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Basics of the Operator Method

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Non-perturbative Description of Quantum Systems

Part of the book series: Lecture Notes in Physics ((LNP,volume 894))

Abstract

The majority of approximate methods for the solution of Schrödinger equation (SE) demonstrated in Chap. 1 is not sufficiently universal and theirs applications in the case of system with many degrees of freedom are bound up with serious difficulties. In the following chapters we consider the method which is proposed as an universal procedure for transformation of the perturbation series and its further applications for various quantum systems. We call this technique operator method (OM) in the sense that all calculations are reduced to the algebraic manipulations with the operator matrix elements without solving any differential or integral equations, in zeroth-order approximation as well as in calculating the successive approximations. The results prove that the OM zeroth-order approximation provides an uniformly fitted estimation of the SE eigenvalues and eigenfunctions in the entire range of the Hamiltonian parameters.

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References

  1. I.D. Feranchuk, L.I. Komarov, Phys. Lett. A, 88, 211 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  2. I.D. Feranchuk, L.I. Komarov, J. Phys. C. Solid State Phys. 15, 1965 (1982)

    Article  ADS  Google Scholar 

  3. I.D. Feranchuk, L.I. Komarov, J. Phys. A: Math. Gen. 17, 3111 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  4. I.G. Holliday, P. Suranyi, Phys. Lett. B 85, 421 (1979)

    Article  ADS  Google Scholar 

  5. W.E. Caswell, Ann. Phys. (N.Y.) 123, 153 (1979)

    Google Scholar 

  6. I.G. Holliday, P. Suranyi, Phys. Rev. D 21, 1529 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  7. I.K. Dmitrieva, G.I. Plindov, Phys. Lett. A 79, 47 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  8. K. Bhattacharyya, J. Phys. B: At. Mol. Phys. 14, 783 (1981)

    Article  ADS  Google Scholar 

  9. J. Austin, J. Killingbeck, J. Phys. A: Math. Gen. 15, L433, (1982)

    Article  MathSciNet  Google Scholar 

  10. C.K. Au, G.W. Roders, J. Aharonov, Phys. Lett. A 95, 287 (1983)

    Article  ADS  Google Scholar 

  11. F.M. Fernandez, E.A. Castro, Phys. Lett. A 88, 4 (1982)

    Article  ADS  Google Scholar 

  12. F.M. Fernandez, E.A. Castro, Phys. Lett. A 91, 339 (1982)

    Article  ADS  Google Scholar 

  13. F.M. Fernandez, A.M. Meson, E.A. Castro, Phys. Lett. A 104, 401 (1984)

    Article  ADS  Google Scholar 

  14. F.M. Fernandez, A.M. Meson, E.A. Castro, Phys. Lett. A 111, 104 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  15. F.M. Fernandez, A.M. Meson, E.A. Castro, Phys. Lett. A 112, 107 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  16. F.M. Fernandez, A.M. Meson, E.A. Castro, Mol. Phys. 59, 365 (1986)

    Article  ADS  Google Scholar 

  17. C.C. Gerry, S. Silverman, Phys. Lett. A 97, 481 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  18. W. Witschel, Phys. Lett. A 97, 315 (1983)

    Article  ADS  Google Scholar 

  19. H. Mitter, K. Yamazaki, J. Phys. A: Math. Gen. 17, 1215 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  20. B. Bidyut, B. Sikha, R. Raykumar, Fizika 16, 161 (1984)

    Google Scholar 

  21. K. Yamazaki, J. Phys. A: Math. Gen. 17, 345 (1984)

    Article  ADS  Google Scholar 

  22. C.-S. Hsue, J.L. Chern, Phys. Rev. D 29, 643 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  23. B. Rath, J. Math. Phys. 31, 1175 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  24. K.L. Chan, S.K. Lee, K.L. Liu, K. Young, C.K. Au, Phys. Lett. A 162, 227 (1992)

    Article  ADS  Google Scholar 

  25. I.D. Feranchuk, S.I. Fisher, L.I. Komarov, J. Phys. C: Solid State Phys. 17, 4309 (1984)

    Article  ADS  Google Scholar 

  26. C.Z. An, I.D. Feranchuk, Mol. Phys. 64, 589 (1988)

    Article  ADS  Google Scholar 

  27. I.D. Feranchuk, L.X. Hai, Phys. Lett. A 137, 385 (1989)

    Article  ADS  Google Scholar 

  28. L.I. Komarov, T.S. Romanova, J. Phys. B: At. Mol. Phys. 18, 859 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  29. I.D. Feranchuk, S.I. Fisher, L.I. Komarov, J. Phys. C: Solid State Phys. 18, 5083 (1985)

    Article  ADS  Google Scholar 

  30. C.Z. An, I.D. Feranchuk, L.I. Komarov, L.I. Nakhamchik, J. Phys. A: Math. Gen. 19, 1583 (1986)

    Article  ADS  MATH  Google Scholar 

  31. I.D. Feranchuk, L.I. Komarov, I.V. Nechipor, J. Phys. A: Math. Gen. 20, 3849 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  32. C.Z. An, I.D. Feranchuk, L.I. Komarov, Phys. Lett. A 125, 123 (1987)

    Article  ADS  Google Scholar 

  33. I.D. Feranchuk, L.I. Komarov, Phys. Lett. A 106, 109 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  34. I.D. Feranchuk, V.N. T’ok, Chem. Phys. Lett. 150, 78 (1988)

    Google Scholar 

  35. I.D. Feranchuk, V.S. Kuz’min, A.P. Ulyanenkov, Chem. Phys. 157, 61 (1991)

    Google Scholar 

  36. I.D. Feranchuk, A.P. Ulyanenkov, NIM B 88, 369 (1994)

    Article  ADS  Google Scholar 

  37. I.D. Feranchuk, L.I. Komarov, A.P. Ulyanenkov, J. Phys. A: Math. Gen. 29, 4035 (1996)

    Article  ADS  MATH  Google Scholar 

  38. I.D. Feranchuk, L.I. Komarov, A.P. Ulyanenkov, J. Phys. B: At. Mol. Opt. Phys. 35, 3957 (2002)

    Article  ADS  Google Scholar 

  39. I.D. Feranchuk, L.I. Gurskii, L.I. Komarov, O.M. Lugovskaya, F. Burgäzy, A.P. Ulyanenkov, Acta Crystallogr. A 58, 370 (2002)

    Article  Google Scholar 

  40. L.V. Hoang, L.I. Komarov, T.S. Romanova, J. Phys. A: Math. Gen. 22, 1543 (1989)

    Article  ADS  MATH  Google Scholar 

  41. L.I. Gurskii, L.I. Komarov, A.M. Solodukhin, Int. J. Quantum Chem. 72, 499 (1999)

    Article  Google Scholar 

  42. I.D. Feranchuk, A.V. Leonov, Phys. Lett. A 375, 385 (2011)

    Article  ADS  MATH  Google Scholar 

  43. I.D. Feranchuk, V.V. Triguk, Phys. Lett. A 375, 2550 (2011)

    Article  ADS  MATH  Google Scholar 

  44. I.D. Feranchuk, L.I. Komarov, I.V. Nechipor, A.P. Ulyanenkov, Ann. Phys. (N Y) 238, 370 (1995)

    Article  ADS  MATH  Google Scholar 

  45. A. Duncan, H.F. Jones, Nucl. Phys. B 320, 189 (1989)

    Article  ADS  Google Scholar 

  46. V.V. Kudrashov, V.I. Reshetnyak, E.A. Tolkachov, Dokl. Belarusian Acad. Sci. (in Russian) 31, 135 (1987)

    Google Scholar 

  47. F.T. Hioe, D. MacMillen, E.W. Montroll, Phys. Rep. 335, 307 (1978)

    MathSciNet  Google Scholar 

  48. F.M. Fernandez, E.A. Castro, Phys. Lett. A 88, 5083 (1982)

    Article  Google Scholar 

  49. I.I. Gegusin, L.I. Leontieva, J. Phys. C: Condens. Matter 2, 5869 (1990)

    Google Scholar 

  50. S. Srivastava, Vishamitter, Mol. Phys. 72, 1285 (1991)

    Article  ADS  Google Scholar 

  51. F.A. Kaempfer, Concepts in Quantum Mechanics (Academic, New York, 1965)

    Google Scholar 

  52. D.A. Kirzhnitz, Field Methods of Many–Particle Theory (in Russian) (Gosatomizdat, Moscow, 1963)

    Google Scholar 

  53. A.J. Barnes, W.J. Orville-Thomas, (ed.), Vibrational Spectroscopy. Modern Trends (Oxford, Amsterdam, Elsevier, New York, 1977)

    Google Scholar 

  54. C. Hayashi, Nonlinear Oscillations in Physical Systems (McGraw-Hill, New York, 1964)

    MATH  Google Scholar 

  55. I.S. Gradshtein, I.M. Ryzhik, Tables of Integrals, Sums, Series and Products (FIZMATGIZ, Moscow, 1963)

    Google Scholar 

  56. A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1976)

    Google Scholar 

  57. N.H. March, W.H. Young, S. Sampanthar, The Many-Body Problem in Quantum Mechanics (University Press, Cambridge, 1967)

    Google Scholar 

  58. M.L. Goldberger, K.M. Watson, Collision Theory (Wiley, New York, 1964)

    MATH  Google Scholar 

  59. V.G. Baryshevsky, High-Energy Nuclear Optics of Polarised Media (World Scientific, Singapore, 2012)

    Book  Google Scholar 

  60. P.E. Shanley, Phys. Lett. A 117, 161 (1986)

    Article  ADS  Google Scholar 

  61. R.J. Damburg, V.G. Kolosov, J. Phys. B: At. Mol. 9, 3149 (1976)

    Article  ADS  Google Scholar 

  62. H.K. Shepard, Phys. Rev. D 27, 1288 (1983)

    Article  ADS  Google Scholar 

  63. D. Farrelly, W.P. Reinhardt, J. Phys. B: At. Mol. 16, 2103 (1983)

    Article  ADS  Google Scholar 

  64. P.M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953)

    MATH  Google Scholar 

  65. J.E. Drummond, J. Phys. A: Math. Gen. 14, 1651 (1981)

    Article  ADS  Google Scholar 

  66. N.N. Bogoliubov, Physica 26, 1 (1960)

    Article  ADS  Google Scholar 

  67. G. Ross, Grand Unified Theories (Westview Press, USA, 1984)

    Google Scholar 

  68. L.D. Landau, E.M. Lifshitz, Quantum Mechanics (Fizmatgiz, Moscow, 2004)

    Google Scholar 

  69. M.O. Scully, M.S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, 1997)

    Book  Google Scholar 

  70. A. Bohm, Quantum Mechanics: Foundations and Applications (Springer, New York, 1986)

    Book  MATH  Google Scholar 

  71. C.G. Callan, S. Coleman, Phys. Rev. D 16, 1762, (1977)

    Article  ADS  Google Scholar 

  72. F.M. Fernandez, E.A. Castro, A.M. Meson, J. Phys. A: Math. Gen. 7, 1389 (1985)

    Article  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Physics Department, Belarusian State University, Minsk, Belarus

    Ilya Feranchuk

  2. Physics Department, Belarusian National Technical University, Minsk, Belarus

    Alexey Ivanov

  3. Department of Physics, Ho Chi Minh City University of Pedagogy, Ho Chi Minh City, Vietnam

    Van-Hoang Le

  4. Atomicus GmbH, Karlsruhe, Germany

    Alexander Ulyanenkov

Authors
  1. Ilya Feranchuk
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  2. Alexey Ivanov
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  3. Van-Hoang Le
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  4. Alexander Ulyanenkov
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Feranchuk, I., Ivanov, A., Le, VH., Ulyanenkov, A. (2015). Basics of the Operator Method. In: Non-perturbative Description of Quantum Systems. Lecture Notes in Physics, vol 894. Springer, Cham. https://doi.org/10.1007/978-3-319-13006-4_2

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