Fractional Schrödinger Equation in the Presence of the Linear Potential

Abstract

Abstract:In this paper, we consider the time-dependent Schrödinger equation:with the Riesz space-fractional derivative of order 0 < α ≤ 2 in the presence of the linear potential V(x) = βx. 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.

An Investigation on Reliable Analytical and Numerical Methods for the Riesz Fractional Nonlinear Schrödinger Equation inQuantum Mechanics

“…. Now substituting Equation (3.10) into Equation (3.5) and equating the coefficients of like powers of p, we obtain the governing equation forû 1 (k, t) as follows 11) and thus, the general equations for u j (k, t) is represented by 12) where N m û 0 (k, t) ,û 1 (k, t) , . .…”

Qualitative Analysis of Multi-Terms Fractional Order Delay Differential Equations via the Topological Degree Theory

“…Now we provide stability results for the problem (5). Here we say that the goal of stability analysis of time delay problems/systems is to find the region in the delay parameter space where the considered problem/system is still stable.…”

“…Nowadays it has become an important tool due to its wide range of applications in various scientific disciplines such as biology, chemistry, physics, dynamical systems, electrodynamics, etc. (see [1][2][3][4][5][6] and references therein). Its importance can also be explored in other fields like fluid dynamics traffic models, oscillation due to earthquakes, flow in porous media due to seepage, etc.…”

“…Therefore investigation of problems under nonlocal initial or boundary conditions is one of the important areas of research in recent times (for detail we refer to [24,25]). Inspired by the aforesaid work, this research aims to study Equation (5) under generalized nonlocal integral boundary condition as:…”

Effective Potential from the Generalized Time-Dependent Schrödinger Equation

“…Different generalizations of the standard Schrödinger equation by introducing either time fractional derivatives or space fractional derivatives have been introduced for this reason [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. It has been shown that their solutions can be represented by using the Mittag-Leffler (M-L) and Fox H-functions.…”