Nonlinear Dynamics Governing Quantum Transition Behavior

Abstract

In quantum mechanics, the only known quantitative index describing a 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 statetransition 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. The proposed nonlinear analysis of transition behavior will be applied to the transition in a hydrogen atom from the 1s state to the 2s state.

Invariant Eigen-Structure in Complex-Valued Quantum Mechanics

“…The first source comes from the quantum systems with complex eigenvalues, as discussed in the previous sections. The second possible source of the complex energy originates from the entanglement of eigenstates with real eigenvalues [26]. In a state transition process, both the initial and final energy levels are real but the intermediate energy level may be complex.…”

“…By using the complex-space formulation of fractal spacetime [21][22], many quantum phenomena, such as tunneling [23], wave-particle duality [10], spin [24,25], state transition [26], path integral [12], quantum chaos [27], uncertainty principle [11] and molecular dynamics [28,29], have been modeled exactly in the framework of complex-extended Hamilton mechanics, wherein quantum motions are described by complex-valued nonlinear differential equations. Apparently, quantum motions within complex space provide us with a better understanding of the quantum world in terms of a deterministic interpretation than a probabilistic one which is the only vision given by quantum mechanics.…”

Strong chaos in one-dimensional quantum system

“…According to quantum Hamilton mechanics [4][5][6][7][8], the quantum Hamiltonian can be written as (1) where is what we call quantum potential :…”