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Physical Review A

Regular, irregular, and scarred wave functions in the two-center shell model

Debabrata Biswas, Santanu Pal, A. K. Chaudhuri

Published May 1, 1992• 3 citations
Publisher

Abstract

The two-center shell-model potential admits a two-component eigenfunction due to the presence of the spin-orbit coupling. We carry out a detailed numerical study of its variation with the shape of the potential and also investigate the applicability of the criteria commonly used to distinguish regular and irregular states in systems that possess a clear classical analog. Our results corroborate similar studies on spectral fluctuations. The occurrence of scarred states and a Gaussian amplitude distribution strongly indicate the presence of an underlying classical dynamics. Moreover, since the spin-up and spin-down spatial components of each eigenfunction are significantly dissimilar, there seem to be two sets of closed orbits that give rise to the same spectrum but influence the wave functions in different ways.

DOI
10.1103/physreva.45.6185

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The two-center shell-model potential admits a two-component eigenfunction due to the presence of the spin-orbit coupling. We carry out a detailed numerical study of its variation with the shape of the potential and also investigate the applicability of the criteria commonly used to distinguish regular and irregular states in systems that possess a clear classical analog. Our results corroborate similar studies on spectral fluctuations. The occurrence of scarred states and a Gaussian amplitude distribution strongly indicate the presence of an underlying classical dynamics. Moreover, since the spin-up and spin-down spatial components of each eigenfunction are significantly dissimilar, there seem to be two sets of closed orbits that give rise to the same spectrum but influence the wave functions in different ways.
Quantum–classical correspondence for the average shape of eigenfunctions and the local spectral density of states are well-known facts. In this paper, the fluctuations of the quantum wavefunctions around the classical value are discussed. A simple random matrix model leads to a Gaussian distribution of the amplitudes whose width is determined by the classical shape of the eigenfunction. To compare this prediction with numerical calculations in chaotic models of coupled quartic oscillators, we develop a rescaling method for the components. The expectations are broadly confirmed, but deviations due to scars are observed. This effect is much reduced when both Hamiltonians have chaotic dynamics.
We propose a picture of Wigner function scars as a sequence of concentric rings along a two-dimensional surface inside a periodic orbit. This is verified for a two-dimensional plane that contains a classical hyperbolic orbit of a Hamiltonian system with 2 degrees of freedom. The stationary wave functions are the familiar mixture of scarred and random waves, but the spectral average of the Wigner functions in part of the plane is nearly that of a harmonic oscillator and individual states are also remarkably regular. These results are interpreted in terms of the semiclassical picture of chords and centers.
The question of classical correspondence in the case of the Dirac equation has not been fully resolved as yet. In this paper, we consider Dirac particles in cavities of various shapes and provide numerical evidence of the influence of periodic orbits in the quantal density of states, d(E). Interestingly, the orbit lengths seem to be the same as in the (scalar) spinless case, though the associated phases in the periodic orbit sum for d(E) are in general different. We also study the spectral fluctuations and find that statistical measures reflect an order-to-chaos transition in the underlying classical dynamics, similar to that seen in the scalar case. We provide a detailed study of the wave functions as well and find, apart from other characteristic features, the existence of scarred states. Typical irregular eigenfunction components, however, exhibit contour splitting and have a Gaussian amplitude distribution.
The Husimi distribution for the eigenstates of a classically nonintegrable two-dimensional system with mixed phase space was calculated in order to analyze its structure in terms of classical trajectories. Besides eigenfunctions concentrated on invariant tori or with support on chaotic regions of phase space, we also found others that combine different classical invariant sets (like a stable and an unstable periodic orbit) and families of eigenstates scarred by classically unstable periodic orbits.
The average shape of the spectral local density of states (LDOS) and eigenfunctions (EFs) has been studied numerically for a conservative dynamical model (three-orbital Lipkin-Meshkov-Glick model) that can exhibit strong chaos in the classical limit. Attention is paid to the comparison of the shape of the LDOS with that known for random matrix models, as well as to the shape of the EFs, for different values of the perturbation strength. The classical counterparts of the LDOS has also been studied and found to be in remarkable agreement with the quantum calculations. Finally, by making use of a generalization of Brillouin-Wigner perturbation expansion, the form of the long tails of the LDOS and EFs is given analytically and confirmed numerically.
The localization of eigenfunctions around classical periodic orbits is studied numerically for the H-atom in a strong magnetic field by calculating their Husimi distribution in phase space. In contrast to the configuration space representation, the phase space distributions are simply structured: about 90% of eigenstates may be unambigously related to fixed points and invariant manifolds of periodic orbits, indicating that scars are the rule rather than the exception. In order to measure the influence of one particular orbit, we calculate the integrals of the energetically lowest 500 Husimi distributions along the orbit. Their incoherent superposition defines the scar strength distribution for the particular periodic orbit which is analyzed by Fourier transformation. The Husimi distribution at (q, p) in phase space may be represented as a scalar product of the wavefunction with a coherent state of the unperturbed system, i.e., a radial Gaussian wave packet located at (q, p) in the (regularized) Coulomb system. This simplifies the actual calculation of the Husimi distribution and allows to treat their incoherent superposition within Gutzwillers theory extended to matrix elements of an operator A, if we choose A to be the projector on a coherent state.