Abstract
We analyze the generalized time-dependent Schrodinger equation for the force free case, as a generalization, for example, of the standard time-dependent Schrodinger equation, time fractional Schrodinger equation, distributed order time fractional Schrodinger equation, and tempered in time Schrodinger equation. We relate it to the corresponding standard Schrodinger equation with effective potential. The general form of the effective potential that leads to a standard time-dependent Schrodinger equation with the same solution as the generalized one is derived explicitly. Further, effective potentials for several special cases, such as Dirac delta, power-law, Mittag-Leffler and truncated power-law memory kernels, are expressed in terms of the Mittag-Leffler functions. Such complex potentials have been used in the transport simulations in quantum dots, and in simulation of resonant tunneling diode.
Highlights
In our recent work [15], we considered time-dependent Schrödinger equation with memory kernel and we show that such generalized equation contains many already investigated special cases, such as standard Schrödinger equation, time fractional Schrödinger equation with Caputo fractional derivative, and distributed order Schrödinger equation
The analyzis performed here has a particular case the one presented in Reference [5] for a time fractional Schrödinger equation which may correspond to a choice of γ(t)
Note that for α → 1 the effective potential (10) becomes zero as it is expected, since the time fractional Schrödinger equation turns to standard Schrödinger equation, i.e., the Caputo time fractional derivative becomes ordinary derivative
Summary
A series of interesting and unusual properties of the state-of-the art low-dimensional quantum systems requires new approaches in mathematical modeling by means of the Schrödinger equation. The effective potential approach is of great importance for elucidating for example, dissipative quantum transport processes in quantum dots [17,18] This motivates us to further extend and upgrade the quantum transport modeling by means of generalized Schrödinger equation. We derive a general form of the imaginary effective potential which, at same boundary conditions, relates the standard Schrödinger equation to the generalized Schrödinger equation with a memory kernel. We show that by using the appropriately derived effective potential, one may consider a generalized Schrödinger equation with a memory kernel instead of the standard time-dependent. The advantage of this approach is that the solutions of generalized Schrödinger equation can be represented in an elegant manner by means of the M-L and Fox H-functions, enabling a comprehensive mathematical modeling for a wide class of problems.
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