The time-dependent Schrödinger equation describes dynamical processes of one-electron species in terms of complex wave functions. The functions are inherently complex; therefore zeros occur only when both the real and imaginary parts of the wave functions vanish. If this happens at isolated points rather than on nodal surfaces one can show that the zeros must correspond to vortices. An imaging theorem is given which shows how such vortices can be seen experimentally. Since the theorem requires time propagation from microscopic to macroscopic scales, a method is developed that does just the same. Examples of vortices that emerge in dynamical processes are given. The vortices that we find are linked to the hydrodynamic interpretation of Schrödinger's time-dependent equation.
The time-dependent Schrödinger equation describes dynamical processes of one-electron species in terms of complex wave functions. The functions are inherently complex; therefore zeros occur only when both the real and imaginary parts of the wave functions vanish. If this happens at isolated points rather than on nodal surfaces one can show that the zeros must correspond to vortices. An imaging theorem is given which shows how such vortices can be seen experimentally. Since the theorem requires time propagation from microscopic to macroscopic scales, a method is developed that does just the same. Examples of vortices that emerge in dynamical processes are given. The vortices that we find are linked to the hydrodynamic interpretation of Schrödinger's time-dependent equation.
In quantum theory, vortex lines arise in the hydrodynamic interpretation of the wave equation. In this interpretation, which is originally due to Madelung, the flow of the probability density for a single particle is described in terms of the hydrodynamic variables. For the sake of simplicity, the standard time-dependent Schrödinger equation, and the related vortex lines embedded in the probability fluid of the quantum particle, are considered here. A vortex line in this case is simply the curve defined by equating the wave function to zero. The linearity of the Schrödinger equation enables us to obtain a large family of exact time-dependent analytic solutions for the wave functions with vortex lines. Moreover, the method is general enough to allow for various initial configurations of the vortex lines. Although the equation of motion of the quantum mechanical probability fluid is different in its literal form from the equations describing the real physical fluid, we believe that the evolution of the vorticity in the quantum and in the real fluid share the same qualitative features that can be described in terms of the topology of the vortex lines configurations. The general phenomena such as the switch-over, creation and annihilation of vortices can be observed in the quantum mechanical fluid. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Vortices are usually associated with systems containing large numbers of particles. Of particular topical interest though are those formed within atomic-scale wave functions and observed in macroscopic systems such as superfluids and quantum condensates. We uncover them here in one of the most fundamental quantum systems consisting of just one electron and two protons. Moreover, the results of novel simulations of the dynamics of this system reveal previously unknown mechanisms of angular momentum transfer and new ways to image atomic-scale quantized vortices at macroscopic distances. Probing of vortices and vortex-driven dynamics in quantum systems is thereby illustrated.
A formalism which is equivalent to the preceding stationary treatment is provided by the time-dependent Schrödinger equation. In the Schrödinger picture of quantum mechanics, operators are stationary and wave functions are time-dependent. The term ‘transient’ usually refers to a time developing phenomenon which dies out after a sufficiently long time lag has elapsed. Such are the envelopes of experimentally encoded time signals as a linear combination of damped exponentials with constant amplitudes such that all complex frequencies must have the negative imaginary parts. A more general meaning of the term transient in sequence-to-sequence transformations has been encountered in quantum-mechanical signal processing. The term unstable or secular transients has been used whenever referring to a time signal in which one or more fundamental frequencies are exponentially diverging. Any finite or infinite sum of harmonics would diverge if at least one of the complex fundamental frequencies is secular.
The present paper deals with some kind of quantum ``velocity'' which is introduced by the method of hydrodynamical analogy. It is found that this ``velocity'' is in general irrotational, namely, a vorticity vanishes, and then a velocity potential must exist in quantum mechanics. In some elementary examples of stable systems we will see what the ``velocities'' are. In particular, the two-dimensional flows of these examples can be expressed by complex velocity potentials whose real and imaginary parts are the velocity potentials and stream functions, respectively.
The time-dependent Schrödinger equation (TDSE) is a wave equation that is used to understand the behavior of small particles such as electrons or nuclei. The equation must be "solved" in the sense that a function must be found that can be plugged into the equation and that leaves the left-hand side equal to the right-hand side. The TDSE can be understood by first studying the classical wave equation that is used to "solve" the behavior of a guitar string or a water wave. The TDSE is a bit different in that it equates a double space derivative with a single time derivative (the classical equation equates two double derivatives); it also implicitly includes the symbol i , which stands for the square root of -1. These differences give rise to different sorts of solutions, including complex solutions with real and imaginary parts. In the case of the TDSE, the solution or "wave" does not directly show the motion of the particle. Instead, it shows where the particle is likely to be found. More specifically, it is a probability wave for particle location. To obtain the precise probability for the particle to be found at a specific position x at a specific time t , the wave function Ψ ( x , t ) must be squared via a special procedure called taking the square modulus. The wave contains information about the frequency, wavelength, position, and momentum of the particle. In the following chapters, we will be showing how to find the wave solution in different circumstances (different potentials), and how to extract the information about momentum, etc., once the wave function is known.
Calculations of the time‐dependent wave function for proton impact on atomic hydrogen using the Lattice‐Time‐Dependent‐Schrödinger‐Equation (LTDSE) method [1] follow the wave function from microscopic to macroscopic times. Isolated zeros, now identified as vortices, appear when the target and projectile nuclei are separated by a few atomic units [2]. Such structure has apparently been observed frequently in ab. initio. calculations. Our work shows that such structures persist to macroscopic distances and appear as “holes” in electron momentum distributions. They can, in principle, be observed in reaction microscope studies. Such observation is formally justified by the “imaging theorem”, which can be derived from first principles. Similarly, two‐electron momentum distributions as observed, e.g., in (e,2e) coincidence measurements may show isolated zeros [3]. Calculations using correlated wave functions support the interpretation that a minima in (e,2e) for helium targets corresponds to a vortex. In this case the “imaging theorem” can be applied to argue that there must be vortices in the two‐electron wave function[4]. A general discussion of vortices in quantum mechanics will be illustrated using exact LTDSE calculations, and, for interpretive purposes, the time‐dependent first Born approximation. These calculations show that the plane wave B1 amplitudes have no vortices, but the time‐dependent B1 amplitudes do. It will be further shown that angular momentum transfer is the key to forming vortices in the time‐dependent theory.