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Journal of Physics B Atomic Molecular and Optical Physics

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

T. S. Monteiro

Published Dec 28, 1992• 2 citations
Publisher

Abstract

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.

DOI
10.1088/0953-4075/25/24/001

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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.
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.
A discussion regarding the energy levels spectrum of quantum systems whose classical analogous has states of chaotic motion is presented. The chaotic dynamics of the classical underlying system has its manifestation in the wave functions (in the form of “scars”) and in the energy levels (in the form of “statistical repulsion” of the energy levels). The above mentioned signatures are named “quantum chaos”. A typical study of quantum chaos requires finding accurate energy eigenvalues of highly excited states, to calculate the nearest neighbors spacing between levels, to perform the “unfolding” of the spectrum in order to separate the fluctuations, and finally to find the probability distribution of the unfolded spectrum. This is exemplified by the hydrogen atom under uniform magnetic field and a quadrupole electric field.
A hydrogen atom moving across a magnetic field is considered in a wide region of magnitudes of magnetic field and atom momentum. We solve the Schr\"odinger equation of the system numerically using an imaginary time method and find wave functions of the lowest states of atom. We calculate the energy and the mean electron-nucleus separation as a function of atom momentum and magnetic field. All the results obtained could be summarized as a phase diagram on the ``atom-momentum -- magnetic-field'' plane. There are transformations of wave-function structure at critical values of atom momentum and magnetic field that result in a specific behavior of dependencies of energy and mean interparticle separation on the atom momentum P.P.P. We discuss a transition from the Zeeman regime to the high magnetic field regime. A qualitative analysis of the complicated behavior of wave functions vs PPP based on the effective potential examination is given. We analyze a sharp transition at the critical momentum from a Coulomb-type state polarized due to atom motion to a strongly decentered (Landau-type) state at low magnetic fields. A crossover occurring at intermediate magnetic fields is also studied.
A simple and accurate variational method is formulated and applied to the ground and excited states of the hydrogen atom as well as the hydrogen molecule ion in a homogeneous magnetic field of arbitrary strength. For the ground state, the electronic wave function o is assumed to be a function of a single, appropriately chosen variable R which contains a variational parameter. The functional form of o, and the corresponding energy, are determined by solving a Schr6dinger- like equation, in which the variational parameter appears explicitly in the kinetic and potential energy terms. The variational parameter is optimized by minimizing the lowest eigenvalue of the equation. Excited states may also be obtained by a slight modification of the method, and explicit calculations are carried out for the m1 = - 1 first excited state of hydrogen and the ( )-state of H2 +. Numerical comparisons are made with some recent papers in the literature. Subject headings: atomic processes - magnetic fields - molecular processes - quantum mechanics
Many electron calculations on a simplest realistic two electron system i.e. H2molecule was applied and as the consequence correlation effects was reflected accurately in the wavefunctions of H2. Zanardi’s entanglement measurement, demonstrated that the maximum of entanglement for the ground state happens when U =J and this resolved the controversial conclusion of U = 0 for maximum entanglement. It was shown that the ground and third excited states are maximally entangled. These maximally entangled states and also the minimally entangled states are correlated to their spin’s property. The wavefunctions of the not magnetic (S = 0) ground and excited states explicitly depend on correlation parameters whereas the first excited states which is magnetic (S2 = 2 and Sz≠0) is not entangled. The second excited state is not magnetic but its wavefunction does not depend on correlation parameters therefore it is a moderately entangled state. In any case, by switching on a magnetic field an entangled state with Sz = 0 can be extracted from a not entangled degenerate magnetic state. We suggest that in a realistic molecular scale system, there is two criteria for finding maximally entangled electronic states, first the system should be in moderately correlated regime and second the system should have a non-magnetic (Sz = 0) electronic state.