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Journal of Mathematical Physics

The time-dependent Schrödinger equation in three dimensions under geometric constraints

Irina Petreska, Antonio S. M. de Castro, Trifce Sandev, Ervin Kaminski Lenzi

Published Mar 1, 2019• 16 citations
Publisher

Abstract

We consider a quantum motion governed by the time-dependent Schrödinger equation on a three dimensional comb structure. We derive the corresponding fractional Schrödinger equations for the reduced probability density functions by projection of the three dimensional comb dynamics in the two- and one-dimensional configuration space. This represents another physical example of a system where fractional calculus emerges. We give closed-form solutions of the corresponding equations for the reduced probability density functions in terms of the Fox H-function, by using the Green’s function approach.

DOI
10.1063/1.5079226

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We consider a quantum motion governed by the time-dependent Schrödinger equation on a three dimensional comb structure. We derive the corresponding fractional Schrödinger equations for the reduced probability density functions by projection of the three dimensional comb dynamics in the two- and one-dimensional configuration space. This represents another physical example of a system where fractional calculus emerges. We give closed-form solutions of the corresponding equations for the reduced probability density functions in terms of the Fox H-function, by using the Green’s function approach.
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Quantum dynamics of a particle on a two-dimensional comb structure is considered. This dynamics of a Hamiltonian system with a topologically constrained geometry leads to the non-Markovian behavior. In the framework of a rigorous analytical consideration, it is shown how a fractional time derivative appears for the relevant description of this non-Markovian quantum mechanics in the framework of fractional time Schrödinger equations. Analytical solutions for the Green functions are obtained for both conservative and periodically driven in time Hamiltonian systems.
Application of the fractional calculus to quantum processes is presented. In particular, the quantum dynamics is considered in the framework of the fractional time Schrödinger equation (SE), which differs from the standard SE by the fractional time derivative: partial differential/partial differentialt --> partial differential(alpha)/partial differentialt(alpha). It is shown that for alpha=1/2 the fractional SE is isospectral to a comb model. An analytical expression for the Green's functions of the systems are obtained. The semiclassical limit is discussed.
The ‘diffraction in space’ and the ‘diffraction in time’ phenomena are considered in regard to a continuously open, and a closed shutter that is opened at an instant in time, respectively. The purpose of this is to provide a background to the principal theme of this article, which is to extend the ‘quantum shutter problem’ for the case when the wave function is determined by the fundamental solution to a partial differential equation with a fractional derivative of space or of time. This involves the development of Green’s function solutions for the space- and time-fractional Schrödinger equation and the time-fractional Klein–Gordon equation (for the semi-relativistic case). In each case, the focus is on the development of primarily one-dimensional solutions, subject to an initial condition which controls the dynamical behaviour of the wave function. Coupled with variations in the fractional order of the fractional derivatives, illustrative example results are provided that are based on presenting space-time maps of the wave function; specifically, the probability density of the wave function. In this context, the paper provides a case study of fractional quantum mechanics and control using fractional calculus.
The time fractional Schrodinger equation (TFSE) for a nonrelativistic particle is derived on the basis of the Feynman path integral method by extending it initially to the case of a "free particle" obeying fractional dynamics, obtained by replacing the integer order derivatives with respect to time by those of fractional order.The equations of motion contain quantities which have "fractional" dimensions, chosen such that the "energy" has the correct dimension [𝑀𝐿 2 /𝑇 2 ].The action 𝑆 is defined as a fractional time integral of the Lagrangian, and a "fractional Planck constant" is introduced.The TFSE corresponds to a "subdiffusion" equation with an imaginary fractional diffusion constant and reproduces the regular Schrodinger equation in the limit of integer order.The present work corrects a number of errors in Naber's work.The correct continuity equation for the probability density is derived and a Green function solution for the case of a "free particle" is obtained.The total probability for a "free" particle is shown to go to zero in the limit of infinite time, in contrast with Naber's result of a total probability greater than unity.A generalization to the case of a particle moving in a potential is also given.
The 2D space-fractional Schrödinger equation in the time-independent and time-dependent cases for the scattering problems in the fractional quantum mechanics is studied. We define the Green's functions for the two cases and give the mathematical expression of them in infinite series form and in terms of some special functions. The asymptotic formulas of the Green's functions are also given, and applied to get the approximate wave functions for the fractional quantum scattering problems. These results contain those in the standard (integer) quantum mechanics as special cases, and can be applied to study the complex quantum systems.