Finite systems on phase space

International Journal of Modern Physics B

Volumn 20, Issue 11-13, 2006, Pages 1956-1967

  Wolf, Kurt Bernardo ^a  

^a INSTITUTO DE CIENCIAS FÍSICAS   (Mexico)

Author keywords

Difference equations; Finite signal analysis; Fractional Fourier transforms; Hamiltonian systems; Harmonic oscillators

Indexed keywords

EID: 33744540016     PISSN: 02179792     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0217979206034431     Document Type: Conference Paper

References (63)

1. K.B. Wolf, Integral Transforms in Science and Engineering (Plenum Publ. Corp., New York, 1979), Chapter 2.
2. J.H. McClellan and T.W. Parks, Eigenvalue and eigenvector decomposition of the discrete Fourier transform, IEEE Trans.AU 2066-741972;
3. T.S. Santhanam and A.R. Tekumalla, Quantum mechanics in finite dimensions, Found. Phys.6583-5891976;
4. F.A. Grünbaum, The eigenvectors of the discrete Fourier transform: A version of the Hermite functions, J. Math. Anal. Appl.88355-3631982;
5. M.L. Mehta, Eigenvalues and eigenvectors of the finite Fourier transform, J. Math. Phys.28781-7851987;
6. C. Belingeri and P.E. Ricci, A generalization of the discrete Fourier transform, Int. Transf. Spec. Funct.3165-1741995;
7. M, J. Phys. A302509-25171997;
8. S.-C. Pei and M.-H. Yeh, Improved discrete fractional transform, Opt. Lett.221047-10491997;
9. T.S. Santhanam, Finite-space quantum mechanics and Krawtchuk functions, in: Proceedings of the Workshop on Special Functions and Differential Equations, Ed. by K. Srinivasa Rao, R. Jagannathan, J. Van der Jeugt, and G. Van der Berghe (Allied, Publ., Delhi, 1998);
10. M.S. Richman, T.W. Parks and R.G. Shenoy, Discrete-time, discrete-frequency analysis, IEEE Trans. Signal Processing461517-15271998;
11. V.B. Matveev, Intertwining relations between the Fourier transform and discrete Frourier transform, the related functional identities and beyond, Inverse Problems17633-6582001.
12. H.M. Ozaktas, Z. Zalevsky, and M. Alper Kutay, The Fractional Fourier Transform (John Wiley, Chichester, 2000);
13. L. Barker, Ç. Çandan, T. Hakioǧlu, A. Kutay, and H.M. Ozaktas, The discrete harmonic oscillator, Harper's equation, and the discrete fractional Fourier transform, J. Phys. A332209-22222000.
14. T. Hakioǧlu, Finite-dimensional Schwinger basis, deformed symmetries, Wigner function, and an algebraic approach to quantum phase, J. Phys. A316975-69941998;
15. T. Hakioǧlu, Linear canonical transforms and quantum phase: a unified canonical and algebraic approachJ. Phys. A324111-41301999;
16. T. Hakioǧlu and K.B. Wolf, The canonical Kravchuk basis for discrete quantum mechanics, J. Phys. A333313-33232000.
17. T. Opatrný, Mean value and uncertainty of optical phase - a simple mechanical analogy, J. Phys. A277201-72081994;
18. T. Opatrný, Number-phase uncertainty relations, J. Phys. A286961-69751995;
19. G.W. Forbes and M.A. Alonso, Measures of spread for periodic distributions and the associated uncertainty relations Am. J. Phys.69340-3472001.
20. C. Aragone, G. Guerri, S. Salamó and J.L. Trani, Intelligent spin states, J. Phys. A7L149-L1511974;
21. C. Brif, SU(2) and SU(1,1) algebra eigenstates: a unified analytic approach to coherent and intelligent states, Int. J. Theor. Phys.361651- 16821997;
22. C. Brif and A. Mann, Phase space formulation of quantum mechanics and quantum state reconstruction for physical systems with Lie algebras, Phys. Rev. A594264-42741999;
23. S.D. Bartlett, H. de Guise, and B.C. Sanders, Quantum encodings in spin systems and harmonic oscillators, Phys. Rev. A650523162002;
24. A.B. Klimov, L.L. Sánchez-Soto, and H. de Guise, Multicomplementary operators via finite Fourier transform, J. Phys. A374097-41062005.
25. S. Chountasis, A. Vourdas, and C. Bendjaballah, Fractional Fourier operators and generalized Wigner functions, Phys. Rev. A603467-34731999;
26. A. Vourdas, Quantum systems with finite Hilbert space and Chebyshev polynomials, J. Comp. Appl. Math.133657-6642001;
27. A. Vourdas, Coding with finite quantum systems, Phys. Rev. A650423212002;
28. S. Zhang and A. Vourdas, Phase space methods for particles on a circle, J. Math. Phys.445084-50942003;
29. S. Zhang and A. Vourdas, Analytic representation of finite quantum systems, J. Phys. A378349-83632004;
30. A. Vourdas, Quantum systems with finite Hilbert space, Rep. Prog. Phys.67267-3452004.
31. N.M. Atakishiyev, G.S. Pogosyan, and K.B. Wolf, Contraction of the finite one-dimensional oscillator, Int. J. Mod. Phys. A18317-3272003;
32. N.M. Atakishiyev, G.S. Pogosyan, and K.B. Wolf, Contraction of the finite radial oscillator, Int. J. Mod. Phys. A18329-3412003.
33. N.M. Atakishiyev and S.K. Suslov, Difference analogs of the harmonic oscillator, Theor. Math. Phys.851055-10621991;
34. R. Askey, N.M. Atakishiyev, and S.K. Suslov, Fourier transformations for difference analogs of the harmonic oscillator. In: Proceedings of the XV Workshop on High Energy Physics and Field Theory (Protvino, Russia, 6-10 July, 1992), Ed. by A.P. Samokhin and G.L. Rcheulishvili (IHEP, Serpukhov, 1995), pp. 140-144.
35. N.M. Atakishiyev and K.B. Wolf, Fractional Fourier-Kravchuk transform, J. Opt. Soc. Am. A141467-14771997;
36. N.M. Atakishiyev, L.E. Vicent and K.B. Wolf, Continuous vs. discrete fractional Fourier transforms. J. Comp. Appl. Math. 10773-951999.
37. N.M. Atakishiyev, G.S. Pogosyan, L.E. Vicent, and K.B. Wolf, Finite two-dimensionsal oscillator. I: The Cartesian model. J. Phys. A349381-93982001;
38. N.M. Atakishiyev, G.S. Pogosyan, L.E. Vicent and K.B. Wolf, Finite two-dimensionsal oscillator. II: The radial model, J. Phys. A349399-94152001.
39. E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev.40749-7591932;
40. M. Hillery, R.F. O'Connell, M.O. Scully, and E.P. Wigner, Distribution functions in physics: Fundamentals, Phys. Rep. 106121-1671984;
41. H.-W. Lee, Theory and application of the quantum phase-space distribution functions, Phys. Rep.259147-2111995.
42. N.M. Atakishiyev, S.M. Chumakov and K.B. Wolf, Wigner distribution function for finite systems, J. Math. Phys.396247-62611998,
43. S.T. Ali, N.M. Atakishiyev, S.M. Chumakov and K.B. Wolf, The Wigner function for general Lie groups and the wavelet transform, Ann. Henri Poincaré1685-7142000.
44. N.M. Atakishiyev, G.S. Pogosyan and K.B. Wolf, Finite models of the oscillator, Phys. Part. Nuclei, Suppl. 336521-5552005.
45. V. Namias, The fractional order Fourier transform and its applications to quantum mechanics, J. Inst. Maths. Applics.25241-2651980;
46. V. Namias, Fractionalization of Hankel transforms, J. Inst. Maths. Applics.26187-1971980.
47. K.B. Wolf, The Heisenberg-Weyl group in quantum mechanics. In: Group Theory and its Applications, Vol. III, Ed. by E.M. Loebl (Academic Press, New York, 1975).
48. E.P. Wigner, Do the equations of motion determine the quantum mechanical commutation relations? Phys. Rev.77711-7121950.
49. L.C. Biedenharn and J.D. Louck, Angular Momentum in Quantum Physics, Encyclopedia of Mathematics and Its Applications, Ed. by G.-C. Rota (Addison-Wesley Publ. Co., Reading, Mass., 1981).
50. M. Krawtchouk, Sur une generalization des polinômes d'Hermite, C.R. Acad. Sci. Paris189620-6221929;
51. A.F. Nikiforov, S.K. Suslov, V.B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable, Springer Seris in Computational Physics (Springer-Verlag, Heidelberg, 1991), Eqs. (5.2.13) and (5.2.14);
52. N.M. Atakishiyev and K.B. Wolf, Approximation on a finite set of points through Kravchuk functions, Rev. Mex. Fís.40366-3771994;
53. R. Koekoek and R.F. Swarttow, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue; Report No. 98-17 (Delft University of Technology, Delft, 1998), p. 34.
54. R. Gilmore, Lie Groups, Lie Algebras, and some of Their Applications (John Wiley & Sons, New York, 1974).
55. S.M. Chumakov, A.B. Klimov and K.B. Wolf, On the connection of two Wigner functions for spin systems, Phys. Rev. A61034101(3)2000;
56. cf: J.-P. Amiet and S. Weigert, Contracting the Wigner-kernel of a spin to the Wigner-kernel of a particle, Phys. Rev. A630121022001.
57. K.B. Wolf, Wigner distribution function for paraxial polychromatic optics, Opt. Commun. 132343-3521996.
58. L.M. Nieto, N.M. Atakishiyev, S.M. Chumakov, and K.B. Wolf, Wigner distribution function for Euclidean systems. , J. Phys. A313875-38951998.
59. N.M. Atakishiyev, E.I. Jafarov, Sh.M. Nagiyev and K.B. Wolf, Meixner oscillators, Rev. Mex. Fís.44235-2441998;
60. N.M. Atakishiyev, Sh.M. Nagiyev, L.E. Vicent, and K.B. Wolf, Covariant discretization of axis-symmetric linear optical systems, J. Opt. Soc. Am. A172301-23142000.
61. J.D. Talman, Special Functions - A Group Theoretic Approach (W.A. Benjamin, Inc., New York, 1968); Chapter 13.
62. S.M. Chumakov, A. Frank and K.B. Wolf, Finite Kerr medium: Macroscopic quantum superposition states and Wigner functions on the sphere. Phys. Rev. A601817-18231999.
63. N.M. Atakishiyev, S.M. Chumakov, A.L. Rivera, and K.B. Wolf, On the phase space description of quantum nonlinear dynamics, Phys. Lett. A215128-1341996.