Abstract
Tangles of string typically become knotted, from macroscopic twine down to long-chain macromolecules such as DNA. Here, we demonstrate that knotting also occurs in quantum wavefunctions, where the tangled filaments are vortices (nodal lines/phase singularities). The probability that a vortex loop is knotted is found to increase with its length, and a wide gamut of knots from standard tabulations occur. The results follow from computer simulations of random superpositions of degenerate eigenstates of three simple quantum systems: a cube with periodic boundaries, the isotropic three-dimensional harmonic oscillator and the 3-sphere. In the latter two cases, vortex knots occur frequently, even in random eigenfunctions at relatively low energy, and are constrained by the spatial symmetries of the modes. The results suggest that knotted vortex structures are generic in complex three-dimensional wave systems, establishing a topological commonality between wave chaos, polymers and turbulent Bose–Einstein condensates.
Highlights
Tangles of string typically become knotted, from macroscopic twine down to long-chain macromolecules such as DNA
The spatial distribution of these nodal lines is statistically isotropic at high energies[2], and this irregularity is typical for chaotic systems, whose ergodic dynamics are determined only by the energy and there are no other constants of motion
Vortices and defects which are knotted have been successfully embedded in a controlled way in various 3D fields, such as vortex knots in water[13], knotted defects in liquid crystals[14,15] and knotted optical vortices in laser beams[16], and theoretically as vortex lines in complex scalar fields, including superfluid flows[17], and superpositions of energy eigenstates of the quantum hydrogen atom[18] and other wave fields[19], but rigorous mathematical techniques to resolve the statistical topology of random fields are limited[20]
Summary
Tangles of string typically become knotted, from macroscopic twine down to long-chain macromolecules such as DNA. The results follow from computer simulations of random superpositions of degenerate eigenstates of three simple quantum systems: a cube with periodic boundaries, the isotropic three-dimensional harmonic oscillator and the 3-sphere In the latter two cases, vortex knots occur frequently, even in random eigenfunctions at relatively low energy, and are constrained by the spatial symmetries of the modes. Following the hydrodynamic interpretation of single-particle quantum mechanics[3], the zeros of 3D complex-valued scalar fields are in general lines; vortex filaments around which the phase, local velocity and probability current (in a quantum wavefunction) circulate[4,5,6]. Further information and illustrations of these eigenfunction systems appear in Supplementary Note 1 and Supplementary Figs 1–4
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