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 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.
The mechanism of the transition of a dynamical system from quantum to classical mechanics is of continuing interest. Practically it is of importance for the interpretation of multi-particle coincidence measurements performed at macroscopic distances from a microscopic reaction zone. Here we prove the generalized iimaging theorem which shows that the spatial wave function of any multi-particle quantum system, propagating over distances and times large on an atomic scale but still microscopic, and subject to deterministic external fields and particle interactions, becomes proportional to the initial momentum wave function where the position and momentum coordinates define a classical trajectory. Currently, the quantum to classical transition is considered to occur via decoherence caused by stochastic interaction with an environment. The imaging theorem arises from unitary Schroedinger propagation and so is valid without any environmental interaction. It implies that a simultaneous measurement of both position and momentum will define a unique classical trajectory, whereas a less complete measurement of say position alone can lead to quantum interference effects.
The mechanism of the transition of a dynamical system from quantum to classical mechanics is of continuing interest. Practically it is of importance for the interpretation of multi-particle coincidence measurements performed at macroscopic distances from a microscopic reaction zone. Here we prove the generalized imaging theorem which shows that the spatial wave function of any multi-particle quantum system, propagating over distances and times large on an atomic scale but still microscopic, and subject to deterministic external fields and particle interactions, becomes proportional to the initial momentum wave function where the position and momentum coordinates define a classical trajectory . Currently, the quantum to classical transition is considered to occur via decoherence caused by stochastic interaction with an environment. The imaging theorem arises from unitary Schrödinger propagation and so is valid without any environmental interaction. It implies that a simultaneous measurement of both position and momentum will define a unique classical trajectory, whereas a less complete measurement of say position alone can lead to quantum interference effects.
The physical meaning of Schrödinger’s wave function remains afoundational interpretive question in quantum mechanics. This paperseparates two questions that are often introduced together: what thewave function describes during propagation, and why experimentaloutcomes obey Born statistics. Rather than taking
∣Ψ∣2 as the starting definition of
Ψ, we begin from theoperational structure of early electron-scattering and transmissionexperiments surrounding the emergence of wave mechanics and Born’s1926 collision theory. Davisson and Kunsman measured angle-dependent collected current,Ramsauer measured energy-dependent electron transmission throughgases, and Dymond later resolved the energy and angular redistributionof electrons after collisions. In each case, the apparatus returnedphysical signals---charge, current, or flux---from which reproducibleoutcome distributions were reconstructed. From this common structure we propose a deliberately limitedinterpretation: for a single nonrelativistic particle,
Ψ may beread first as a complex-amplitude representation of thespatial--directional organization of quantum propagation. Thisproposal is anchored not only in the experimental sequence but also inthe mathematics of the wave function. Its Fourier representationresolves propagation into wave-vector components, while theSchrödinger continuity relation defines a conserved density--currentpair; in polar form
Ψ=Rexp(iS/ℏ),the associated current is proportional to
R2∇S. These facts show that
Ψ contains both spatial weighting anddirectional phase structure before any additional ontological claim ismade. We do not infer a material wave medium, deterministic particletrajectories, hidden variables, or an alternative dynamics. Nor do wederive the Born rule. Instead, we isolate a separate mapping problem:how does a structured propagation state, through a specific detectorinteraction, become the observed statistics of localized or collectedparticle events? In scattering, any acceptable account must recover the standard fluxand cross-section relations. The contribution of the present paper istherefore interpretive and programmatic: it identifies a constrainedphysical reading of
Ψ and states precisely what remains to beexplained.