Abstract
An analytical and local expression of the semiclassical Wigner distribution is performed using the Airy uniform approximation in the neighbourhood of a turning point. At the classical limit the one-dimensional Wigner distribution leads to the microcanonical distribution. For the two-dimensional systems the variables separable problems (nonchaotic dynamics) and the variables nonseparable problems (possibility of chaos) are not locally distinguishable by using the uniform approximation. Meanwhile the examination of a section of the phase space allows us to see that the apparent complexity of the Wigner distribution is not a proof of chaos.
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References
Abramowitz, M., Stegun, I. (1965): Handbook of Mathematical Functions (Dover Publications, New York)
Balazs, N.L., Zipfel, G.G. (1973): Quantum oscillations in the semiclassical fermion μ-space density. Ann. Phys. 77, 139–156
Berry, M.V. (1977): Semiclassical mechanics in phase space: a study of Wigner's function. Phil. Trans. R. Soc. London A 287, 237–271
Berry, M.V., Mount, K.E. (1972): Semiclassical approximations in wave mechanics. Rep. Prog. Phys. 35, 315–397
Dando, P.A., Monteiro, T.S. (1994): Quantum surfaces of section for the diamagnetic hydrogen atom: Husimi functions versus Wigner functions. J. Phys. B 27, 2681–2692
Ford, J. (1989): Quantum chaos, is there any? in: Directions in Chaos, Hao Bai-Lin ed. (World Scientific, Singapore) Vol 2
Ford, J., Ilg, M. (1992): Eigenfunctions, eigenvalues, and time evolution of finite, bounded, undriven quantum systems are not chaotic. Phys. Rev. A 45, 6165–6173
Ford, J., Mantica, G. (1992): Does quantum mechanics obey the correspondence principle? Is it complete? Am. J. Phys. 60, 1086–1098
Grémaud, B., Delande, D., Gay, J.C. (1993): Origin of narrow resonances in the diamagnetic Rydberg spectrum. Phys. Rev. Lett. 70, 1615–1618
Heller, E.J. (1976): Wigner phase space method: analysis for semiclassical applications. J. Chem. phys. 65, 1289–1298
Heller, E.J. (1984): Bound-state eigenfunctions of classically chaotic hamiltonian systems: scars of periodic orbits. Phys. Rev. Lett. 53, 1515–1518
McDonald, S.W., Kaufman, A.N. (1979): Spectrum and eigenfunctions for a hamiltonian with stochastic trajectories. Phys. Rev. Lett. 42, 1189–1191
Meredith, D.C. (1992): Semiclassical wavefunctions of nonintegrable systems and localization on periodic orbits. J. Stat. Phys. 68, 97–130
Ozorio de Almeida, A.M., Hannay, J.H. (1982): Geometry of two dimensional tori in phase space: projections, sections and the Wigner function. Ann. Phys. 138, 115–154
Percival, I.C. (1973): Regular and irregular spectra. J. Phys. B 6, L229–L232
Pomphrey, N. (1974): Numerical identification of regular and irregular spectra. J. Phys. B 7, 1909–1915
Tabor, M. (1989): Chaos and integrability in semiclassical mechanics. in Chaos and integrability in non-linear dynamics, Tabor, M. ed. (Wiley, New York)
Tomsovic, S., Heller, E.J. (1993): Semiclassical construction of chaotic eigenstates. Phys. Rev. Lett. 70, 1405–1408
Voros, A. (1976): Semi-classical approximations. Ann. Inst. Henri Poincaré A XXIV, 31–90
Wigner, E.P. (1932): On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749–759
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© 1999 Springer-Verlag
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Soares, M., Vallée, O., de Izarra, C. (1999). A study of the analytical and local semiclassical Wigner distribution. In: Leach, P.G.L., Bouquet, S.E., Rouet, JL., Fijalkow, E. (eds) Dynamical Systems, Plasmas and Gravitation. Lecture Notes in Physics, vol 518. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0105942
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DOI: https://doi.org/10.1007/BFb0105942
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