ABSTRACT:The hydrogen molecule ion is a two-center force system expressed under the prolate spheroidal coordinates, whose quantum motions and quantum trajectories have never been addressed in the literature before. The momentum operators in this coordinate system are derived for the first time from the Hamilton equations of motion and used to construct the Hamiltonian operator. The resulting Hamiltonian comprises a kinetic energy T and a total potential V Total consisting of the Coulomb potential and a quantum potential. It is shown that the participation of the quantum potential and the accompanied quantum forces in the force interaction within H 2 þ is essential to develop an electronic motion consistent with the prediction of the probability density function |W| 2 . The motion of the electron in H 2 þ can be either described by the Hamilton equations derived from the Hamiltonian H ¼ T K þ V Total or by the Lagrange equations derived from the Lagrangian H ¼ T K À V Total . Solving the equations of motion with different initial positions, we show that the solutions yield an assembly of electronic quantum trajectories whose distribution and concentration reconstruct the r and p molecular orbitals in H 2 þ .

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 Schrö dinger 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) ¼ A/r 2 À 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.

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