Correction to: Scientific Reports https://doi.org/10.1038/s41598-020-76512-5, published online 10 November 2020


The original version of this Article contained errors in the Formalism section, where Equation 1 and the subsequent formula were incorrect and Equations 2–4 were omitted.


“The wavepacket propagation in the two-dimensional fractional Schrodinger equation formalism can be studied by using:

$$i\frac{\partial }{\partial t}\psi \left( {x,y,t} \right) = \left[ {{\beta }\left( { - \Delta} \right)^{\alpha /2} - \gamma \left| {\psi \left( {x,y,t} \right)} \right|^{2} + V\left( {x,y} \right)} \right]\psi \left( {x,y,t} \right)$$
(1)

where \({\alpha }\), \(\beta\), and \({\gamma }\) are the fractional derivative order, Laplacian coefficient, and nonlinear interaction strength, respectively. We also have \(\Delta = \partial^{2} /\partial x^{2} + \partial^{2} /\partial y^{2}\).”


now reads:


“The wavepacket propagation in the two-dimensional fractional Schrodinger equation formalism can be studied by using:

$$i\frac{\partial \psi }{{\partial t}} = \left[ {{\upbeta }{\text{Q}}_{{\text{R}}} \left( {{\text{x}},{\text{y}},t,\alpha } \right)\left( { - \Delta^{2} } \right)^{{\frac{\alpha }{2}}} - \gamma \left| {\psi \left( {x,y,t} \right)} \right|^{2} + {\text{M}}\left( {{\text{x}},{\text{y}},p_{x} ,p_{y} ,t,\alpha } \right)} \right]\psi \left( {x,y,t} \right)$$
(1)

where \({\alpha }\), \({\upbeta}\), and \({\gamma }\) are the fractional derivative order, Laplacian coefficient, and nonlinear interaction strength, respectively. We also have \(\Delta^{2} = \partial^{2} /\partial x^{2} + \partial^{2} /\partial y^{2}\). Also \({\text{Q}}_{{\text{R}}}\), and M are real and complex functions, respectively. Let us assume that

$${\text{M}}\left( {{\text{x}},{\text{y}},p_{x} ,p_{y} ,t,\alpha } \right) = {\text{V}}\left( {x,y} \right) + i{\upbeta }{\text{Q}}_{{\text{I}}} \left( {{\text{x}},{\text{y}},t,\alpha } \right)\left( { - \Delta^{2} } \right)^{{\frac{\alpha }{2}}}$$
(2)

where the geometrical potential \({\text{V}}\left( {x,y} \right)\) is defined for the double slit problem, and \({\upbeta }{\text{Q}}_{{\text{I}}} \left( {{\text{x}},{\text{y}},t,\alpha } \right)\) is a real function that determines amplitude of imaginary part of the potential. By substituting Eq. (2) in Eq. (1), we have

$$i\frac{\partial \psi }{{\partial t}} = \left[ {{\upbeta }{\text{Q}}\left( {{\text{x}},{\text{y}},t,\alpha } \right)\left( { - \Delta^{2} } \right)^{{\frac{\alpha }{2}}} - \gamma \left| {\psi \left( {x,y,t} \right)} \right|^{2} + {\text{V}}\left( {{\text{x}},{\text{y}}} \right)} \right]\psi \left( {x,y,t} \right)$$
(3)

where \({\text{Q}}\left( {{\text{x}},{\text{y}},t,\alpha } \right) = {\text{Q}}_{R} \left( {{\text{x}},{\text{y}},t,\alpha } \right) + {\text{ iQ}}_{{\text{I}}} \left( {{\text{x}},{\text{y}},t,\alpha } \right)\).


Note that when \({\text{Q}}\left( {{\text{x}},{\text{y}},t,\alpha } \right) = {\text{exp}}\left( {2\pi i} \right) = 1\), we obtain the usual NFSE. In this paper however, we investigated a more general case of \({\text{Q}}\left( {{\text{x}},{\text{y}},t,\alpha } \right)\) as follows

$${\text{Q}}\left( {{\text{x}},{\text{y}},t,\alpha } \right) = \exp \left( {\frac{i\pi \alpha }{{2\left| {g\left( {{\text{x}},{\text{y}},t} \right)} \right|}}\left( {\left| {g\left( {{\text{x}},{\text{y}},t} \right)} \right| - g\left( {{\text{x}},{\text{y}},t} \right)} \right)} \right)$$
(4)

where \(g\left( {{\text{x}},{\text{y}},t} \right) = - i\frac{{\Delta \psi \left( {x,y,t} \right)}}{{\psi \left( {x,y,t} \right)}}\) is a real function.”


As a result, Equations 5–8 were originally listed as Equations 2–5.


The original Article has been corrected.