Quantum streamlines are investigated in the framework of the quantum Hamilton–Jacobi formalism. The local structures of the quantum momentum function (QMF) and the Pólya vector field near a stagnation point or a pole are analyzed. Streamlines near a stagnation point of the QMF may spiral into or away from it, or they may become circles centered on this point or straight lines. Additionally, streamlines near a pole display east-west and north-south opening hyperbolic structure. On the other hand, streamlines near a stagnation point of the Pólya vector field for the QMF display general hyperbolic structure, and streamlines near a pole become circles enclosing the pole. Furthermore, the local structures of the QMF and the Pólya vector field around a stagnation point are related to the first derivative of the QMF; however, the magnitude of the asymptotic structures for these two fields near a pole depends only on the order of the node in the wave function. Two nonstationary states constructed from the eigenstates of the harmonic oscillator are used to illustrate the local structures of these two fields and the dynamics of the streamlines near a stagnation point or a pole. This study presents the abundant dynamics of the streamlines in the complex space for one-dimensional time-dependent problems.
Quantum streamlines are investigated in the framework of the quantum Hamilton–Jacobi formalism. The local structures of the quantum momentum function (QMF) and the Pólya vector field near a stagnation point or a pole are analyzed. Streamlines near a stagnation point of the QMF may spiral into or away from it, or they may become circles centered on this point or straight lines. Additionally, streamlines near a pole display east-west and north-south opening hyperbolic structure. On the other hand, streamlines near a stagnation point of the Pólya vector field for the QMF display general hyperbolic structure, and streamlines near a pole become circles enclosing the pole. Furthermore, the local structures of the QMF and the Pólya vector field around a stagnation point are related to the first derivative of the QMF; however, the magnitude of the asymptotic structures for these two fields near a pole depends only on the order of the node in the wave function. Two nonstationary states constructed from the eigenstates of the harmonic oscillator are used to illustrate the local structures of these two fields and the dynamics of the streamlines near a stagnation point or a pole. This study presents the abundant dynamics of the streamlines in the complex space for one-dimensional time-dependent problems.
Quantum streamlines are investigated in the framework of the quantum Hamilton-Jacobi formalism. The local structures of the quantum momentum function (QMF) and the Polya vector field near a stagnation point or a pole are analyzed. Streamlines near a stagnation point of the QMF may spiral into or away from it, or they may become circles centered on this point or straight lines. Additionally, streamlines near a pole display east-west and north-south opening hyperbolic structure. On the other hand, streamlines near a stagnation point of the Polya vector field for the QMF display general hyperbolic structure, and streamlines near a pole become circles enclosing the pole. Furthermore, the local structures of the QMF and the Polya vector field around a stagnation point are related to the first derivative of the QMF; however, the magnitude of the asymptotic structures for these two fields near a pole depends only on the order of the node in the wave function. Two nonstationary states constructed from the eigenstates of the harmonic oscillator are used to illustrate the local structures of these two fields and the dynamics of the streamlines near a stagnation point or a pole. This study presents the abundant dynamics of the streamlines in the complex space for one-dimensional time-dependent problems.
One-dimensional time-independent scattering problems are investigated in the framework of the quantum Hamilton-Jacobi formalism. The equation for the local approximate quantum trajectories near the stagnation point of the quantum momentum function is derived, and the first derivative of the quantum momentum function is related to the local structure of quantum trajectories. Exact complex quantum trajectories are determined for two examples by numerically integrating the equations of motion. For the soft potential step, some particles penetrate into the nonclassical region, and then turn back to the reflection region. For the barrier scattering problem, quantum trajectories may spiral into the attractors or from the repellers in the barrier region. Although the classical potentials extended to complex space show different pole structures for each problem, the quantum potentials present the same second-order pole structure in the reflection region. This paper not only analyzes complex quantum trajectories and the total potentials for these examples but also demonstrates general properties and similar structures of the complex quantum trajectories and the quantum potentials for one-dimensional time-independent scattering problems.
The time-dependent scattering of a wave packet from a Gaussian barrier is investigated computationally in the complex z-plane. The initial wave packet and the potential energy are obtained through analytic continuation from functions specified on the real-axis. The wave packet is then propagated on the two-dimensional grid. For a low initial wave packet energy, the time evolution is followed by plotting the following functions: |ψ(z,t)|, real(ψ(z,t)), and the quantum momentum function (QMF), p(z,t). In the reflected packet, an important role is played by ripples (quasi-nodes) forming above the real axis. As these quasi-nodes move down across the real axis, they are 'detected' as 'interference oscillations' in the density. In contrast, the component of the packet below the real axis makes a significant contribution to the transmitted packet. Vector maps of the QMF show hyperbolic flow around quasi-nodes and counterclockwise circular flow around transient stagnation points, where the QMF vanishes. However, when the Pólya vector field (defined by P(z,t) = p*(z,t)) is plotted, circular counterclockwise flow is obtained near the quasi-nodes. The real and imaginary parts of the quantum action function S(z,t) are plotted and the vorticity, defined by the curl of the Pólya field, is used to pinpoint regions of nonanalyticity in the QMF.
Quantum trajectories in complex space in the framework of the quantum Hamilton-Jacobi formalism are investigated. For time-dependent problems, the complex quantum trajectories determined from the exact analytical wave function for the free Gaussian wave packet and the coherent state in the harmonic potential are used to demonstrate that the information transported by the particles in the complex space can be used to synthesize the time-dependent wave function on the real axis. For time-independent problems, the exact complex quantum trajectories for the Eckart potential are obtained by numerically integrating the equations of motion. The unusual structure of the total potential (the sum of the classical and the quantum potentials) for the stationary states for the Eckart potential is pointed out. The variations of the complex-valued kinetic energy, classical potential, and quantum potential along the complex quantum trajectories are analyzed. This paper not only analyzes complex quantum trajectories for time-dependent and time-independent systems but also provides a unified description for complex quantum trajectories for one-dimensional problems in the quantum Hamilton-Jacobi formalism.
Wave-packet interference is investigated within the complex quantum Hamilton-Jacobi formalism using a hydrodynamic description. Quantum interference leads to the formation of the topological structure of quantum caves in space-time Argand plots. These caves consist of the vortical and stagnation tubes originating from the isosurfaces of the amplitude of the wave function and its first derivative. Complex quantum trajectories display counterclockwise helical wrapping around the stagnation tubes and hyperbolic deflection near the vortical tubes. The string of alternating stagnation and vortical tubes is sufficient to generate divergent trajectories. Moreover, the average wrapping time for trajectories and the rotational rate of the nodal line in the complex plane can be used to define the lifetime for interference features.
In this article, we develop quantum mechanics upon the framework of the quantum mechanical Hamilton-Jacobi theory. We will show, that the Schroedinger point of view and the Hamilton-Jacobi point of view are fully equivalent in their description of physical systems, but differ in their descriptive manner. As a main result, a wave function in Hamilton-Jacobi theory can be decomposed into travelling waves in any point in space, not only asymptotically. The well known WKB-theory will be a special result of the more general theory, we will develop below. By the example of the linear potential and the harmonic oscillator, we will discuss quantum mechanics from the Hamilton-Jacobi point of view. Soft boundary value problems as the connection problem can be solved exactely. Quantizised energies and Maslov-indices can be calculated directely without orthonormalizing wave-functions. Also, we will focus on trajectory themes, which, in contrast to the Schroedinger point of view, follow naturally from the quantum mechanical action function.