Knotted Zeros in the Quantum States of Hydrogen

C. Colijn

Louis Hirsch Kauffman, Samuel J. Lomonaco

In 2001, Michael Berry4 published the paper ”Knotted Zeros in the Quantum States of Hydrogen” in Foundations of Physics. In this paper we show how to place Berry’s discovery in the context of general knot theory and in the context of our formulations for quantum knots. Berry gave a time independent wave function for hydrogen, as a map from three space R3 to the complex plane and such that the inverse image of 0 in the complex plane contains a knotted curve in R3. We show that for knots in R3 this is a generic situation in that every smooth knot K in R3 has a smooth classifying map f : R3 −→ C (the complex plane) such that f−1(0) = K. This leaves open the question of characterizing just when such f are wave-functions for quantum systems. One can compare this result with the work of Mark Dennis and his collaborators and with the work of Lee Rudolph. Our approach provides great generality to the structure of knotted zeros of a wavefunction and opens up many new avenues for research in the relationships of quantum theory and knot theory. We show how this classifying construction can be related our previous work on two dimensional and three dimensional mosaic and lattice quantum knots.

Samuel J. Lomonaco, Louis Hirsch Kauffman

Caroline Colijn, Edward R. Vrscay

Qian Shang-Wu, Gu Zhi-Yu

This article discusses the variation of the knotted picture of the quantum pure state with the variation of the complex coefficients α and β. It is shown that there are three kinds of link that correspond to three different ranks of the matrix of covariance correlation tensor, i.e., the zero rank corresponds to trivial link, the rank one corresponds to the two-component link with two crossings, and the rank three corresponds to the two-component link with four crossings.

Leandra Vranješ Markić, Jordi Boronat