Quantum bifurcation in a coulombic‐like potential

Abstract

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.

Trajectory description of the quantum–classical transition for wave packet interference

“…Based on the complex quantum HamiltonJacobi formalism [36,37], the complex QTM has been used to analyze both stationary bound and scattering state problems [38][39][40][41][42][43][44][45][46][47][48][49][50][51][52]. Quantum interference demonstrated by the head-on collision of two Gaussian wave packets has been thoroughly analyzed using complex quantum trajectories [53][54][55].…”

Quantum‐classical transition of the dissipative wave packet dynamics for barrier scattering

“…For example, a detailed analysis has been carried out for a driven triple‐well potential using quantum trajectories . Complex‐valued quantum trajectories have been employed to study quantum systems, including Coulombic systems, the hydrogen molecule ion, the interference fringes in slit experiments, the electric, magnetic, and thermal effects on a quantum dot, and chaotic trajectories in complex space . In addition, the quantum trajectory formulation has been employed to study quantum dissipative systems .…”

Quantum‐classical transition equation with complex trajectories

“…In addition to computational applications, the complex quantum hydrodynamic representation leads to novel trajectory‐based pictures of quantum mechanics. For example, complex quantum trajectories determined from the analytical form of the wave function have been analyzed for several stationary and nonstationary state problems . Quantum interference demonstrated by the head‐on collision of two Gaussian wave packets has been explored in the complex plane .…”

“…For example, complex quantum trajectories determined from the analytical form of the wave function have been analyzed for several stationary and nonstationary state problems. [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32] Quantum interference demonstrated by the head-on collision of two Gaussian wave packets has been explored in the complex plane. [33][34][35] Several studies have been dedicated to issues concerning the probability density in complex space and probability conservation along complex quantum trajectories.…”