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International Journal of Nonlinear Sciences and Numerical Simulation

Nonlinear Dynamics Governing Quantum Transition Behavior

C.-D. Yang, Shiang-Yi Han, F.-B. Hsiao

Published Jan 1, 2007• 11 citations
Publisher

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 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. 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. Introduction of complex mechanics In conventional quantum mechanics (QM), a particle’s position and momentum in a quantum system can only be obtained in terms of an expectation value through a quantum operator. To reflect this situation, one can imagine that there is a bee in a house with no window. Suppose that the bee has a

DOI
10.1515/ijnsns.2007.8.3.397

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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 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. 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. Introduction of complex mechanics In conventional quantum mechanics (QM), a particle’s position and momentum in a quantum system can only be obtained in terms of an expectation value through a quantum operator. To reflect this situation, one can imagine that there is a bee in a house with no window. Suppose that the bee has a
In this work we will examine an on-shell action, its basic properties and its relation to transition amplitude. Derivatives of on-shell action with respect to position and time are equal to momentum and energy. On-shell action of a system is sufficient for determining the trajectory describing time evo- lution of the system. On-shell action can be computed as (off-shell) action of specific physical trajectory connecting initial position in initial time with final position in final time but it can also be found from solutions to two Hamilton-Jacobi equations, one in initial time and position variables and the other in final time and position variables. In quantum mechanics a transition amplitude is directly proportional to the complex exponential of the on-shell action. Work with on-shell action is demonstrated on examples such as free particle, harmonic oscillator and partly also on central force problem.
The introduction of nonlinearities in the Schrödinger equation has been considered in the literature as an effective manner to describe the action of external environments or mean fields. Here, in particular, we explore the nonlinear effects induced by subtracting a term proportional to Bohm’s quantum potential to the usual (linear) Schrödinger equation, which generates the so-called “classical” Schrödinger equation. Although a simple nonlinear transformation allows us to recover the well-known classical Hamilton–Jacobi equation, by combining a series of analytical results (in the limiting cases) and simulations (whenever the analytical treatment is unaffordable), we find an analytical explanation to why the dynamics in the nonlinear “classical” regime is still strongly nonclassical. This is even more evident by establishing a one-to-one comparison between the Bohmian trajectories associated with the corresponding wave function and the classical trajectories that one should obtain. Based on these observations, it is clear that the transition to a fully classical regime requires extra conditions in order to remove any trace of coherence, which is the truly distinctive trait of quantum mechanics. This behavior is investigated in three paradigmatic cases, namely, the dispersion of a free propagating localized particle, the harmonic oscillator, and a simplified version of Young’s two-slit experiment.
Abstract A quantum‐classical transition equation in complex space is derived in the framework of the complex quantum Hamilton–Jacobi formalism. The transition equation is obtained by subtracting a complex‐valued quantum potential term from the complex‐extended time‐dependent Schrödinger equation (TDSE). It is shown that the nonlinear transition equation is equivalent to a linear scaled TDSE with a rescaled Planck's constant. Employing the quantum momentum function defined by the gradient of the complex action, we can analyze the quantum‐classical transition of physical systems using complex transition trajectories. Complex transition trajectories are presented for the free Gaussian wave packet, the harmonic oscillator, and the Morse potential. This study demonstrates that the transition equation provides a continuous description for the quantum‐classical transition of physical systems in complex space.
State transition in the multiple-levels system has the great potential applications in the quantum technology. In this article we employ a deterministic approach in complex space to analyze the dynamics of the 1s-2p electron transition in the hydrogen atom. The electron’s spin motion is embodied in the framework of quantum Hamilton mechanics that allows us to examine the transition dynamics more precisely. The transition is driven by an oscillating electric field in the z-direction. The electron’s transition process can be visualized by monitoring its motion in the complex space. The quantum potential and the total energy proposed in this paper provide new indices to observe the dynamic changes of electrons in the transition process.
The vast majority of the literature dealing with quantum dynamics is concerned with linear evolution of the wave function or the density matrix. A complete dynamical description requires a full understanding of the evolution of measured quantum systems, necessary to explain actual experimental results. The dynamics of such systems is intrinsically nonlinear even at the level of distribution functions, both classically as well as quantum mechanically. Aside from being physically more complete, this treatment reveals the existence of dynamical regimes, such as chaos, that have no counterpart in the linear case. Here, we present a short introductory review of some of these aspects, with a few illustrative results and examples.
This paper introduces the Quantum State Transition Path (QSTP) framework, a novel approach to understanding and characterizing quantum phase transitions. The core premise of QSTP posits that quantum phase transitions are governed not merely by initial and final states, but crucially by a unique, deterministic trajectory – the transition path itself. This path, denoted as [Pq], represents the quantum mechanical process through which the system evolves from its initial state to its final state, incorporating all intermediate quantum states along the way. The framework leverages quantum simulation techniques to explore and identify these transition paths, potentially revealing new insights into quantum dynamics and offering a pathway toward understanding phenomena currently considered intractable with traditional approaches. This work lays the foundation for a more rigorous and predictive model of quantum phase transitions, with potential implications for a range of scientific and technological applications.