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A fractional Schrödinger equation and its solution. (English) Zbl 1197.81126
Summary: This paper presents a fractional Schrödinger equation and its solution. The fractional Schrödinger equation may be obtained using a fractional variational principle and a fractional Klein-Gordon equation; both methods are considered here. We extend the variational formulations for fractional discrete systems to fractional field systems defined in terms of Caputo derivatives to obtain the fractional Euler-Lagrange equations of motion. We present the Lagrangian for the fractional Schrödinger equation of order $\alpha$. We also use a fractional Klein-Gordon equation to obtain the fractional Schrödinger equation which is the same as that obtained using the fractional variational principle. As an example, we consider the eigensolutions of a particle in an infinite potential well. The solutions are obtained in terms of the sines of the Mittag-Leffler function.

MSC:
81Q05Closed and approximate solutions to quantum-mechanical equations
26A33Fractional derivatives and integrals (real functions)
35R11Fractional partial differential equations
70H03Lagrange’s equations
49S05Variational principles of physics
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References:
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