"Repulsion of Energy Levels" in Complex Atomic Spectra

Rosenzweig, N.; Porter, Charles E.

doi:10.1103/physrev.120.1698

Cited by 440 publications

(340 citation statements)

References 12 publications

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“…Note that in common random matrices, e.g., those of the Gaussian orthogonal ensemble, the diagonal matrix elements fluctuate similarly to the offdiagonal ones, whereas the diagonal matrix elements of the many-body Hamiltonian increase monotonically. 1 Studies of experimental data for the energy levels in heavy nuclei [5] and complex atoms [6,7] agree with the Wigner statistics. They have been observed in numerical calculations for the atom of cerium (Ce) [8] and the nuclear sd shell model [9][10][11][12].…”

Section: Introductionmentioning

confidence: 90%

“…(6). We also use the similarity between the eigenvalue density ρ(E) and that of the basis state energies E k .…”

Section: Equilibrium Brought By the Interaction Between Particlesmentioning

confidence: 99%

“…(5). The spectral statistics were checked for many experimental and calculated complex atomic spectra [6][7][8][33][34][35], as well as for molecular vibronic spectra [36]. On the inset in Fig.…”

Section: Energy Levelsmentioning

confidence: 99%

“…Nevertheless, they display a number of chaotic properties described above. The first insight into quantum chaotic properties of complex atoms was given by Rosenzweig and Porter [6] who analysed experimental spectra of some neutral atoms and showed that the spectral statistics of heavy open-shell atoms are similar to those of compound nuclei. That analysis was later extended and refined in [7].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Quantum chaos in many-body systems: what can we learn from the Ce atom?

Flambaum

Gribakina

Gribakin

et al. 1999

Physica D: Nonlinear Phenomena

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Results of an extensive study of a real quantum chaotic many-body system -the Ce atom -are presented. We discuss the origins of the quantum chaotic behaviour of the system, analyse statistical and dynamical properties of the multi-particle chaotic eigenstates and consider matrix elements or transition amplitudes between them. We show that based on the universal properties of the chaotic eigenstates a statistical theory of finite few-particle systems with strong interaction can be developed. We also discuss such important physical effects as enhancement of weak perturbations in many-body quantum chaotic systems, distribution of single-particle occupation numbers and its deviations from the standard Fermi-Dirac shape, and ways of introducing statistical temperature-based description in such systems.

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Section: Introductionmentioning

confidence: 90%

“…(6). We also use the similarity between the eigenvalue density ρ(E) and that of the basis state energies E k .…”

Section: Equilibrium Brought By the Interaction Between Particlesmentioning

confidence: 99%

Section: Energy Levelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Quantum chaos in many-body systems: what can we learn from the Ce atom?

Flambaum

Gribakina

Gribakin

et al. 1999

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

show abstract

“…observed dependance of P (s) on the number m of basis matrices in linear combination is reminiscent of the early work of Rosenzweig and Porter [26] (RP) on the nearest neighbor spacing distribution of superpositions of independent spectra. Although the spectra of basis matrices H i (x) are not strictly independent and are added together instead of superposed ("superposed" here means "combined into a single list"), we see the same qualitative behavior as described by RP: a single basis matrix has level repulsion, but a sufficiently large number combined have Poisson statistics.…”

Section: E Basis Matrices: How Many Conservation Laws?mentioning

confidence: 70%

Integrable matrix theory: Level statistics

2016

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We study level statistics in ensembles of integrable $N\times N$ matrices linear in a real parameter $x$. The matrix $H(x)$ is considered integrable if it has a prescribed number $n>1$ of linearly independent commuting partners $H^i(x)$ (integrals of motion) $\left[H(x),H^i(x)\right] = 0$, $\left[H^i(x), H^j(x)\right]$ = 0, for all $x$. In a recent work, we developed a basis-independent construction of $H(x)$ for any $n$ from which we derived the probability density function, thereby determining how to choose a typical integrable matrix from the ensemble. Here, we find that typical integrable matrices have Poisson statistics in the $N\to\infty$ limit provided $n$ scales at least as $\log{N}$; otherwise, they exhibit level repulsion. Exceptions to the Poisson case occur at isolated coupling values $x=x_0$ or when correlations are introduced between typically independent matrix parameters. However, level statistics cross over to Poisson at $ \mathcal{O}(N^{-0.5})$ deviations from these exceptions, indicating that non-Poissonian statistics characterize only subsets of measure zero in the parameter space. Furthermore, we present strong numerical evidence that ensembles of integrable matrices are stationary and ergodic with respect to nearest neighbor level statistics.Comment: 18 pages, 26 figures, discussion on number variance added; published versio

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Statistical Properties of Energy Levels in Non‐Born‐Oppenheimer Systems

Zimmermann¹,

Cederbaum²,

Köppel³

1988

Ber Bunsenges Phys Chem

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Basic ideas and methods of the statistical analysis of energy spectra are reviewed. Complex nuclear, atomic and molecular spectra show typical fluctuation patterns which can be modeled with the aid of random matrices. Calculations on model systems and theoretical results show the existence of two universality classes of spectral fluctuations which depend on whether there are strong or only weak couplings between the various degrees of freedom. This correspondence also provides a link between quantum spectral statistics and classical dynamics. We illustrate statistical methods by analyzing model spectra obtained from a Hamiltonian which describes non‐Born‐Oppenheimer coupling of nuclear and electronic motion in molecules.

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"Repulsion of Energy Levels" in Complex Atomic Spectra

Cited by 440 publications

References 12 publications

Quantum chaos in many-body systems: what can we learn from the Ce atom?

Quantum chaos in many-body systems: what can we learn from the Ce atom?

Integrable matrix theory: Level statistics

Statistical Properties of Energy Levels in Non‐Born‐Oppenheimer Systems

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