Quantum bifurcation in a coulombic-like potential

Because of the lack of nonlinear dynamics, up to now no bifurcation phenomenon in its original sense has been discovered directly in quantum mechanical systems. Based on the formalism of complex-valued quantum mechanics, this article derives the nonlinear Hamilton equations from the Schrödinger equation to provide the necessary mathematic framework for the analysis of quantum bifurcation. This new approach makes it possible to identify quantum bifurcation by the direct evidence of the sudden change of fixed points and their surrounding trajectories. As a practical application of the proposed approach, we consider the quantum motion in a Coulombic-like potential modeled by V(r) = A/r² - B/r, where the first term describes the centrifugal trend and the second deals with the Coulombic attraction. As the bifurcation parameter evolves, we demonstrate how local and global bifurcations in quantum dynamics can be identified by inspecting the changes of fixed points and their surrounding trajectories.