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Register a new userBohmian Trajectories of the Time-oscillating Schrodinger Equations
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Bohmian mechanics is a non-relativistic quantum theory based on a particle approach. In this paper we study the Schrodinger equation with rapidly oscillating potential and the associated Bohmian trajectory. We prove that the corresponding Bohmian trajectory converges locally in measure, and the limit coincides with the Bohmian trajectory for the effective Schr{o}dinger equation on a finite time interval. This is beneficial for the efficient simulation of the Bohmian trajectories in oscillating potential fields.
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Quantum trajectories are Markov processes that describe the time-evolution of a quantum system undergoing continuous indirect measurement. Mathematically, they are defined as solutions of the so-called Stochastic Schrodinger Equations, which are nonlinear stochastic differential equations driven by Poisson and Wiener processes. This paper is devoted to the study of the invariant measures of quantum trajectories. Particularly, we prove that the invariant measure is unique under an ergodicity condition on the mean time evolution, and a purification condition on the generator of the evolution. We further show that quantum trajectories converge in law exponentially fast towards this invariant measure. We illustrate our results with examples where we can derive explicit expressions for the invariant measure.
For multi-time wave functions, which naturally arise as the relativistic particle-position representation of the quantum state vector, the analog of the Schrodinger equation consists of several equations, one for each time variable. This leads to the question of how to prove the consistency of such a system of PDEs. The question becomes more difficult for theories with particle creation, as then different sectors of the wave function have different numbers of time variables. Petrat and Tumulka (2014) gave an example of such a model and a non-rigorous argument for its consistency. We give here a rigorous version of the argument after introducing an ultraviolet cut-off into the creation and annihilation terms of the multi-time evolution equations. These equations form an infinite system of coupled PDEs; they are based on the Dirac equation but are not fully relativistic (in part because of the cut-off). We prove the existence and uniqueness of a smooth solution to this system for every initial wave function from a certain class that corresponds to a dense subspace in the appropriate Hilbert space.
The Darwin-Howie-Whelan equations are commonly used to describe and simulate the scattering of fast electrons in transmission electron microscopy. They are a system of infinitely many envelope functions, derived from the Schrodinger equation. However, for the simulation of images only a finite set of envelope functions is used, leading to a system of ordinary differential equations in thickness direction of the specimen. We study the mathematical structure of this system and provide error estimates to evaluate the accuracy of special approximations, like the two-beam and the systematic-row approximation.
We solve the time-dependent Schrodinger equation describing the emission of electrons from a metal surface by an external electric field $E$, turned on at $t=0$. Starting with a wave function $psi(x,0)$, representing a generalized eigenfunction when $E=0$, we find $psi(x,t)$ and show that it approaches, as $ttoinfty$, the Fowler-Nordheim tunneling wavefunction $psi_E$. The deviation of $psi$ from $psi_E$ decays asymptotically as a power law $t^{-frac32}$. The time scales involved for typical metals and fields of several V/nm are of the order of femtoseconds.
A Bose-Einstein condensate (BEC) confined in a one-dimensional lattice under the effect of an external homogeneous field is described by the Gross-Pitaevskii equation. Here we prove that such an equation can be reduced, in the semiclassical limit and in the case of a lattice with a finite number of wells, to a finite-dimensional discrete nonlinear Schrodinger equation. Then, by means of numerical experiments we show that the BECs center of mass exhibits an oscillating behavior with modulated amplitude; in particular, we show that the oscillating period actually depends on the shape of the initial wavefunction of the condensate as well as on the strength of the nonlinear term. This fact opens a question concerning the validity of a method proposed for the determination of the gravitational constant by means of the measurement of the oscillating period.
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