It is shown that "repulsion of energy levels" of the same symmetry type occurs in complex atomic spectra. Thus, for the elements Hf, Ta, W, Re, Os, and Ir, for which the spin-dependent forces are relatively strong, the spacings between neighboring odd-parity levels of the same $J$ value follow the Wigner distribution (approximately). For the elements Sc, Ti, V, Cr, Mn, Fe, Co, and Ni, for which the spin-dependent forces are relatively weak, a similar distribution is obtained for the odd-parity levels having fixed values for $S$, $L$, and $J$. (When the quantum numbers $S$ and $L$ are disregarded, the same levels give rise to a distribution of spacings which is approximated by a random superposition of a number of appropriately weighted Wigner distributions.) For the elements Y, Zr, Nb, Mo, Ru, Rh, and Pd, for which the spin-dependent forces are of intermediate strength, the empirical distribution of spacings between the odd levels of the same $J$ value has a character which is intermediate between the Wigner and exponential distributions. All of these observations are explained in terms of a statistical model for the Hamiltonian matrix in the $S$, $L$, $J$, $\pi$ representation. Quantitative results are obtained for a relatively simple form of the model which depends on only two parameters, viz., the dimensionality $N$ and the ratio $\mu$ of the dispersions of the normal distributions for the off-diagonal and diagonal matrix elements. The transition from the exponential to the Wigner distribution occurs, roughly speaking, in the range of $N{\mu }^2$ from 0 to 1. The results of this work suggest that an empirical study of the distribution of the spacing between the energy levels of a complex quantum system may be capable of pointing to the existence of constants of the motion beyond those which are already known.

• Received 2 August 1960

DOI:https://doi.org/10.1103/PhysRev.120.1698