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Journal of Physics B Atomic Molecular and Optical Physics

Phase space scars and quantization of non-integrable systems: the diamagnetic hydrogen atom

Wolfgang Schweizer, M. Schaich, Wolfgang Jans

Published Jul 14, 1994• 6 citations
Publisher

Abstract

We present here a study of the quantum phase space localization (Wigner functions) in dependence of semiclassical quantization rules for the hydrogen atom in magnetic fields. We consider primarily two energy regions. In the first the corresponding classical system is near-integrable and in the second near fully chaotic. We study phase space localization (scars) close to stable and unstable trajectories, tori and the invariant manifold associated with the almost circular, unstable periodic trajectory. We find that these classical structures are an important element for the structure of the wavefunctions and the semiclassical quantization predictions are in good agreement with the quantal results.

DOI
10.1088/0953-4075/27/13/005

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Quantum phase-space behaviour for the H2molecule in a magnetic field

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The first study of the quantum dynamics for a highly excited ('Rydberg') H 2 molecule in a magnetic field is presented and compared with the well studied atomic case. Wavefunctions from an accurate quantal calculation have been used to obtain the corresponding Wigner functions by means of a transformation to a basis where the latter are analytical. Unlike the atom, the dynamics of the excited electron in the molecular case experiences two, interfering, energy regimes. This study consider the case near electronic scaled energies -0.5 and -2.3, which would correspond to predominantly regular motion in the atomic case. In the molecular case there is evidence for chaotic behaviour in this energy range. Quantum states for the molecule may divide into those where the underlying dynamics is a perturbed version of the hydrogen atom case, and others where different configuration-space and phase-space structures are apparent.

Journal of Physics B Atomic Molecular and Optical Physics•Dec 28, 1992•2 citations
We present here a study of the quantum phase space localization (Wigner functions) in dependence of semiclassical quantization rules for the hydrogen atom in magnetic fields. We consider primarily two energy regions. In the first the corresponding classical system is near-integrable and in the second near fully chaotic. We study phase space localization (scars) close to stable and unstable trajectories, tori and the invariant manifold associated with the almost circular, unstable periodic trajectory. We find that these classical structures are an important element for the structure of the wavefunctions and the semiclassical quantization predictions are in good agreement with the quantal results.
We compare quantum phase space distributions of individual quantum states of the diamagnetic hydrogen atom obtained by means of Wigner functions with those given by Husimi functions. The comparison is carried out at effective h(cross) approximately 0.035 and at a fixed scaled energy ( epsilon =-0.316) which corresponds to an especially interesting mixed classical phase-space structure. The object of the comparison is to establish which of the two distributions best correlates with the classical Poincare surface of section for a representative set of single states. As expected, the states investigated display the strongest positive intensity on a single local invariant phase-space structure (such as a scar or torus) and at this level there is little difference between the Husimi and the Wigner function. However the weak structures (fringes of the Wigner function or zeros of the Husimi function) are shown here to be radically different for the Wigner relative to the Husimi representation. In both cases the weak structures permeate all of phase space. In addition they show different character for integrable and chaotic dynamics and so reflect the global structure of phase-space to a much greater extent than the localized strong structure. The Wigner functions of selected states are shown to have additional dynamically significant structures which are not apparent in the Husimi functions. These include negative intensity features seen in the Wigner-but not the Husimi-functions which are very well correlated with structures of the classical Poincare surfaces of section. Despite its complex oscillatory nature the Wigner function delineates better classical features, for example showing the outline of islands of stability avoided by states supported by chaotic regions with greater sharpness than the Husimi function.
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.
For pt.I, see ibid, p.5799-5810. On the basis of the complex WKB method, new spectral series for perturbed states of an 'anisotropic' hydrogen atom, taking into account the electron's spin polarization in a homogeneous magnetic field are constructed. These series correspond to conditionally periodic motions of a classical electron in the plane orthogonal to the field direction. The semiclassical wavefunctions are regular in the entire configuration space including the focal points and have the 'superscar' properties near the domain of classical motion, i.e. near the projection of stable two-dimensional Lagrangian tori on the configuration space. Both for Schrodinger and Pauli operators the analytical description of the semiclassical wavefunctions 'superscar' structure is obtained. This structure is non-uniform with respect to the space coordinates and has singularities with respect to the parameter h(cross),h(cross) to 0, near focal points. In the limit case of strong magnetic fields explicit analytical formulae for the semiclassical energy levels are obtained.
Abstract We explore the semi-classical structure of the Wigner functions Ψ(q,p) representing bound energy eigenstates | Ψ 〉 for systems with fdegrees of freedom. If the classical motion is integrable, the classical limit of Ψ is a delta function on the f-dimensional torus to which classical trajectories corresponding to |Ψ〉 are confined in the 2f-dimensional phase space. In the semi-classical limit of Ψ (ℏ small but not zero) the delta function softens to a peak of order Ψ−f and the torus develops fringes of a characteristic ‘Airy’ form. Away from the torus,Ψ can have semi-classical singularities that are not delta functions; these are discussed (in full detail when f = 1) using Thom's theory of catastrophes. Brief consideration is given to problems raised when is calculated in a representation based on operators derived from angle coordinates and their conjugate momenta. When Ψ the classical motion is non-integrable, the phase space is not filled with tori and existing semi-classical methods fail. We conjecture that (a) For a given value of non-integrability parameter ⋲,the system passes through three semi-classical régimes as ℏ diminishes. (b) For states |Ψ〉 associated with regions in phase space filled with irregular trajectories, Ψ will be a random function confined near that region of the ‘energy shell’ explored by these trajectories (this region has more thanks dimensions). (c) For ⋲ ≠ 0, ℏ blurs the infinitely fine classical path structure, in contrast to the integrable case ⋲ = 0, where ℏ imposes oscillatory quantum detail on a smooth classical path structure.
The periodic-orbit quantization approach of Gutzwiller, which is a semiclassical approximation of Feynman's path-integral formalism, is used to calculate finite-resolution densities of states for a Hamiltonian system whose classical motion is dominated by chaos, viz., the hydrogen atom in a uniform magnetic field. In spectral regions where the resolution obtained is larger than the mean level spacing, it is possible to extract approximate eigenvalues of the quantized system. The particular role of stable periodic orbits and their direct quantization are elaborated. Remarkable agreement with exact quantum calculations results.
The first study of the quantum dynamics for a highly excited ('Rydberg') H 2 molecule in a magnetic field is presented and compared with the well studied atomic case. Wavefunctions from an accurate quantal calculation have been used to obtain the corresponding Wigner functions by means of a transformation to a basis where the latter are analytical. Unlike the atom, the dynamics of the excited electron in the molecular case experiences two, interfering, energy regimes. This study consider the case near electronic scaled energies -0.5 and -2.3, which would correspond to predominantly regular motion in the atomic case. In the molecular case there is evidence for chaotic behaviour in this energy range. Quantum states for the molecule may divide into those where the underlying dynamics is a perturbed version of the hydrogen atom case, and others where different configuration-space and phase-space structures are apparent.