NLSENonlinear Schrodinger Equation
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Notice that the NLSE above cannot be obtained by a naive scaling of the wavefunction
The NLSE has the complex coefficients a/ (in the kinetic terms), and -i[beta]/[?
and = complex, the actual numerical values in the apparent nonlinear equation, in general, would have still been different than those present in the NLSE.
in this very special case, the NLSE would be obtained from a linear Schrodinger equation after scaling the wavefunction [psi] [right arrow] [[psi].
0] and, consequently, the NLSE cannot be obtained from the ordinary (linear) Schrodinger equations after a naive scaling, with a complex exponent, [psi] [right arrow] [[psi].
Therefore, the NLSE based on a fractal Brownian motion with a complex valued diffusion constant 2mD = [?
A Fractal Scale Calculus description of our NLSE was developed later on by Cresson [20] who obtained, on a rigorous mathematical footing, the same functional form of our NLSE equation above ( although with different complex numerical coefficients) by using Nottale's fractal scale-calculus that obeyed a quantum bialgebra.
Notice that the NLSE (34) obeys the homogeneity condition [psi] [right arrow] [lambda] [psi] for any constant [lambda].
The classic Gross-Pitaveskii NLSE (of the 1960's), based on a quartic interaction potential energy, relevant to Bose-Einstein condensation, contains the nonlinear cubic terms in the Schrodinger equation, after differentiation, ([psi]*[psi])[psi].
However, in the fractal-based NLSE there is no discrepancy between the quantum-mechanical energy functional and the field theory energy functional.
This is why we push forward the NLSE derived from the fractal Brownian motion with a complex-valued diffusion coefficient.
Hence, the plane-wave is a solution to our fractal-based NLSE (when U = 0) with a real-valued energy and has the correct energy-momentum dispersion relation.