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Physical Review A

Quantum chaos of the hydrogen atom in a generalized van der Waals potential

K. Ganesan, M. Lakshmanan

Published Aug 1, 1993• 24 citations
Publisher

Abstract

The quantum manifestations of chaos in the hydrogen atom in a generalized van der Waals potential, which includes the celebrated quadratic-Zeeman-effect problem under an appropriate limiting condition, are studied in detail. Using group-theoretical methods, we derive the matrix elements in an algebraic form. As the Hamiltonian is scale invariant, we introduce an appropriate scaling parameter. By considering the unperturbed hydrogen-atom problem, we demonstrate that suitable changes in the scaling parameter stabilize different parts of the spectrum depending upon the parameter's value. For the present generalized van der Waals potential problem, we utilize this property of the scaling parameter effectively to improve the convergence of eigenvalues while diagonalizing the matrices for various parametric values. Then, we vary one of the system parameters in the range [0,3], study the level statistics, and observe a GOE-Poisson-Brody-Poisson-Brody-Poisson-GOE-type (where GOE means Gaussian orthogonal ensemble) of transition regime hitherto unidentified in any of the perturbed hydrogen-atom problems. Our results are not only in agreement with random-matrix-theory predictions but also justify classical and semiclassical investigations.

DOI
10.1103/physreva.48.964

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quant-pharXiv:2607.03786•Jul 7, 2026
The quantum manifestations of chaos in the hydrogen atom in a generalized van der Waals potential, which includes the celebrated quadratic-Zeeman-effect problem under an appropriate limiting condition, are studied in detail. Using group-theoretical methods, we derive the matrix elements in an algebraic form. As the Hamiltonian is scale invariant, we introduce an appropriate scaling parameter. By considering the unperturbed hydrogen-atom problem, we demonstrate that suitable changes in the scaling parameter stabilize different parts of the spectrum depending upon the parameter's value. For the present generalized van der Waals potential problem, we utilize this property of the scaling parameter effectively to improve the convergence of eigenvalues while diagonalizing the matrices for various parametric values. Then, we vary one of the system parameters in the range [0,3], study the level statistics, and observe a GOE-Poisson-Brody-Poisson-Brody-Poisson-GOE-type (where GOE means Gaussian orthogonal ensemble) of transition regime hitherto unidentified in any of the perturbed hydrogen-atom problems. Our results are not only in agreement with random-matrix-theory predictions but also justify classical and semiclassical investigations.
We study signatures of quantum chaos in (1+1)D quantum field theory (QFT) models. Our analysis is based on the method of Hamiltonian truncation, a numerical approach for the construction of low-energy spectra and eigenstates of QFTs that can be considered as perturbations of exactly solvable models. We focus on the double sine-Gordon, also studying the massive sine-Gordon and ϕ^{4} model, all of which are nonintegrable and can be studied by this method with sufficiently high precision from small to intermediate perturbation strength. We analyze the statistics of level spacings and of eigenvector components, which are expected to follow random matrix theory predictions. While level spacing statistics are close to the Gaussian orthogonal ensemble (GOE) as expected, on the contrary, the eigenvector components follow a distribution markedly different from the expected Gaussian. Unlike in the typical quantum chaos scenario, the transition of level spacing statistics to chaotic behavior takes place already in the perturbative regime. Moreover, the distribution of eigenvector components does not appear to change or approach Gaussian behavior, even for relatively large perturbations. Our results suggest that these features are independent of the choice of model and basis.
A recent proposal by Hallam et al. suggested using the chaotic properties of the semiclassical equations of motion, obtained by the time dependent variational principle (TDVP), as a characterization of quantum chaos. In this paper, we calculate the Lyapunov spectrum of the semiclassical theory approximating the quantum dynamics of a strongly interacting Rydberg atom array, which lead to periodic motion. In addition, we calculate the effect of quantum fluctuations around this approximation, and obtain the escape rate from the periodic orbit. We compare this rate to the rate extracted from the exact solution of the quantum theory, and find an order of magnitude discrepancy. We conclude that in this case, chaos in the TDVP equations does not correpond to phsyical properties of the system. Our result complement those of Ho et al. regarding the escape rate from the semiclassical periodic orbit.
Whether the many-body mobility edges can exist in a one-dimensional interacting quantum system is a controversial problem, mainly hampered by the limited system sizes amenable to numerical simulations. We investigate the transition from chaos to localization by constructing a combined random matrix, which has two extremes, one of Gaussian orthogonal ensemble and the other of Poisson statistics, drawn from different distributions. We find that by fixing a scaling parameter, the mobility edges can exist while increasing the matrix dimension D→∞D\rightarrow\inftyD→∞, depending on the distribution of matrix elements of the diagonal uncorrelated matrix. By applying those results to a specific one-dimensional isolated quantum system of random diagonal elements, we confirm the existence of a many-body mobility edge, connecting it with results on the onset of level repulsion extracted from ensembles of mixed random matrices.
It is a well-established fact that statistical properties of energy-level spectra are the most efficient tool to characterize nonintegrable quantum systems. The statistical behavior of different systems such as complex atoms, atomic nuclei, two-dimensional Hamiltonians, quantum billiards, and noninteracting many bosons has been studied. The study of statistical properties and spectral fluctuations in interacting many-boson systems has developed interest in this direction. We are especially interested in weakly interacting trapped bosons in the context of Bose-Einstein condensation (BEC) as the energy spectrum shows a transition from a collective nature to a single-particle nature with an increase in the number of levels. However this has received less attention as it is believed that the system may exhibit Poisson-like fluctuations due to the existence of an external harmonic trap. Here we compute numerically the energy levels of the zero-temperature many-boson systems which are weakly interacting through the van der Waals potential and are confined in the three-dimensional harmonic potential. We study the nearest-neighbor spacing distribution and the spectral rigidity by unfolding the spectrum. It is found that an increase in the number of energy levels for repulsive BEC induces a transition from a Wigner-like form displaying level repulsion to the Poisson distribution for P(s)P(s)P(s). It does not follow the Gaussian orthogonal ensemble prediction. For repulsive interaction, the lower levels are correlated and manifest level-repulsion. For intermediate levels P(s)P(s)P(s) shows mixed statistics, which clearly signifies the existence of two energy scales: external trap and interatomic interaction, whereas for very high levels the trapping potential dominates, generating a Poisson distribution. Comparison with mean-field results for lower levels are also presented. For attractive BEC near the critical point we observe the Shnirelman-like peak near s=0s=0s=0, which signifies the presence of a large number of quasidegenerate states.
We study the emergence of generic quantum chaos from an arithmetic billiard in the hyperbolic plane. Starting from an arithmetic triangle, we introduce small area-preserving geometric perturbations and compute long consecutive sequences of eigenvalues and eigenstates. The spectral statistics show a perturbation-dependent crossover from Poisson-like arithmetic behaviour to the Gaussian orthogonal ensemble (GOE), with the crossover scale moving to higher wavenumbers as the perturbation decreases. To characterize the nearest-neighbour spacing distribution, we compare the data with Brody and generalized-gamma fits. The generalized-gamma form is used only as a two-parameter diagnostic, allowing the small- and intermediate-spacing structure to vary independently from the large-spacing tail, rather than as a universal crossover law. For the weakest perturbation, we also compare the data with an effective block-GOE model, which diagnoses residual block-like structure deep into the computed spectral range. We complement the spectral analysis with Poincaré-Husimi representations of the eigenstates and quantify phase-space localization using an entropy-based measure. Its distribution is well described by Beta distributions whose width decays as a power law in the wavenumber. Before the spectral crossover, the localization statistics retain memory of the near-arithmetic regime. After the GOE regime is reached, the localization-fluctuation widths collapse onto a common decay law, sigma(Beta) ~ k^{-eta}, with eta approximately 0.37--0.38. These results provide a combined spectral and eigenfunction-level picture of the arithmetic-to-GOE crossover in hyperbolic quantum billiards.