in 4 and 5 dimensions. The algebra is realized by introducing two complex scalar and one (complex) spinor fields and the explicit transformation properties have been found.     

A quantum model of option pricing: When Black-Scholes meets Schrödinger and its semi-classical limit

The Black-Scholes equation can be interpreted from the point of view of quantum mechanics, as the imaginary time Schrödinger equation of a free particle. When deviations of this state of equilibrium are considered, as a product of some market imperfection, such as: Transaction cost, asymmetric information issues, short-term volatility, extreme discontinuities, or serial correlations; the classical non-arbitrage assumption of the Black-Scholes model is violated, implying a non-risk-free portfolio. From Haven (2002) [1] we know that an arbitrage environment is a necessary condition to embedding the Black-Scholes option pricing model in a more general quantum physics setting. The aim of this paper is to propose a new Black-Scholes-Schrödinger model based on the endogenous arbitrage option pricing formulation introduced by Contreras et al. (2010) [2]. Hence, we derive a more general quantum model of option pricing, that incorporates arbitrage as an external time dependent force, which has an associated potential related to the random dynamic of the underlying asset price. This new resultant model can be interpreted as a Schrödinger equation in imaginary time for a particle of mass 1/σ² with a wave function in an external field force generated by the arbitrage potential. As pointed out above, this new model can be seen as a more general formulation, where the perfect market equilibrium state postulated by the Black-Scholes model represent a particular case. Finally, since the Schrödinger equation is in place, we can apply semiclassical methods, of common use in theoretical physics, to find an approximate analytical solution of the Black-Scholes equation in the presence of market imperfections, as it is the case of an arbitrage bubble. Here, as a numerical illustration of the potential of this Schrödinger equation analogy, the semiclassical approximation is performed for different arbitrage bubble forms (step, linear and parabolic) and compare with the exact solution of our general quantum model of option pricing.     

On Quasi-Exact Solvability of the Schrödinger Equation for a Free Particle on the Surface of a Spindle Torus

We show that the Schrödinger equation for a free particle on the surface of a spindle torus is quasi-exactly solvable. Our result complements former ones in an interesting way: it is known that the Schrödinger equation for a free particle on a ring torus is non-solvable, whereas it is exactly solvable for a particle on a horn torus.     

String Without Strings

Scale invariance provides a principled reason for the physical importance of Hilbert space, the Virasoro algebra, the string mode expansion, canonical commutators and Schrödinger evolution of states, independent of the assumptions of string theory and quantum theory. The usual properties of dimensionful fields imply an infinite, projective tower of conformal weights associated with the tangent space to scale-invariant spacetimes. Convergence and measurability on this tangent tower are guaranteed using a scale-invariant norm, restricted to conformally self-dual vectors. Maps on the resulting Hilbert space are correspondingly restricted to semi-definite conformal weight. We find the maximally- and minimally-commuting, complete Lie algebras of definite-weight operators. The projective symmetry of the tower gives these algebras central charges, giving the canonical commutator and quantum Virasoro algebras, respectively. Using a continuous, m-parameter representation for rank-m tower tensors, we show that the parallel transport equation for the momentum-vector of a particle is the Schrödinger equation, while the associated definite-weight operators obey canonical commutation relations. Generalizing to the set of integral curves of general timelike, self-dual vector-valued weight maps gives a lifting such that the action of the curves parallel transports arbitrary tower vectors. We prove that the full set of Schrödinger-lifted integral curves of a general self-dual map gives an immersion of its 2-dim parameter space into spacetime, inducing a Lorentzian metric on the parameter space. This immersion is shown to satisfy the full variational equations of open string.     

Time-Dependent Transport in Nanoscale Devices

A method for simulating ballistic time-dependent device transport, which solves the time-dependent Sehrǒdinger equation using the finite difference time domain （FDTD） method together with Poisson＇s equation, is described in detail. The effective mass Schrǒdinger equation is solved. The continuous energy spectrum of the system is discretized using adaptive mesh, resulting in energy levels that sample the density-of-states. By calculating time evolution of wavefunctions at sampled energies, time-dependent transport characteristics such as current and charge density distributions are obtained. Simulation results in a nanowire and a coaxially gated carbon nanotube field-effect transistor （CNTFET） are presented. Transient effects, e.g., finite rising time, are investigated in these devices.     

The difference between Schrödinger equation derived from Schrödinger map and Landau-Lifshitz equation

We provide an explicit blow up solution of Schrödinger equation derived from Schrödinger map. Consequently we show the non-equivalence between the Schrödinger equation and Landau-Lifshitz equation. We also find that two class of equivariant solutions of Landau-Lifshitz equation are static.     

Exact solution of the Schrödinger equation for a particle in a regular N-simplex

We present the exact solution for the Schrödinger equation for a particle inside an N-dimensional regular simplex shaped enclosure. This result extends and unifies the earlier results for equilateral triangle and K-tetrahedron billiards.     

Derivation of quantum mechanics from the Boltzmann equation for the Planck aether

The Planck aether hypothesis assumes that space is densely filled with an equal number of locally interacting positive and negative Planck masses obeying an exactly nonrelativistic law of motion. The Planck masses can be described by a quantum mechanical two-component nonrelativistic operator field equation having the form of a two-component nonlinear Schrödinger equation, with a spectrum of quasiparticles obeying Lorentz invariance as a dynamic symmetry for energies small compared to the Planck energy. We show that quantum mechanics itself can be derived from the Newtonian mechanics of the Planck aether as an approximate solution of Boltzmann's equation for the locally interacting positive and negative Planck masses, and that the validity of the nonrelativistic Schrödinger equation depends on Lorentz invariance as a dynamic symmetry. We also show how the many-body Schrödinger wave function can be factorized into a product of quasiparticles of the Planck aether with separable quantum potentials. Finally, we present a possible explanation of wave function collapse as a kind of enhanced gravitational collapse in the presence of the negative Planck masses.     

A parallel algorithm for solving the 3d Schrödinger equation

We describe a parallel algorithm for solving the time-independent 3d Schrödinger equation using the finite difference time domain (FDTD) method. We introduce an optimized parallelization scheme that reduces communication overhead between computational nodes. We demonstrate that the compute time, t, scales inversely with the number of computational nodes as t ∝ (N_nodes)^{−0.95 ± 0.04}. This makes it possible to solve the 3d Schrödinger equation on extremely large spatial lattices using a small computing cluster. In addition, we present a new method for precisely determining the energy eigenvalues and wavefunctions of quantum states based on a symmetry constraint on the FDTD initial condition. Finally, we discuss the usage of multi-resolution techniques in order to speed up convergence on extremely large lattices.     

Symmetries of the Free Schrödinger Equation

An algorithm is proposed for studying the symmetry properties of equations used in theoretical and mathematical physics. The application of this algorithm to the free Schrödinger equation permits one to establish that, in addition to the known Galilei symmetry, the free Schrödinger equation possesses also relativistic symmetry in some generalized sense. This property of the free Schrödinger equation provides an extension of the equation into the relativistic domain of the free particle motion under study.     

Algebraic approach and coherent states for a relativistic quantum particle in cosmic string spacetime

We study a relativistic quantum particle in cosmic string spacetime in the presence of a magnetic field and a Coulomb-type scalar potential. It is shown that the radial part of this problem possesses the su(1,1)$su(1,1)$ symmetry. We obtain the energy spectrum and eigenfunctions of this problem by using two algebraic methods: the Schrödinger factorization and the tilting transformation. Finally, we give the explicit form of the relativistic coherent states for this problem.     

Klein-Gordon-Wheeler-DeWitt-Schrödinger equation

We start from the Einstein-Hilbert action for the gravitational field in the presence of a “point particle” source, and cast the action into the corresponding phase space form. The dynamical variables of such a system satisfy the point particle mass shell constraint, the Hamilton and the momentum constraints of the canonical gravity. In the quantized theory, those constraints become operators that annihilate a state. A state can be represented by a wave functional Ψ that simultaneously satisfies the Klein-Gordon and the Wheeler-DeWitt-Schrödinger equation. The latter equation, besides the term due to gravity, also contains the Schrödinger like term, namely the derivative of Ψ with respect to time, that occurs because of the presence of the point particle. The particle?s time coordinate, X⁰, serves the role of time. Next, we generalize the system to p-branes, and find out that for a quantized spacetime filling brane there occurs an effective cosmological constant, proportional to the expectation value of the brane?s momentum, a degree of freedom that has two discrete values only, a positive and a negative one. This mechanism could be an explanation for the small cosmological constant that drives the accelerated expansion of the universe.     

ON FINITE NORMAL SERIES SOLUTIONS OF THE SCHRÖDINGER EQUATION WITH IRREGULAR SINGULARITY

We compute all potentials with the following property: The one-dimensional nonrelativistic Schrödinger equation for these potentials has irregular singular points at infinity and/or zero and is solved by a finite normal series. We restrict to expansion order zero, discuss some properties of the potentials obtained and, as an application, calculate for some given potentials exact solutions and energies. The aim of this paper is to provide a tool for finding exact solutions of the Schrödinger equation for a large class of singular potentials.     

Diffusive, super-diffusive and ballistic transport in the long-range correlated 1D Anderson model

In this paper, we study the 1D Anderson model with long-range correlated on-site energies. This diagonal-correlated disorder is considered in such a way that the random sequence of site energies ε_n has a 1/k^α power spectrum, where k is the wave-vector of the modulations on the random sequence landscape. Using the Runge-Kutta method to solve the time-dependent Schrödinger equation, we compute the participation number and the Shannon entropy for an initially localized wave packet. We observe that strong correlations can induce ballistic transport associated with the emergence of low-energy extended states, in agreement with previous works in this model. We further identify an intermediate regime with super-diffusive spreading of the wave-packet.     

Lévy processes and Schrödinger equation

We analyze the extension of the well known relation between Brownian motion and the Schrödinger equation to the family of the Lévy processes. We consider a Lévy-Schrödinger equation where the usual kinetic energy operator-the Laplacian-is generalized by means of a selfadjoint, pseudodifferential operator whose symbol is the logarithmic characteristic of an infinitely divisible law. The Lévy-Khintchin formula shows then how to write down this operator in an integro-differential form. When the underlying Lévy process is stable we recover as a particular case the fractional Schrödinger equation. A few examples are finally given and we find that there are physically relevant models-such as a form of the relativistic Schrödinger equation-that are in the domain of the non stable Lévy-Schrödinger equations.     

Effective Field Theories for Disordered Systems from a Generalized Phase Formulation

We consider a spinless particle moving in a d-dimensional box, subject to periodic boundary conditions, and in the presence of a random potential. Introducing the logarithm of the wave function transforms the time-independent Schrödinger equation into a stochastic differential equation with the random potential acting as the source. Using this as our starting point we write functional integral representations for the disorder averaged density of states, the two point correlator of the absolute value of the wave function, and inverse participation ratios. We also show how a deterministic or random magnetic field can be included in the formalism.