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Journal of Physics A Mathematical and General

Semiclassical study of particle motion in two-dimensional and three-dimensional elliptical boxes. II

Robert Arvieu, Yves Ayant

Published Apr 1, 1987• 19 citations
Publisher

Abstract

For pt.I see ibid., vol.20, p.397-409, 1987. The spectrum of an ellipsoidal box of prolate and oblate symmetry are calculated by an extract diagonalisation and by semiclassical methods. The influence of a separatrix in phase space is analysed with the help of the WKB phases derived in an earlier paper. Differences between the prolate the oblate L z =0 states are explained using this separatrix. It is found that the EBK method, improved by the uniform approximation, leads to a spectrum in very good agreement with the wave mechanical results. The spectrum of a two-dimensional elliptical membrane ('billiard' box) is also studied with similar conclusions.

DOI
10.1088/0305-4470/20/5/023

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The Zeeman effect for the 'anisotropic hydrogen atoms' in the complex WKB approximation: II. Quantization of two-dimensional Lagrangian tori (with focal points) for the Pauli operator with spin-orbit interaction

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Semiclassical quantization in a prolate cavity via the adiabatic switching method: evolution of the classical actions

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Invariance of the actions which are the classical counterparts of the quantum numbers is very often postulated in adiabatic switching calculations but never checked carefully. In the particular case of a free particle moving in a prolate cavity deformed adiabatically at constant volume, time evolution of the classical actions is presented. Starting from semiclassical energies in a spherical cavity, the classical actions are calculated for each trajectory at each step of the switching procedure. This can be performed for any deformation since the system is separable in suitable coordinates. Invariance of the actions is not well verified for few values of the deformation. If it is easily understood why the actions invariance fails when the semiclassical solution cross the separatrix for the L z =0 levels, such failures are in principle not expected for the L z not=0 levels for which the trajectories belong to only one topology. The intrinsic frequencies of the motion are thus calculated for each trajectory at each step of the switching. Then, it is shown that failures in the action invariance occur exactly for values of the deformation for which two intrinsic frequencies become commensurate. For these particular values, the semiclassical trajectories are resonant for the L z not=0 levels and periodic for the L z =0 levels. By an appropriate canonical transformation on the action variables, a better invariant than the actions themselves has been built in the neighbourhood of each resonance. Apart from the particular values of the deformation where the radial and the angular frequencies become commensurate, action invariance is verified with a very good accuracy during the adiabatic switching.

Journal of Physics A Mathematical and General•Sep 21, 1993•1 citation
For pt.I see ibid., vol.20, p.397-409, 1987. The spectrum of an ellipsoidal box of prolate and oblate symmetry are calculated by an extract diagonalisation and by semiclassical methods. The influence of a separatrix in phase space is analysed with the help of the WKB phases derived in an earlier paper. Differences between the prolate the oblate L z =0 states are explained using this separatrix. It is found that the EBK method, improved by the uniform approximation, leads to a spectrum in very good agreement with the wave mechanical results. The spectrum of a two-dimensional elliptical membrane ('billiard' box) is also studied with similar conclusions.
The authors compare three problems of quantum mechanics (or more generally of wave mechanics) which reduce to the same problem in classical mechanics and which can also be treated semiclassically by the Einstein, Brillouin and Keller (1958) semiquantisation rules. The free particle motion in an ellipsoidal oblate or prolate cavity (a deformed nucleus) is compared to that in a plane elliptical billiard box. Separation of variables is performed in appropriate coordinate systems. The presence of a separatrix in phase space is exhibited, which is connected to a potential barrier that is different for each problem. The uniform approximation is used to calculate WKB phase rules appropriate to each symmetry. An important difference results between the prolate and the oblate systems.
The exact and semiclassical quantum mechanics of the elliptic billiard is investigated. The classical system is integrable and exhibits a separatrix, dividing the phase space into regions of oscillatory and rotational motion. The classical separability carries over to quantum mechanics, and the Schrödinger equation is shown to be equivalent to the spheroidal wave equation. The quantum eigenvalues show a clear pattern when transformed into the classical action space. The implication of the separatrix on the wave functions is illustrated. A uniform WKB quantization taking into account complex orbits is shown to be adequate for the semiclassical quantization in the presence of a separatrix. The pattern of states in classical action space is nicely explained by this quantization procedure. We extract an effective Maslov phase varying smoothly on the energy surface, which is used to modify the Berry-Tabor trace formula, resulting in a summation over non-periodic orbits. This modified trace formula produces the correct number of states, even close to the separatrix. The Fourier transform of the density of states is explained in terms of classical orbits, and the The semiclassical quantization of a Hamiltonian system is deeply connected to the structure of its phase space. The generic Hamiltonian system contains a complicated mixture of near-integrable and
The correspondence between the quantum spectra and the classical orbits of the two-dimensional elliptic billiards is investigated. In order to study systems without analytic solutions in the future, an expansion method for stationary states (EMSS) to get highly accurate numerical solutions is applied and tested. The closed-orbit theory was extended to the open orbit case. By comparing the Fourier-transformed spectra with the classical orbits, we find that the peak positions in the quantum spectra match with the lengths of the classical orbits with small numerical differences. This is another example showing that semiclassical method provides a bridge between quantum and classical mechanics.
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.
Invariance of the actions which are the classical counterparts of the quantum numbers is very often postulated in adiabatic switching calculations but never checked carefully. In the particular case of a free particle moving in a prolate cavity deformed adiabatically at constant volume, time evolution of the classical actions is presented. Starting from semiclassical energies in a spherical cavity, the classical actions are calculated for each trajectory at each step of the switching procedure. This can be performed for any deformation since the system is separable in suitable coordinates. Invariance of the actions is not well verified for few values of the deformation. If it is easily understood why the actions invariance fails when the semiclassical solution cross the separatrix for the L z =0 levels, such failures are in principle not expected for the L z not=0 levels for which the trajectories belong to only one topology. The intrinsic frequencies of the motion are thus calculated for each trajectory at each step of the switching. Then, it is shown that failures in the action invariance occur exactly for values of the deformation for which two intrinsic frequencies become commensurate. For these particular values, the semiclassical trajectories are resonant for the L z not=0 levels and periodic for the L z =0 levels. By an appropriate canonical transformation on the action variables, a better invariant than the actions themselves has been built in the neighbourhood of each resonance. Apart from the particular values of the deformation where the radial and the angular frequencies become commensurate, action invariance is verified with a very good accuracy during the adiabatic switching.