Properties of a fractional derivative Schroedinger type wave equation and a new interpretation of the charmonium spectrum

Abstract

Based on the Caputo fractional derivative the classical, non relativistic Hamiltonian is quantized leading to a fractional Schroedinger type wave equation. The free particle solutions are localized in space. Solutions for the infinite well potential and the radial symmetric ground state solution are presented. It is shown, that the behaviour of these functions may be reproduced with a ordinary Schroeodinger equation with an additional potential, which is of the form V ~ x for $α<1$, corresponding to the confinement potential, introduced phenomenologically to the standard models for non relativistic interpretation of quarkonium-spectra. The ordinary Schroedinger equation is triple factorized and yields a fractional wave equation with internal SU(3) symmetry. The twofold iterated version of this wave equation shows a direct analogy to the fractional Schroedinger equation derived. The angular momentum eigenvalues are calculated algebraically. The resulting mass formula is applied to the charmonium spectrum and reproduces the experimental masses with an accuracy better than 0.1%. Extending the standard charmonium spectrum, three additional particles are predicted and associated with $Σ_c^0(2455)$ and Y(4260) observed recently and one X(4965), not yet observed. The root mean square radius for $Σ_c^0(2455)$ is calculated to be <r>~0.3[fm]. The derived results indicate, that a fractional wave equation may be an appropriate tool for a description of quark-like particles.