纠缠量子本征函数中的涡旋结
Vortex knots in tangled quantum eigenfunctions
摘要：
从宏观的小分子到DNA这样的长链大分子，缠结的弦通常会打结。在这里，我们证明了打结也发生在量子波函数，其中纠缠的细丝是漩涡（节线/相位奇点）。一个涡环打结的概率被发现增加其长度，和一个广泛的范围内的结从标准制表发生。这些结果来自三个简单量子系统简并本征态的随机叠加的计算机模拟：具有周期边界的立方体，各向同性三维谐振子和3球。在后两种情况下，涡结频繁发生，即使在相对低的能量随机本征函数，并受到空间对称性的模式。结果表明，打结涡结构在复杂的三维波系统中是普遍的，建立波混沌，聚合物和湍流玻色-爱因斯坦凝聚体之间的拓扑共性。
英文摘要：
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 3-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 3-dimensional wave systems, establishing a topological commonality between wave chaos, polymers and turbulent Bose-Einstein condensates.
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