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Time-Independent Fractional Schrödinger Equation

WORLD SCIENTIFIC eBooksMay 23, 20180 citations

An iteration algorithm for the time-independent fractional Schrödinger equation with Coulomb potential

Marwan Al‐Raeei, Moustafa Sayem El‐Daher

PramanaOct 29, 202010 citations

On the consistency of the solutions of the space fractional Schrödinger equation

Selçuk Ş. Bayın

Recently, it was pointed out that the solutions found in the literature for the space fractional Schrödinger equation in a piecewise manner are wrong, except the case with the delta potential. We re-analyze this problem and show that an exact and a proper treatment of the relevant integral prove otherwise. We also discuss effective potential approach and present a free particle solution for the space and time fractional Schrödinger equation in general coordinates in terms of Fox's H-functions.Recently, it was pointed out that the solutions found in the literature for the space fractional Schrödinger equation in a piecewise manner are wrong, except the case with the delta potential. We re-analyze this problem and show that an exact and a proper treatment of the relevant integral prove otherwise. We also discuss effective potential approach and present a free particle solution for the space and time fractional Schrödinger equation in general coordinates in terms of Fox's H-functions.

arXiv:1203.4556Journal of Mathematical PhysicsApr 1, 201266 citations

Separation of Variables

Steven M. Pascal

The time-dependent Schrödinger equation (TDSE) contains two variables: position and time. Therefore, the solutions are functions of both space and time (x and t, if we limit ourselves to particles moving in one dimension). It turns out that the easiest way to "solve" such an equation of two variables is to perform a separation of variables. We show that if the potential does not change with time, then certain special solutions (eigenfunctions) can be written as a product of a space function and a time function: Ψ(x,t) = ψ(x) f(t). By assuming a "separable" solution of this form, and plugging into the TDSE, the TDSE itself can be separated into two equations: the time-independent Schrödinger equation (TISE) with solution ψ(x), and the space-independent Schrödinger equation (SISE) with solution f(t). Furthermore, f(t) always equals e -iωt , while ψ(x) depends on the potential. So, for potentials that do not vary with time, we need only solve the much simpler TISE in order to investigate the energy, momentum, and so on of the particle. This is why, for the remainder of this book, we only use the TISE.The time-dependent Schrödinger equation (TDSE) contains two variables: position and time. Therefore, the solutions are functions of both space and time (x and t, if we limit ourselves to particles moving in one dimension). It turns out that the easiest way to "solve" such an equation of two variables is to perform a separation of variables. We show that if the potential does not change with time, then certain special solutions (eigenfunctions) can be written as a product of a space function and a time function: Ψ(x,t) = ψ(x) f(t). By assuming a "separable" solution of this form, and plugging into the TDSE, the TDSE itself can be separated into two equations: the time-independent Schrödinger equation (TISE) with solution ψ(x), and the space-independent Schrödinger equation (SISE) with solution f(t). Furthermore, f(t) always equals e -iωt , while ψ(x) depends on the potential. So, for potentials that do not vary with time, we need only solve the much simpler TISE in order to investigate the energy, momentum, and so on of the particle. This is why, for the remainder of this book, we only use the TISE.

CRC Press eBooksJul 12, 20230 citations

Time dependent solutions for fractional coupled Schrödinger equations

Ervin Kaminski Lenzi, Antonio S. M. de Castro, R. S. Mendes

Applied Mathematics and ComputationNov 7, 20186 citations

Time fractional Schrödinger equation: Fox's H-functions and the effective potential

Selçuk Ş. Bayın

After introducing the formalism of the general space and time fractional Schrödinger equation, we concentrate on the time fractional Schrödinger equation and present new results via the elegant language of Fox's H-functions. We show that the general time dependent part of the wave function for the separable solutions of the time-fractional Schrödinger equation is the Mittag-Leffler function with an imaginary argument by two different methods. After separating the Mittag-Leffler function into its real and imaginary parts, in contrast to existing works, we show that the total probability is ⩽1 and decays with time. Introducing the effective potential approach, we also write the Mittag-Leffler function with an imaginary argument as the product of its purely decaying and purely oscillating parts. In the light of these, we reconsider the simple box problem.After introducing the formalism of the general space and time fractional Schrödinger equation, we concentrate on the time fractional Schrödinger equation and present new results via the elegant language of Fox's H-functions. We show that the general time dependent part of the wave function for the separable solutions of the time-fractional Schrödinger equation is the Mittag-Leffler function with an imaginary argument by two different methods. After separating the Mittag-Leffler function into its real and imaginary parts, in contrast to existing works, we show that the total probability is ⩽1 and decays with time. Introducing the effective potential approach, we also write the Mittag-Leffler function with an imaginary argument as the product of its purely decaying and purely oscillating parts. In the light of these, we reconsider the simple box problem.

arXiv:1103.3295Journal of Mathematical PhysicsJan 1, 201355 citations

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