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Time-Dependent Quantum Theory

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Part of the Graduate Texts in Physics book series (GTP)

Abstract

In this chapter, we continue laying the foundations for the later chapters by reviewing some basic properties of the time-dependent Schrödinger equation and the corresponding time-evolution operator, respectively its position space matrix element, the propagator. After the discussion of two analytically solvable cases, we will consider general ways to rewrite, respectively solve the time-dependent Schrödinger equation. Formulating the solution with the help of the Feynman path integral will allow us to consider the approximate time-dependent semiclassical approach, using classical trajectory information for the construction of the propagator. The last part of this chapter is then dealing with different numerical techniques for the full quantum as well as the semiclassical solution of the time-dependent Schrödinger equation, that will be referred to in later chapters.

Keywords

Coherent State Semiclassical Approximation Interaction Picture Symplectic Integration Classical Path 
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.

References

  1. 1.
    E. Schrödinger, Ann. Phys. (Leipz.) 81, 109 (1926) zbMATHCrossRefGoogle Scholar
  2. 2.
    E. Schrödinger, Ann. Phys. (Leipz.) 79, 489 (1926) zbMATHCrossRefGoogle Scholar
  3. 3.
    J.S. Briggs, J.M. Rost, Eur. Phys. J. D 10, 311 (2000) ADSCrossRefGoogle Scholar
  4. 4.
    V.A. Mandelshtam, J. Chem. Phys. 108, 9999 (1998) ADSCrossRefGoogle Scholar
  5. 5.
    R.P. Feynman, A.R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965). For errata see, e.g. http://www.oberlin.edu/physics/dstyer/FeynmanHibbs/ zbMATHGoogle Scholar
  6. 6.
    E. Schrödinger, Naturwissenschaften 14, 664 (1926) ADSzbMATHCrossRefGoogle Scholar
  7. 7.
    E.J. Heller, J. Chem. Phys. 62, 1544 (1975) ADSCrossRefGoogle Scholar
  8. 8.
    W. Kinzel, Phys. Bl. 51, 1190 (1995) CrossRefGoogle Scholar
  9. 9.
    F. Grossmann, J.M. Rost, W.P. Schleich, J. Phys. A 30, L277 (1997) MathSciNetADSzbMATHCrossRefGoogle Scholar
  10. 10.
    R.P. Feynman, Rev. Mod. Phys. 20, 367 (1948) MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    P.A.M. Dirac, The Principles of Quantum Mechanics, 4th edn. (Oxford, London, 1958) zbMATHGoogle Scholar
  12. 12.
    L.S. Schulman, Techniques and Applications of Path Integration (Dover, Mineola, 2005), pp. 22–26 zbMATHGoogle Scholar
  13. 13.
    J.H. van Vleck, Proc. Natl. Acad. Sci. USA 14, 178 (1928) ADSzbMATHCrossRefGoogle Scholar
  14. 14.
    M.C. Gutzwiller, J. Math. Phys. 8, 1979 (1967) ADSCrossRefGoogle Scholar
  15. 15.
    S. Grossmann, Funktionalanalysis II (Akademie Verlag, Wiesbaden, 1977) zbMATHGoogle Scholar
  16. 16.
    D.J. Tannor, Introduction to Quantum Mechanics: A Time-Dependent Perspective (University Science Books, Sausalito, 2007). And Refs. therein Google Scholar
  17. 17.
    W.R. Salzman, J. Chem. Phys. 85, 4605 (1986) ADSCrossRefGoogle Scholar
  18. 18.
    M.H. Beck, A. Jäckle, G.A. Worth, H.D. Meyer, Phys. Rep. 324, 1 (2000) ADSCrossRefGoogle Scholar
  19. 19.
    D. Kohen, F. Stillinger, J.C. Tully, J. Chem. Phys. 109, 4713 (1998) ADSCrossRefGoogle Scholar
  20. 20.
    M. Tinkham, Group Theory and Quantum Mechanics (McGraw-Hill, New York, 1964) zbMATHGoogle Scholar
  21. 21.
    H. Sambe, Phys. Rev. A 7, 2203 (1973) ADSCrossRefGoogle Scholar
  22. 22.
    M. Kleber, Phys. Rep. 236, 331 (1994) ADSCrossRefGoogle Scholar
  23. 23.
    M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965) Google Scholar
  24. 24.
    W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in Fortran, 2nd edn. (Cambridge University Press, Cambridge, 1992) zbMATHGoogle Scholar
  25. 25.
    J.H. Shirley, Phys. Rev. 138, B979 (1965) ADSCrossRefGoogle Scholar
  26. 26.
    J.C. Light, in Time-Dependent Quantum Molecular Dynamics, ed. by J. Broeckhove, L. Lathouwers (Plenum, New York, 1992), p. 185 CrossRefGoogle Scholar
  27. 27.
    U. Peskin, N. Moiseyev, Phys. Rev. A 49, 3712 (1994) ADSCrossRefGoogle Scholar
  28. 28.
    W. Witschel, J. Phys. A 8, 143 (1975) MathSciNetADSCrossRefGoogle Scholar
  29. 29.
    J.A. Fleck, J.R. Morris, M.D. Feit, Appl. Phys. 10, 129 (1976) ADSCrossRefGoogle Scholar
  30. 30.
    M.D. Feit, J.A. Fleck, A. Steiger, J. Comput. Phys. 47, 412 (1982) MathSciNetADSzbMATHCrossRefGoogle Scholar
  31. 31.
    M. Braun, C. Meier, V. Engel, Comput. Phys. Commun. 93, 152 (1996) ADSCrossRefGoogle Scholar
  32. 32.
    M. Frigo, S.G. Johnson, Proc. IEEE 93(2), 216 (2005). Special issue on Program Generation, Optimization, and Platform Adaptation CrossRefGoogle Scholar
  33. 33.
    A. Vibok, G.G. Balint-Kurti, J. Phys. Chem. 96, 8712 (1992) CrossRefGoogle Scholar
  34. 34.
    A. Askar, A.S. Cakmak, J. Chem. Phys. 68, 2794 (1978) MathSciNetADSCrossRefGoogle Scholar
  35. 35.
    C. Leforestier, R.H. Bisseling, C. Cerjan, M.D. Feit, R. Friesner, A. Guldberg, A. Hammerich, G. Jolicard, W. Karrlein, H.D. Meyer, N. Lipkin, O. Roncero, R. Kosloff, J. Comput. Phys. 94, 59 (1991) MathSciNetADSzbMATHCrossRefGoogle Scholar
  36. 36.
    H. Tal-Ezer, R. Kosloff, J. Chem. Phys. 81, 3967 (1984) ADSCrossRefGoogle Scholar
  37. 37.
    S.K. Gray, D.W. Noid, B.G. Sumpter, J. Chem. Phys. 101, 4062 (1994) ADSCrossRefGoogle Scholar
  38. 38.
    M.L. Brewer, J.S. Hulme, D.E. Manolopoulos, J. Chem. Phys. 106, 4832 (1997) ADSCrossRefGoogle Scholar
  39. 39.
    H. Yoshida, Phys. Lett. A 150, 262 (1990) MathSciNetADSCrossRefGoogle Scholar
  40. 40.
    W.H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1990) zbMATHGoogle Scholar
  41. 41.
    E.J. Heller, in Chaos and Quantum Physics, ed. by M.J. Giannoni, A. Voros, J. Zinn-Justin. Les Houches Session LII (Elsevier, Amsterdam, 1991), pp. 549–661 Google Scholar
  42. 42.
    J.R. Klauder, in Random Media, ed. by G. Papanicolauou (Springer, New York, 1987), p. 163 CrossRefGoogle Scholar
  43. 43.
    F. Grossmann, J.A.L. Xavier, Phys. Lett. A 243, 243 (1998) MathSciNetADSzbMATHCrossRefGoogle Scholar
  44. 44.
    M.F. Herman, E. Kluk, Chem. Phys. 91, 27 (1984) CrossRefGoogle Scholar
  45. 45.
    E.J. Heller, J. Chem. Phys. 75, 2923 (1981) MathSciNetADSCrossRefGoogle Scholar
  46. 46.
    K.G. Kay, Chem. Phys. 322, 3 (2006) ADSCrossRefGoogle Scholar
  47. 47.
    K.G. Kay, J. Chem. Phys. 100, 4377 (1994) ADSCrossRefGoogle Scholar
  48. 48.
    F. Grossmann, J. Chem. Phys. 125, 014111 (2006) ADSCrossRefGoogle Scholar
  49. 49.
    E. Kluk, M.F. Herman, H.L. Davis, J. Chem. Phys. 84, 326 (1986) ADSCrossRefGoogle Scholar
  50. 50.
    F. Grossmann, M.F. Herman, J. Phys. A, Math. Gen. 35, 9489 (2002) MathSciNetADSzbMATHCrossRefGoogle Scholar
  51. 51.
    W.P. Schleich, Quantum Optics in Phase Space (Wiley-VCH, Berlin, 2001) zbMATHCrossRefGoogle Scholar
  52. 52.
    E.N. Economou, Green’s Functions in Quantum Physics, 3rd edn. (Springer, Berlin, 2006) Google Scholar
  53. 53.
    G.L. Ingold, in Coherent Evolution in Noisy Environments, ed. by A. Buchleitner, K. Hornberger. Lecture Notes in Physics (Springer, Berlin, 2002), pp. 1–53 CrossRefGoogle Scholar
  54. 54.
    L.E. Reichl, The Transition to Chaos, 2nd edn. (Springer, New York, 2004) zbMATHCrossRefGoogle Scholar
  55. 55.
    G.D. Billing, The Quantum Classical Theory (Oxford University Press, New York, 2003) zbMATHGoogle Scholar
  56. 56.
    K.J. Schafer, in Strong Field Laser Physics, ed. by T. Brabec. Springer Series in Optical Sciences, vol. 134 (Springer, Berlin, 2009), Chap. 6, pp. 111–145 CrossRefGoogle Scholar
  57. 57.
    N. Takemoto, A. Shimshovitz, D.J. Tannor, J. Chem. Phys. 137, 011102 (2012) ADSCrossRefGoogle Scholar
  58. 58.
    F. Grossmann, Comments At. Mol. Phys. 34, 141 (1999) Google Scholar
  59. 59.
    D.V. Shalashilin, M.S. Child, J. Chem. Phys. 113, 10028 (2000) ADSCrossRefGoogle Scholar
  60. 60.
    W.H. Miller, J. Phys. Chem. B 106, 8132 (2002) CrossRefGoogle Scholar
  61. 61.
    M. Mizrahi, J. Math. Phys. 16, 2201 (1975) MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikTechnische Universität DresdenDresdenGermany

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