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Instanton versus traditional WKB approach to the Landau-Zener problem

  • V. A. Benderskii
  • E. V. Vetoshkin
  • E. I. Kats
Atoms, Spectra, Radiation

Abstract

Different theoretical approaches to the famous two-state Landau-Zener problem are briefly discussed. Apart from traditional methods of the adiabatic perturbation theory, the Born-Oppenheimer approximation with geometric phase effects, the two-level approach, and the momentum space representation, the problem is treated semiclassically in the coordinate space. In the framework of the instanton approach, we present a full and unified description of the 1D Landau-Zener problem of level crossing. The method enables us to accurately treat all four transition points (appearing at two-level crossing), while the standard WKB approach takes only two of them into account. The latter approximation is adequate for calculating the transition probability or for studying scattering processes, but it does not work in finding the corresponding chemical reaction rates, in which all four transition points can often be relevant in the typical range of parameters. Applications of the method and of the results may concern various systems in physics, chemistry, and biology.

Keywords

Perturbation Theory Transition Point Momentum Space Space Representation Coordinate Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed. (Nauka, Moscow, 1989; Pergamon, New York, 1977).Google Scholar
  2. 2.
    E. E. Nikitin, Opt. Spektrosk. 13, 761 (1962); Discuss. Faraday Soc. 33, 14 (1962).Google Scholar
  3. 3.
    Yu. N. Demkov, Zh. Éksp. Teor. Fiz. 45, 195 (1963) [Sov. Phys. JETP 18, 138 (1964)].Google Scholar
  4. 4.
    G. Hertzberg and H. C. Longuet-Higgins, Discuss. Faraday Soc. 35, 77 (1963).Google Scholar
  5. 5.
    E. E. Nikitin, Chem. Phys. Lett. 2, 402 (1968).CrossRefADSGoogle Scholar
  6. 6.
    M. S. Child, Adv. At. Mol. Phys. 14, 225 (1978).Google Scholar
  7. 7.
    H.-W. Lee and T. F. George, Phys. Rev. A 29, 2509 (1984).ADSGoogle Scholar
  8. 8.
    S. Griller and C. Gonera, Phys. Rev. A 63, 052101 (2001).Google Scholar
  9. 9.
    A. M. Dykhne, Zh. Éksp. Teor. Fiz. 38, 570 (1960) [Sov. Phys. JETP 11, 411 (1960)].zbMATHGoogle Scholar
  10. 10.
    A. M. Dykhne, Zh. Éksp. Teor. Fiz. 41, 1324 (1961) [Sov. Phys. JETP 14, 941 (dy1961)].Google Scholar
  11. 11.
    A. M. Dykhne and A. V. Chaplik, Zh. Éksp. Teor. Fiz. 43, 889 (1962) [Sov. Phys. JETP 16, 631 (dy1962)].Google Scholar
  12. 12.
    G. V. Dubrovskii, Zh. Éksp. Teor. Fiz. 46, 863 (1964) [Sov. Phys. JETP 19, 591 (dy1964)].Google Scholar
  13. 13.
    L. P. Kotova, Zh. Éksp. Teor. Fiz. 55, 1375 (1968) [Sov. Phys. JETP 28, 719 (dy1968)].Google Scholar
  14. 14.
    P. Pechukas, T. F. George, and K. Morokuma, J. Chem. Phys. 64, 1099 (1976).ADSGoogle Scholar
  15. 15.
    J. P. Davis and P. Pechukas, J. Chem. Phys. 64, 3129 (1976).CrossRefADSGoogle Scholar
  16. 16.
    J.-T. Hwang and P. Pechukas, J. Chem. Phys. 67, 4640 (1977).ADSMathSciNetGoogle Scholar
  17. 17.
    B. M. Garraway and S. Stenholm, Phys. Rev. A 45, 364 (1992).CrossRefADSGoogle Scholar
  18. 18.
    K.-A. Suominen and B. M. Garraway, Phys. Rev. A 45, 374 (1992).ADSGoogle Scholar
  19. 19.
    N. V. Vitanov and B. M. Garraway, Phys. Rev. A 53, 4288 (1996).CrossRefADSGoogle Scholar
  20. 20.
    N. V. Vitanov, Phys. Rev. A 59, 988 (1999).ADSGoogle Scholar
  21. 21.
    N. V. Vitanov and K.-A. Suominen, Phys. Rev. A 59, 4580 (1999).ADSGoogle Scholar
  22. 22.
    J. B. Delos, W. R. Thorson, and S. K. Knudson, Phys. Rev. A 6, 709 (1972).ADSGoogle Scholar
  23. 23.
    C. Zhu, H. Nakamura, N. Re, and V. Aquilanti, J. Chem. Phys. 97, 1892 (1992); C. Zhu and H. Nakamura, J. Chem. Phys. 97, 8497 (1992).ADSMathSciNetGoogle Scholar
  24. 24.
    C. Zhu and H. Nakamura, J. Chem. Phys. 98, 6208 (1993).ADSGoogle Scholar
  25. 25.
    C. Zhu and H. Nakamura, J. Chem. Phys. 101, 4855 (1994); J. Chem. Phys. 101, 10630 (1994).ADSGoogle Scholar
  26. 26.
    V. A. Benderskii, D. E. Makarov, and C. A. Wight, Chemical Dynamics at Low Temperatures (Wiley, New York, 1994).Google Scholar
  27. 27.
    V. A. Benderskii, E. V. Vetoshkin, and H. P. Trommsdorf, Chem. Phys. 244, 273 (1999).Google Scholar
  28. 28.
    V. A. Benderskii and E. V. Vetoshkin, Chem. Phys. 257, 203 (2000).CrossRefGoogle Scholar
  29. 29.
    V. A. Benderskii, E. V. Vetoshkin, and E. I. Kats, Zh. Éksp. Teor. Fiz. 122, 746 (2002) [JETP 95, 645 (2002)].Google Scholar
  30. 30.
    V. L. Pokrovskii and I. M. Khalatnikov, Zh. Éksp. Teor. Fiz. 40, 1713 (1961) [Sov. Phys. JETP 13, 1207 (1961)].Google Scholar
  31. 31.
    B. K. Bykhovskii, E. E. Nikitin, and M. Ya. Ovchinnikova, Zh. Éksp. Teor. Fiz. 47, 750 (1965) [Sov. Phys. JETP 20, 500 (1964)].Google Scholar
  32. 32.
    V. M. Akulin and W. P. Schleich, Phys. Rev. A 46, 4110 (1992).CrossRefADSGoogle Scholar
  33. 33.
    E. E. Nikitin and S. Ya. Umanskii, Theory of Slow Atomic Collisions (Atomizdat, Moscow, 1979; Springer, Berlin, 1984), Springer Ser. Chem. Phys., Vol. 30.Google Scholar
  34. 34.
    W. H. Miller and T. F. George, J. Chem. Phys. 56, 5637 (1972).Google Scholar
  35. 35.
    R. K. Preston, C. Sloane, and W. H. Miller, J. Chem. Phys. 60, 4961 (1974).CrossRefGoogle Scholar
  36. 36.
    F. J. McLafferty and T. G. George, J. Chem. Phys. 63, 2609 (1975).CrossRefADSGoogle Scholar
  37. 37.
    J. C. Slater, Quantum Theory of Molecules and Solids, Vol. 1: Electronic Structure of Molecules (McGraw-Hill, New York, 1963).Google Scholar
  38. 38.
    Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959).CrossRefADSMathSciNetGoogle Scholar
  39. 39.
    C. A. Mead, J. Chem. Phys. 78, 807 (1983).ADSGoogle Scholar
  40. 40.
    M. V. Berry, Proc. R. Soc. London, Ser. A 392, 451 (1984).Google Scholar
  41. 41.
    M. Wilkinson, J. Phys. A 17, 3459 (1984).CrossRefADSMathSciNetGoogle Scholar
  42. 42.
    M. V. Berry, J. Phys. A 18, 15 (1985).CrossRefADSzbMATHMathSciNetGoogle Scholar
  43. 43.
    H. Kuratsui and S. Ida, Prog. Theor. Phys. 74, 439 (1985).ADSGoogle Scholar
  44. 44.
    M. V. Berry, J. Mod. Opt. 34, 1401 (1987).ADSzbMATHMathSciNetGoogle Scholar
  45. 45.
    J. Vidal and J. Wudka, Phys. Rev. A 44, 5383 (1991).CrossRefADSMathSciNetGoogle Scholar
  46. 46.
    A. Mustafazadeh, Phys. Rev. A 55, 1653 (1997).ADSGoogle Scholar
  47. 47.
    R. F. Fox and P. Jung, Phys. Rev. A 57, 2339 (1998).CrossRefADSMathSciNetGoogle Scholar
  48. 48.
    C. A. Mead, Rev. Mod. Phys. 64, 51 (1992).CrossRefADSMathSciNetGoogle Scholar
  49. 49.
    Y. Aharonov and J. Anandan, Phys. Rev. Lett. 58, 1593 (1987).ADSMathSciNetGoogle Scholar
  50. 50.
    J. Samuel and R. Bhachdari, Phys. Rev. Lett. 60, 2339 (1988).CrossRefADSMathSciNetGoogle Scholar
  51. 51.
    N. Mukunda and R. Simon, Ann. Phys. (N.Y.) 228, 20 (1993).Google Scholar
  52. 52.
    M. V. Berry and J. M. Robbins, Proc. R. Soc. London, Ser. A 442, 659 (1993).ADSMathSciNetGoogle Scholar
  53. 53.
    J. Moody, A. Shapere, and F. Wilczek, Phys. Rev. Lett. 56, 893 (1986).CrossRefADSGoogle Scholar
  54. 54.
    D. Suter, G. C. Chingas, R. A. Harris, and A. Pines, Mol. Phys. 61, 1327 (1987).Google Scholar
  55. 55.
    F. Gaitan, Phys. Rev. A 58, 1665 (1998).CrossRefADSMathSciNetGoogle Scholar
  56. 56.
    M. Baer, S. H. Lin, A. Alijah, et al., Phys. Rev. A 62, 032506 (2000).Google Scholar
  57. 57.
    J. Heading, An Introduction to Phase-Integral Methods (Wiley-Interscience, London, 1962).Google Scholar
  58. 58.
    A. M. Polyakov, Nucl. Phys. B 129, 429 (1977).ADSMathSciNetGoogle Scholar
  59. 59.
    S. Coleman, Aspects of Symmetry ((Cambridge Univ. Press, Cambridge, 1985).Google Scholar
  60. 60.
    M. S. Child, Mol. Phys. 20, 171 (1971).Google Scholar
  61. 61.
    M. S. Child, J. Mol. Spectrosc. 53, 280 (1974).CrossRefADSGoogle Scholar
  62. 62.
    T. Holstein, Philos. Mag. 37, 49 (1978).Google Scholar
  63. 63.
    G. Stick and M. Thoss, Phys. Rev. Lett. 78, 578 (1997).ADSMathSciNetGoogle Scholar
  64. 64.
    V. A. Benderskii and E. I. Kats, Phys. Rev. E 65, 036217 (2002).Google Scholar
  65. 65.
    N. T. Maintra and E. J. Heller, Phys. Rev. A 54, 4763 (1996).ADSGoogle Scholar
  66. 66.
    Y. Kayanuma and H. Nakayama, Phys. Rev. B 57, 13099 (1998).Google Scholar
  67. 67.
    K. Saito and Y. Kayanuma, Phys. Rev. A 65, 033407 (2002).Google Scholar
  68. 68.
    M. V. Fedoryuk, Dokl. Akad. Nauk SSSR 158, 540 (1964).zbMATHMathSciNetGoogle Scholar
  69. 69.
    M. V. Fedoryuk, Dokl. Akad. Nauk SSSR 162, 287 (1965).zbMATHGoogle Scholar
  70. 70.
    M. V. Fedoryuk, Usp. Mat. Nauk 2, 1 (1966).Google Scholar
  71. 71.
    A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Ed. by A. Erdelyi (McGraw-Hill, New York, 1953–1955; Nauka, Moscow, 1965–1967), Vols 1–3.Google Scholar
  72. 72.
    F. W. J. Olver, Asymptotics and Special Functions (Academic, New York, 1974; Nauka, Moscow, 1990).Google Scholar
  73. 73.
    F. W. J. Olver, J. Res. Natl. Bur. Stand., Sect. B 63, 131 (1959).zbMATHMathSciNetGoogle Scholar
  74. 74.
    A. V. Shytov, D. A. Ivanov, and M. V. Feigelman, condmat/0110490.Google Scholar
  75. 75.
    V. A. Yurovsky, A. Ben-Reuven, and P. S. Julienne, Phys. Rev. A 65, 043607 (2002).Google Scholar
  76. 76.
    V. L. Pokrovsky and N. A. Sinitsyn, Phys. Rev. B 65, 153105 (2002).Google Scholar
  77. 77.
    N. A. Sinitsyn, cond-mat/0212017.Google Scholar
  78. 78.
    Y. N. Demkov and V. N. Ostrovsky, Phys. Rev. A 61, 032705 (2000).Google Scholar
  79. 79.
    D. A. Garanin and E. M. Chudnovsky, Phys. Rev. B 65, 094423 (2002).Google Scholar
  80. 80.
    H. Eyring, S. H. Lin, and S. M. Lin, Basic Chemical Kinetics (Wiley, New York, 1980; Mir, Moscow, 1983).Google Scholar
  81. 81.
    Yu. A. Bychkov and A. M. Dykhne, Zh. Éksp. Teor. Fiz. 48, 1174 (1965) [Sov. Phys. JETP 21, 783 (1965)].Google Scholar
  82. 82.
    S. Toshev, Phys. Lett. B 198, 551 (1987).ADSGoogle Scholar
  83. 83.
    J. M. Juan and T. F. George, J. Chem. Phys. 68, 3040 (1978).ADSGoogle Scholar
  84. 84.
    A. D. Bandrauk and G. Turcott, J. Chem. Phys. 77, 3867 (1982).CrossRefADSGoogle Scholar
  85. 85.
    H. W. Lee and T. F. George, Phys. Rev. A 35, 4977 (1987).ADSGoogle Scholar
  86. 86.
    V. M. Akulin and N. V. Karlov, Intense Resonant Interactions in Quantum Electronics (Springer, Berlin, 1992).Google Scholar

Copyright information

© MAIK "Nauka/Interperiodica" 2003

Authors and Affiliations

  • V. A. Benderskii
    • 1
    • 2
  • E. V. Vetoshkin
    • 1
  • E. I. Kats
    • 2
    • 3
  1. 1.Institute for Problems of Chemical PhysicsRussian Academy of SciencesChernogolovka, Moscow oblastRussia
  2. 2.Laue-Langevin InstituteGrenobleFrance
  3. 3.Landau Institute for Theoretical PhysicsRussian Academy of SciencesChernogolovka, Moscow oblastRussia

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