Nonlinear dynamics governing quantum transition behavior

In quantum mechanics, the only known quantitative index describing state transition process is the occupation probability. In this paper, we propose three new quantitative indices, namely, complex transition trajectory, quantum potential, and total quantum energy, to give an elaborate description of state-transition behavior, which otherwise cannot be manifested in terms of occupation probability. The new quantitative indices and the nonlinear dynamics governing quantum transition behavior are derived from the quantum Hamilton mechanics that provides a unified description of the entire transition process from an initial state to a final state via an entangled state. Under the framework of quantum Hamilton mechanics, state transition behavior, which was originally represented by the probabilistic view, can now be described deterministically in terms of the continuity of total energy, of quantum potential, and of complex trajectory. The proposed nonlinear analysis of transition behavior is then applied to harmonic oscillator, entangled free-particle motion, and hydrogen atom to demonstrate the usefulness of the new quantitative indices in characterizing the progression of state transition process.