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Fractional Schrödinger wave equation and fractional uncertainty principle. (English) Zbl 1146.35082

Summary: Free particle wavefunction of the fractional Schrödinger wave equation is obtained. The wavefunction of the equation is represented in terms of generalized three-dimensional Green’s function that involves fractional powers of time as variable \(t^\alpha\). It is shown that the wavefunction corresponding to the integral order Schrödinger wave equation follows as special case of that of the corresponding Schrödinger equation with fractional derivatives with respect to time. The wavefunction is obtained using Laplace and Fourier transforms methods and eventually the wavefunction is expressed in terms of Mittag-Leffler function. Heisenberg uncertainty principle is deduced from the wavefunction of the fractional Schrödinger equation using the integral value of fractional parameter \(\alpha=1\).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35A22 Transform methods (e.g., integral transforms) applied to PDEs
43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
35J10 Schrödinger operator, Schrödinger equation
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