Self-Similarity Among Energy Eigenstates

TL;DR

This paper demonstrates that the distribution ratios of energy eigenstates within energy shells exhibit a self-similar invariance across different quantum systems, suggesting a universal property of quantum eigenstates.

Contribution

It introduces the concept of self-similarity in energy eigenstates and provides numerical evidence across diverse quantum models, highlighting its generality.

Findings

Invariant eigenstate ratios in energy shells regardless of shell width or Planck constant

Numerical validation across multiple quantum systems

Proposed universality of self-similarity in quantum eigenstates

Abstract

In a quantum system, different energy eigenstates have different properties or features, allowing us define a classifier to divide them into different groups. We find that the ratio of each type of energy eigenstates in an energy shell $[E_{c}-\Delta E/2,E_{c}+\Delta E/2]$ is invariant with changing width $\Delta E$ or Planck constant $\hbar$ as long as the number of eigenstates in the shell is statistically large enough. We give an argument that such self-similarity in energy eigenstates is a general feature for all quantum systems, which is further illustrated numerically with various quantum systems, including circular billiard, double top model, kicked rotor, and Heisenberg XXZ model.