  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.
