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Journal of Physics A Mathematical and Theoretical

High-energy localized eigenstates of an electronic resonator in a magnetic field

Victor V. Zalipaev, Fedor Vasilievich Kusmartsev, M. M. Popov

Published Jan 29, 2008• 7 citations
Publisher

Abstract

We present a semiclassical analysis of the high-energy eigenstates of an electron inside a closed resonator. An asymptotic method of the construction of the energy spectrum and eigenfunctions, localized in the small neighborhood of a periodic orbit, is developed in the presence of a homogeneous magnetic field and arbitrary scalar potential. The isolated periodic orbit is confined between two interfaces which could be planar, concave or even convex. Such a system represents a quantum electronic resonator, an analog of the well-known high-frequency optical or acoustic resonator with eigenmodes called 'bouncing ball vibrations'. The first step in the asymptotic analysis involves constructing a solitary localized asymptotic solution to the Schrodinger equation (electronic Gaussian beam—wavepackage). Then, the stability of a closed continuous family of periodic trajectories confined between two reflecting surfaces of the resonator boundary was studied. The asymptotics of the eigenfunctions were constructed as a superposition of two electronic Gaussian beams propagating in opposite directions between two reflecting points of the periodic orbits. The asymptotics of the energy spectrum are obtained by the generalized Bohr–Sommerfeld quantization condition derived as a requirement for the eigenfunction asymptotics to be periodic. For one class of periodic orbits, localized eigenstates were computed numerically by the finite element method using FEMLAB and proved to be in a very good agreement with those computed semiclassically.

DOI
10.1088/1751-8113/41/6/065101

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Journal of Physics A Mathematical and General•Oct 21, 1995•9 citations
We present a semiclassical analysis of the high-energy eigenstates of an electron inside a closed resonator. An asymptotic method of the construction of the energy spectrum and eigenfunctions, localized in the small neighborhood of a periodic orbit, is developed in the presence of a homogeneous magnetic field and arbitrary scalar potential. The isolated periodic orbit is confined between two interfaces which could be planar, concave or even convex. Such a system represents a quantum electronic resonator, an analog of the well-known high-frequency optical or acoustic resonator with eigenmodes called 'bouncing ball vibrations'. The first step in the asymptotic analysis involves constructing a solitary localized asymptotic solution to the Schrodinger equation (electronic Gaussian beam—wavepackage). Then, the stability of a closed continuous family of periodic trajectories confined between two reflecting surfaces of the resonator boundary was studied. The asymptotics of the eigenfunctions were constructed as a superposition of two electronic Gaussian beams propagating in opposite directions between two reflecting points of the periodic orbits. The asymptotics of the energy spectrum are obtained by the generalized Bohr–Sommerfeld quantization condition derived as a requirement for the eigenfunction asymptotics to be periodic. For one class of periodic orbits, localized eigenstates were computed numerically by the finite element method using FEMLAB and proved to be in a very good agreement with those computed semiclassically.
We extend the asymptotic boundary layer (ABL) method, originally developed for stable resonator modes, to the description of individual wave functions localized around unstable periodic orbits. The formalism applies to the description of scar states in fully or partially chaotic quantum systems, and also allows for the presence of smooth and sharp potentials, as well as magnetic fields. We argue that the separatrix wave function provides the largest contribution to the scars on a single wave function. This agrees with earlier results on the wave-function asymptotics and on the quantization condition of the scar states. Predictions of the ABL formalism are compared with the exact numerical solution for a strip resonator with a parabolic confinement potential and a magnetic field.
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.
The relation between the solutions of the time-independent Schrödinger equation and the periodic orbits of the corresponding classical system is examined in the case where neither can be found by the separation of variables. If the quasiclassical approximation for the Green's function is integrated over the coordinates, a response function for the system is obtained which depends only on the energy and whose singularities give the approximate eigenvalues of the energy. This response function is written as a sum over all periodic orbits where each term has a phase factor containing the action integral and the number of conjugate points, as well as an amplitude factor containing the period and the stability exponent of the orbit. In terms of the approximate density of states per unit interval of energy, each stable periodic orbit is shown to yield a series of δ functions whose locations are given by a simple quantum condition: The action integral differs from an integer multiple of h by half the stability angle times ℏ. Unstable periodic orbits give a series of broadened peaks whose half-width equals the stability exponent times ℏ, whereas the location of the maxima is given again by a simple quantum condition. These results are applied to the anisotropic Kepler problem, i.e., an electron with an anisotropic mass tensor moving in a (spherically symmetric) Coulomb field. A class of simply closed, periodic orbits is found by a Fourier expansion method as in Hill's theory of the moon. They are shown to yield a well-defined set of levels, whose energy is compared with recent variational calculations of Faulkner on shallow bound states of donor impurities in semiconductors. The agreement is good for silicon, but only fair for the more anisotropic germanium.
Newly developed methods for finding the semiclassical or WKB eigenvalues and eigenfunctions for vector wave fields are applied to spinning particles moving in central potentials subject to spin-orbit forces. The new methods are like the familiar Bohr-Sommerfeld or Einstein-Brillouin-Keller quantization methods for scalar wave fields, but involve additional issues relating to Berry's phase, gauge structures, and monopolelike singularities. All these extra issues occur in the asymptotics of spin-orbit coupling. The role of angular momentum and conservation laws is particularly interesting, due to the role played by gauge fields in the classical-quantum correspondence. The work presented is a contribution to ongoing semiclassical studies of shell structure in nuclei and other fermion systems such as metal clusters, as well as an example of the general methods of semiclassical vector wave quantization.
Gaussian-Klauder coherent states are applied to the bound ``quantum bouncer,'' a gravitating particle above an infinite potential boundary. These Gaussian-Klauder states, originally created for Rydberg atoms, provide an overcomplete set of wave functions that mimic classical trajectories for extended times through the utilization of energy localization. For the quantum bouncer, analytic methods are applied presently to compute first and second moments of position and momentum operators, and from these results, at least two scalings of Gaussian-Klauder parameters are highlighted, one of which tends to remains localized for markedly more bounces than comparable states that are Gaussian in position (by an order of magnitude in some cases). We close with a connection that compares Gaussian-Klauder states and positional Gaussian states directly for the quantum bouncer, relating the two through a known energy-position duality of Airy functions. Our results, taken together, ultimately reemphasize the primacy of energy localization as a key ingredient for long-lived classical correspondence in systems with smooth spectra.
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.