Fractional Schrödinger Equation in the Presence of the Linear Potential

In this paper, we consider the time-dependent Schrödinger equation: i ∂ ψ ( x , t ) ∂ t = 1 2 ( − Δ ) α 2 ψ ( x , t ) + V ( x ) ψ ( x , t ) , x ∈ R , t > 0 with the Riesz space-fractional derivative of order 0 < α ≤ 2 in the presence of the linear potential V ( x ) = β x . The wave function to the one-dimensional Schrödinger equation in momentum space is given in closed form allowing the determination of other measurable quantities such as the mean square displacement. Analytical solutions are derived for the relevant case of α = 1 , which are useable for studying the propagation of wave packets that undergo spreading and splitting. We furthermore address the two-dimensional space-fractional Schrödinger equation under consideration of the potential V ( ρ ) = F · ρ including the free particle case. The derived equations are illustrated in different ways and verified by comparisons with a recently proposed numerical approach

Analytische Lösungen der Strahlungstransfergleichung und ihrer Approximationen

This thesis presents the derivation of analytical expressions of the radiative transfer equation and their approximations in the steady-state and time domains. The diffusion equation, which is the most often used approximation of the radiative transfer equation, is solved analytically for different homogeneous and layered geometries. By applying the integral transform formalism the respective form of the diffusion equation is reduced to an ordinary differential equation. At this stage the solution of the boundary value problem is obtained via standard techniques. The derived solutions were validated against other independent analytical methods found in literature and the Monte Carlo method. Analytical solutions of the simplified spherical harmonics equations are derived for homogeneous media. The Fourier transform is used to convert these coupled diffusion-like equations to a system of ordinary differential equations. The obtained solutions for infinite and semi-infinite media were validated against the finite difference method. The infinite space fluence within the transport theory is derived for an anisotropically scattering medium and different source distributions. Additionally, for the case of isotropic scattering a simplified version is given. Based on results of the infinite space fluence and by applying the eigenvalue method the radiance caused by an isotropic point source in an infinitely extended medium is obtained. The analytical solutions were compared to the Monte Carlo method. Within the stochastic nature of the Monte Carlo simulations an exact agreement was found in the steady-state and time domains

Fractional Schrödinger Equation in the Presence of the Linear Potential

In this paper, we consider the time-dependent Schrödinger equation: i ∂ ψ ( x , t ) ∂ t = 1 2 ( − Δ ) α 2 ψ ( x , t ) + V ( x ) ψ ( x , t ) , x ∈ R , t > 0 with the Riesz space-fractional derivative of order 0 < α ≤ 2 in the presence of the linear potential V ( x ) = β x . The wave function to the one-dimensional Schrödinger equation in momentum space is given in closed form allowing the determination of other measurable quantities such as the mean square displacement. Analytical solutions are derived for the relevant case of α = 1 , which are useable for studying the propagation of wave packets that undergo spreading and splitting. We furthermore address the two-dimensional space-fractional Schrödinger equation under consideration of the potential V ( ρ ) = F · ρ including the free particle case. The derived equations are illustrated in different ways and verified by comparisons with a recently proposed numerical approach

Analytical solution of the correlation transport equation with static background: beyond diffuse correlation spectroscopy

The correlation transport equation (CTE) is the natural generalization of the theory for diffusion correlation spectroscopy and represents a more precise model when dealing with measurements of particle movement in fluids or red blood cell flow in biological tissues. Unfortunately, the CTE is not methodically used due to the complexity of finding solutions. It is shown that actually a very simple modification of the theory/software for the solution of the radiative transport equation allows one to obtain exact solutions of the CTE. The presence of a static background is also taken into account and its influence on the CTE solutions is discussed. The proposed approach permits one to easily work beyond the diffusion regime and potentially for any optical and/or physiological value. The validity of the approach is demonstrated by using "gold standard" Monte Carlo simulations

P3 solution for the total steady-state and time-resolved reflectance and transmittance from a turbid slab

In this paper, we derive some explicit analytical solutions to the P3 equations for the slab geometry that is illuminated by a collimated plane source. The resulting expressions for the total reflectance and transmittance are compared with the corresponding transport theory solution predicted by the Monte Carlo method. Further, for the special case of a non-absorbing anisotropically scattering slab, simple and accurate expressions in the P1 approximation are obtained, yielding for optically thick slabs, the typical behavior of Ohm's law. In view of the time domain, we present an alternative method to the classical frequency-domain approach avoiding the use of complex numbers

Derivation of the correlation diffusion equation with static background and analytical solutions

A new correlation diffusion equation has been derived from a correlation transport equation allowing one to take into account the presence of moving scatterers and static background. Solutions for the reflectance from a semi-infinite medium have been obtained (point-like and ring detectors). The solutions have been tested by comparisons with "gold standard" Monte Carlo (MC) simulations. These formulas suitably describe the electric field autocorrelation function, for Brownian or random movement of the scatterers, even in the case where the probability for a photon to interact with a moving scatterer is very low. The proposed analytical models and the MC simulations show that the "classical" model, often used in diffuse correlation spectroscopy, underestimates the normalized field autocorrelation function for increasing correlation times

Radiative transfer equation for predicting light propagation in biological media: comparison of a modified finite volume method, the monte carlo technique, and an exact analytical solution

International audienceWe examine the accuracy of a modified finite volume method compared to analytical and Monte Carlosolutions for solving the radiative transfer equation. The model is used for predicting light propagation within atwo-dimensional absorbing and highly forward-scattering medium such as biological tissue subjected to a colli-mated light beam. Numerical simulations for the spatially resolved reflectance and transmittance are presentedconsidering refractive index mismatch with Fresnel reflection at the interface, homogeneous and two-layeredmedia. Time-dependent as well as steady-state cases are considered. In the steady state, it is found that themodified finite volume method is in good agreement with the other two methods. The relative differencesbetween the solutions are found to decrease with spatial mesh refinement applied for the modified finite volumemethod obtaining <2.4%. In the time domain, the fourth-order Runge-Kutta method is used for the time semi-discretization of the radiative transfer equation. An agreement among the modified finite volume method, Runge-Kutta method, and Monte Carlo solutions are shown, but with relative differences higher than in the steady state

Simulation of light propagation in biological tissue using a modified finite volume method applied to three-dimensional radiative transport equation

International audienceAn important issue in tissue optics and Optical Tomography is to have an efficient forward solver. In this work, a new numerical algorithm was developed for solving light propagation with the radiative transport equation within a three-dimensional absorbing and a highly forward-scattering medium such as a biological tissue subjected to an incident beam. Both elastically scattered light and fluorescence light were studied. Two steady state problems used to assess the performance and accuracy of the proposed algorithm are presented. We show that it is possible to obtain a good level of accuracy with a deterministic numerical method: relative differences less than 1.7% and 4.5% were obtained when compared against Monte Carlo solutions for problems of elastically scattered light and fluorescence light, respectively