Semiclassical Husimi Function of Simple and Chaotic Systems


We review the semiclassical method proposed in [1], a generalization of this method for n-dimensional system is presented. Using the cited method, we present an analytical method of obtain the semiclassical Husimi Function. The validity of the method is tested using Harmonic Oscillator, Morse Potential and Dikie’s Model as example, we found a good accuracy in the classical limit.

Share and Cite:

A. Oliveira, "Semiclassical Husimi Function of Simple and Chaotic Systems," Journal of Modern Physics, Vol. 3 No. 8, 2012, pp. 694-701. doi: 10.4236/jmp.2012.38094.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] A. C. Oliveira and M. C. Nemes and K. M. F. Romero, “Quantum Time Scales and the Classical Limit: Analytic Results for Some Simple Systems,” Physical Review E, Vol. 68, No. 3, 2003, Article ID: 036214. doi:10.1103/PhysRevE.68.036214
[2] A. Einstein, “Zum Quantensatz von Sommerfeld und Epstein on the Quantum Theory of Sommerfeld and Ep- stein,” Deutsche Physikalische Gesellschaft Verhandlungen, Vol. 19, 1917, p. 82.
[3] M. A. M. de Aguiar, “Einstein and the Quantum Chaos Theory,” Revista de Ensino de Física, Vol. 27, 2005, p. 101.
[4] M. C. Gutzwiller, “Chaos in Classical and Quantum Me- chanics,” Spring-Verlg, New York, 1990.
[5] H. J. StockmannHaake, “Quantum Chaos an Introduc- tion,” Cambridge University Press, New York, 1999.
[6] F. Haake, “Quantum Signatures of Chaos,” Springer- Verlag, Berlin, 2004.
[7] S. W. McDonald and A. N. Kaufmann, “Spectrum and Eigenfunctions for a Hamiltonian with Stochastic Trajec- tories,” Physical Review Letters, Vol. 42, No. 18, 1979, pp. 1189-1191. doi:10.1103/PhysRevLett.42.1189
[8] E. B. Bogomolny, “Fine Structure of the Wave Functions of Quantum Systems,” JETP Letters, Vol. 44, No. 9, 1986, pp. 561-565.
[9] E. B. Bogomolny, “Smoothed Wave Functions of Chaotic Quantum Systems,” Physica D, Vol. 31, No. 2, 1988, pp. 169-189. doi:10.1016/0167-2789(88)90075-9
[10] E. J. Heller, “Bound-State Eigenfunctions of Classically Chaotic Hamiltonian Systems: Scars of Periodic Orbits,” Physical Review Letters, Vol. 53, No. 16, 1984, pp. 1515- 1518. doi:10.1103/PhysRevLett.53.1515
[11] E. J. Heller, “Quantum Chaos and Statistical Nuclear Physics,” Springer, Berlin, 1983.
[12] L. Benet and T. H. Seligman and H. A. Weidenmuller, “Quantum Signatures of Classical Chaos: Sensitivity of Wave Functions to Perturbations,” Physical Review Let- ters, Vol. 71, No. 4, 1993, pp. 529-532. doi:10.1103/PhysRevLett.71.529
[13] M. Srednicki and F. Stiernelof, “Gaussian Fluctuations in Chaotic Eigenstates,” Journal of Physics A, Vol. 29, No. 18, 1996, p. 5817. doi:10.1088/0305-4470/29/18/013
[14] L. Benet and F. M. Izrailev and T. H. Seligman and A. Suarez-Moreno, “Semiclassical Properties of Eigenfunc- tions and Occupation Number Distribution for a Model of Two Interacting Particles,” Physics Letters A, Vol. 277, No. 2, 2000, pp. 87-93. doi:10.1016/S0375-9601(00)00692-7
[15] L. Benet, et al., “Fluctuations of Wavefunctions about Their Classical Average,” Journal of Physics A, Vol. 36, No. 5, 2003, p. 1289. doi:10.1088/0305-4470/36/5/307
[16] P. Bellomo and T. Uzer, “Quantum Scars and Classical Ghosts,” Physical Review A, Vol. 51, No. 2, 1995, pp. 1669-1672. doi:10.1103/PhysRevA.51.1669
[17] W. Zhang, H. Feng and R. Gilmore, “Coherent States: Theory and Some Applications,” Reviews of Modern Physics, Vol. 62, No. 4, 1990, pp. 867-927. doi:10.1103/RevModPhys.62.867
[18] M. Reis and M. C. Nemes and J. G. P. de Faria, “Semi- classical Corrections to the Large-N Limit of Dicke’s Model,” Physical Review E, Vol. 78, No. 3, 2008, Article ID: 036220. doi:10.1103/PhysRevE.78.036220
[19] W. P Scleichr, “Quantum Optics in Pahse Space,” Wiley- VCC, Berlin, 2001.
[20] A. C. Oliveira and M. C. Nemes, “Classical Structures in the Husimi Distributions of Stationary States for H2 and HCl Molecules in the Morse Potential,” Physica Scripta, Vol. 64, No. 4, 2001, p. 279. doi:10.1238/Physica.Regular.064a00279
[21] M. C. Nemes, K. Furuya, G. Q. Pelegrino, A. C. Oliveira, M. Reis and L. SanzM, “Quantum Entanglement and Fixed Point Hopf Bifurcation,” Physics Letters A, Vol. 354, No. 1-2, 2006, pp. 60-66. doi:10.1016/j.physleta.2006.01.028
[22] K. M. F. Romero, M. C. Nemes, J. G. P. de Faria and A. F. R. de T. Piza, “Sensitivity to Initial Conditions in Quan- tum Dynamics: An Analytical Semiclassical Expansion,” Physics Letters A, Vol. 327, No. 2-3, 2004, pp. 129-137. doi:10.1016/j.physleta.2004.04.070
[23] A. C. Oliveira and J. G. P. de Faria and M. C. Nemes, “Quantum-Classical Transition of the Open Quartic Os- cillator: The Role of the Environment,” Physical Review E, Vol. 73, No. 4, 2006, Article ID: 046207. doi:10.1103/PhysRevE.73.046207

Copyright © 2022 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.