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Journal of Physics A Mathematical and General

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

F. Brut

Published Sep 21, 1993• 1 citation
Publisher

Abstract

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.

DOI
10.1088/0305-4470/26/18/036

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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.
The semiclassical method based on the Einstein-Brillouin-Keller quantization of invariant phase-space tori and the hypothesis of adiabatic invariance of classical action variables is applied in the calculation of highly excited energy levels of two-dimensional coupled oscillator systems. The results obtained are in good agreement with quantum-mechanical calculations even in the regions where the corresponding classical dynamics becomes apparently irregular. General limitations of the method are discussed.
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 problem of semiclassical quantization of nonseparable systems with a finite number of degrees of freedom is studied within the framework of Heisenberg matrix mechanics, in extension of previous work on one-dimensional systems. The relationship between the quantum theory and multiply-periodic classical motions is derived anew. A suitably averaged Lagrangian provides a variational basis not only for the Fourier components of the semiclassical equations of motion, but also for the general definition of action variables. A Legendre transformation to the Hamiltonian verifies that these have been properly chosen and therefore provide a basis for the quantization of nonseparable systems. The problem of connection formulas is discussed by a method integral to the present approach. The action variables are shown to be adiabatic invariants of the classical system. An elementary application of the method is given. The methods of this paper are applicable to nondegenerate systems only.
Quantum-mechanical evolution in a slowly varying double-well potential is analyzed in the semiclassical limit to determine the transition probability due to loss of adiabatic invariance of states having energy close to that of a separatrix in the equivalent classical system. Transformation to the rotating-axis basis reduces the problem to the integration of a set of coupled differential equations. In the adiabatic limit the integration is carried out by the method of successive approximations. For adiabatic parameter \ensuremath{\epsilon} of the order of the inverse, 1/N, of the number of quantum states with energy below the separatrix, the spread of the quantum adiabatic invariant is exponentially small in the inverse of the adiabaticity parameter. Surprisingly, the transition to this quantum adiabatic behavior occurs far below the minimum transition frequency of the system.
This paper addresses the general problem of zeroth order representation of resonant dynamics. We investigate the classical, quantum mechanical, and semiclassical transformation properties of two-dimensional isotropic and anisotropic uncoupled harmonic oscillators. The classical and quantal theories are presented in a manner that emphasizes the strong correspondence between the two, and in particular, the SU(2) symmetry exhibited by both the classical and quantum oscillators. The classical canonical transformations relating the action-angle variables appropriate for normal, local, and precessional motion of the isotropic oscillator are derived by explicit calculation of the generating functions. By employing a simple mapping relating the anisotropic and isotropic oscillators, expressions for action-angle variables appropriate for the topology of an arbitrary m:n resonance are determined. The resulting invariant tori are compared with the corresponding quantum mechanical wave functions and phase space densities. The relationship between the classical and quantum mechanical theories is illustrated by determining semiclassical approximations to the unitary transformation matrix elements, which are given in terms of the classical generating functions. Applications to problems of current interest, such as the adiabatic switching method for semiclassical quantization of nonseparable systems, are briefly discussed.
Semiclassical approximations of quantummechanics are known to be invariant with respect to classical canonical transformat ions, even though these are not in general isomorphic to unitary transform- ations in quantum mechanics. However, this rule is not a priori ex- tensive to effects involving t unnelling, since these may be semiclas- sical ly small themselves. An important class of canonical transform - ations b rings the system into action-angle va.riables,which facil i tates the study of tunnell ing near a classical resonance. We verify compu- tational ly that the energy eigenlevels of a resonant system computed in a harmnic oscil l ator basis are in good agreement with the semi- classical values obtained with the use of action-angle variables.