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Journal of Physics A Mathematical and Theoretical

Curvature of wave streamlines

Michael Berry

Published Sep 9, 2013• 8 citations
Publisher

Abstract

Wave streamlines are integral curves of the local wavevector (phase gradient). An exact formula is derived, giving the curvature of streamlines as the component transverse to the local wavevector of the gradient of the logarithm of the local wavenumber. The formula is applied to quantum particles moving in a potential and classical light in the presence of a refractive-index gradient. Three limiting regimes are encompassed. The first is geometrical, in which the bending of streamlines arises solely from the classical force or optical index gradient. The second and third limits concern singularities in the pattern of wave streamlines, of two types: optical vortices, near which the streamlines are asymptotically circular, and phase saddles, near which the streamlines are asymptotically hyperbolic.

DOI
10.1088/1751-8113/46/39/395202

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Wave streamlines are integral curves of the local wavevector (phase gradient). An exact formula is derived, giving the curvature of streamlines as the component transverse to the local wavevector of the gradient of the logarithm of the local wavenumber. The formula is applied to quantum particles moving in a potential and classical light in the presence of a refractive-index gradient. Three limiting regimes are encompassed. The first is geometrical, in which the bending of streamlines arises solely from the classical force or optical index gradient. The second and third limits concern singularities in the pattern of wave streamlines, of two types: optical vortices, near which the streamlines are asymptotically circular, and phase saddles, near which the streamlines are asymptotically hyperbolic.
Abstract The Madelung trajectories associated with a wavefunction are the integral curves (streamlines) of its phase gradient, interpretable in terms of the local velocity (momentum) vector field. The pattern of trajectories provides an immediately visualisable representation of the wave. The patterns can be completely different when the same wave equation describes different physical contexts. For the time-independent Schrödinger or Helmholtz equation, trajectories circulate around the phase singularities (zeros) of the wavefunction; and in the paraxially approximate wave, streamlines spiral slowly in or out of the zeros as well as circulating. But if the paraxial wave equation is reinterpreted as the time-dependent Schrödinger equation, its Madelung trajectories do not circulate around the zeros in spacetime: they undulate while avoiding them, except for isolated trajectories that encounter each zero in a cusp singularity. The different local trajectory geometries are illustrated with two examples; a local model explains the spacetime cusps.
Abstract The present paper concerns a formalism aimed to describe the behavior and trajectories of complex scalar waves with space-varying amplitude in media characterized by variable refractive index. The classical wave equation is transformed into an eikonal-type equation, without imposing the geometrical optics limit, satisfied by an effective refractive index. The effective index combines the refractive properties of the medium with self-refraction phenomena generated by the space dependence of the wave amplitude. It also induces wave packet dispersion, leading to relations between the group and phase velocities that are particularly relevant in the context of de Broglie waves. The equations of motion are provided, and it is shown that the self-refraction term bends the trajectory of the wave and changes its velocity. The dynamics is dictated by the curvature of the trajectory expressed in terms of the gradient of the effective refractive index. The uncovered findings point toward a possible generalization of geometrical optics with relevant implications for its relation to quantum mechanics, general relativity and transformation optics.
Exact numerical calculations are made for the scattering of quantum mechanical particles from a square two-dimensional potential barrier. This treatment is an exact analog of both frustrated total reflection of perpendicularly polarized light and the longitudinal Goos-Hänchen shift. Quantum mechanical streamlines (which are analogous to either classical mechanical trajectories or optical rays) are plotted. These streamlines are smooth, continuous, and have continuous first derivatives even through the classically forbidden region. The streamlines form quantized vortices surrounding each of the nodal points (which result from interference between the incident and reflected waves). Similar vortices occur in reactive collisions of H + H2 (McCullough and Wyatt; Kuppermann, Adams, and Truhlar) and undoubtedly play an important role in molecular collision dynamics. The theory for these vortices is given in a companion paper. A comparison is given between our numerical calculations and the stationary phase approximation (SPA). Although our incident wave packet has a half-width of more than one de Broglie wavelength in contrast to the SPA which replaces the diffuse beams by rays with delta function profiles, the agreement was surprisingly good for both the Goos-Hänchen shifts and for the reflection coefficients. However, we found that the Goos-Hänchen shift for the transmitted beam is significantly smaller than for the reflected beam, although in the stationary phase approximation these two shifts are equal. Furthermore, we found that the scattering from the potential barrier has very little effect on the shape of the wave packets. The power series expansion of the incident Debye-Picht wave packet ψI has an extremely small radius of convergence, whereas the power series for ψI*ψI has a radius of convergence of more than two de Broglie wavelengths. The imaginary velocity viis introduced into the Madelung-Landau-London hydrodynamical formulation of quantum mechanics. The corresponding imaginary streamlines will be considered in a forthcoming paper. The time-independent Schrödinger equation for real wavefunctions is reduced to solving the nonlinear first order partial differential equation: ℏ▿·vi = 2(E − V) + vi2. Here, vi is irrotational. This equation may lead to interesting new methods of solving the Schrödinger equation. It does lead to a generalization of the Prager-Hirschfelder perturbation scheme which invokes an electrostatic analogy. The philosophical implications of the hydrodynamical formulation of quantum mechanics are discussed. Penetration of a one-dimensional square potential barrier is used to demonstrate exactly how tunneling occurs by particles ``riding over the barrier'' à la Bohm. Cases are cited where quantum and classical mechanical motions are identical.
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.
In conventional quantum mechanics the trajectory of a matter wave is described implicitly by the phase of its wavefunction, yet no explicit geometric quantity has been associated with the propagation path itself. This paper introduces the concept of matter wave propagation curvature, denoted K_psi, as an intrinsic geometric measure of the bending of a quantum trajectory in configuration space. Unlike the curvature of space‑time employed in general relativity, K_psi depends solely on the wavefunction's phase structure and is independent of the underlying metric. We derive K_psi from the gradient of the phase of the wavefunction, show that it satisfies a conservation law along stationary paths, and discuss its relationship to the effective potential and external fields. The curvature framework predicts observable deviations in large‑mass interference experiments, suggesting a new route to probe the geometric nature of quantum states and to explore the possibility of a wavefunction‑based geometry. The potential to uncover a deeper layer of quantum reality is highlighted, pointing toward a Nobel‑potential discovery of wavefunction geometry.