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Journal of Physical Studies

Time-dependent Schrödinger equation with one known solution

T V Fityo, V. M. Tkachuk

Published Jan 1, 2005• 1 citation
Publisher

Abstract

We present a method to construct a time-dependent Schrödinger equation with one known solution.Several explicit examples are presented.The quantum Kapitsa effect is discussed.
DOI
10.30970/jps.09.299

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We analyze the generalized time-dependent Schrödinger equation for the force free case, as a generalization, for example, of the standard time-dependent Schrödinger equation, time fractional Schrödinger equation, distributed order time fractional Schrödinger equation, and tempered in time Schrödinger equation. We relate it to the corresponding standard Schrödinger 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.