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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Semiclassical features in the quantum description of a Dirac particle in a cavity

S. C. Phatak, Santanu Pal, Debabrata Biswas

Published Aug 1, 1995• 5 citations
Publisher

Abstract

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.

DOI
10.1103/physreve.52.1333

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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.
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.
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.
We solve the Dirac equation in two spatial dimensions in the setting of resonant tunneling, where the system consists of two symmetric cavities connected by a finite potential barrier. The shape of the cavities can be chosen to yield both regular and chaotic dynamics in the classical limit. We find that certain pointer states about classical periodic orbits can exist, which suppress quantum tunneling, and the effect becomes less severe as the underlying classical dynamics in the cavity is chaotic, leading to regularization of tunneling dynamics even in the relativistic quantum regime. Similar phenomena have been observed in graphene. A physical theory is developed to explain the phenomenon based on the spectrum of complex eigenenergies of the non-Hermitian Hamiltonian describing the effectively open cavity system.
The eigenvalue density of a quantum-mechanical system exhibits oscillations, determined by the closed orbits of the corresponding classical system; this relationship is simple and strong for waves in billiards or on manifolds, but becomes slightly muddy for a Schrodinger equation with a potential, where the orbits depend on the energy. We discuss several variants of a way to restore the simplicity by rescaling the coupling constant or the size of the orbit or both. In each case the relation between the oscillation frequency and the period of the orbit is inspected critically; in many cases it is observed that a characteristic length of the orbit is a better indicator. When these matters are properly understood, the periodic-orbit theory for generic quantum systems recovers the clarity and simplicity that it always had for the wave equation in a cavity. Finally, we comment on the alleged "paradox" that semiclassical periodic-orbit theory is more effective in calculating low energy level...
We study chaotic quantum billiards using both microwave cavities and numerical simulations. For the same geometry, viz., a Sinai billiard, agreement to remarkable precision is found for both the eigenvalue magnitudes and the spatial detail of the eigenfunctions. The association of the eigenfunctions with classical periodic orbits is demonstrated, and scarred states are identified. Desymmetrizing the Sinai billiard by slightly moving the central disk is shown to lead to strong localization of the eigenfunction. The calculated eigenstates of the symmetric billiard show an even- and odd-parity pair whose linear combination gives the localized state.
The eigenvalue density of a quantum-mechanical system exhibits oscillations, determined by the closed orbits of the corresponding classical system; this relationship is simple and strong for waves in billiards or on manifolds, but becomes slightly muddy for a Schrödinger equation with a potential, where the orbits depend on the energy.In special cases the simplicity has been restored by rescaling the size of the orbit, and perhaps a coupling constant at the same time.We point out that the goal can be achieved for any system of this class simply by rescaling the overall coupling constant.In each of these situations we inspect critically the relation between the oscillation frequency and the period of the orbit; in many cases it is observed that a characteristic length of the orbit is a better indicator.When these matters are properly understood, the periodic-orbit theory for generic quantum systems recovers the clarity and simplicity that it always had for the wave equation in a cavity.Finally, we comment on the alleged 'paradox' that semiclassical periodic-orbit theory is more effective in calculating low energy levels than high ones.