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AIP conference proceedings

Origin, Evolution And Imaging Of Vortices In Atomic Processes

J. H. Macek, Floyd “Del” McDaniel, Barney L. Doyle

Published Jan 1, 2011• 2 citations

Abstract

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.

DOI: 10.1063/1.3586074

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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.

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.

Vortices in atomic wave functions are shown to have observable consequences. It is shown that time-dependent electron wave functions in coordinate space go over to electron emission amplitudes in the limit that time becomes infinite. This relation between wave functions and emission amplitudes is called the imaging theorem. According to this theorem vortices in wave functions at small times when particles interact in a complex way appear as vortices in electron momentum distributions where they produce observable features. Conversely, some observable structures may be interpreted in terms of vortices no matter how they are seen or computed. We verify that previously unexplained features in (e,2e) triply differential cross sections can be interpreted in terms of vortices.

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.

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.

We present measurements and calculations of the momentum distribution of electrons emitted during the ion-atom collision

10\text{ }\text{ }\mathrm{keV}/\mathrm{u}

${\mathrm{He}}^{2+}+\mathrm{He}\ensuremath{\rightarrow}{\mathrm{He}}^{+}+{\mathrm{He}}^{2+}+{e}^{\ensuremath{-}}$, which show rich structures for ion scattering angles above 2 mrad arising dominantly from two-electron states. Our calculations reveal that minima in the measured distributions are zeros in the electronic probability density resulting from vortices in the electronic current.