H H Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, UK.
Nat Commun. 2016 Jul 29;7:12346. doi: 10.1038/ncomms12346.
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.
线的缠绕通常会打结，从宏观的麻绳到像 DNA 这样的长链大分子都是如此。在这里，我们证明了缠结也会出现在量子波函数中，其中缠绕的线是旋涡（节线/相位奇点）。发现旋涡环打结的概率随着其长度的增加而增加，并且出现了广泛的标准表格中的结。这些结果来自于对三个简单量子系统的简并本征态的随机叠加的计算机模拟：具有周期性边界的立方体、各向同性的三维谐振子和 3 维球体。在后两种情况下，即使在相对低能的随机本征函数中，旋涡结也经常出现，并且受到模式的空间对称性的限制。这些结果表明，在复杂的三维波系统中，打结的旋涡结构是普遍存在的，这在波混沌、聚合物和湍流玻色-爱因斯坦凝聚物之间建立了拓扑共性。
