Journal of the Korean Chemical Society (대한화학회지)

Korean Chemical Society (대한화학회)
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Multidimetional Uniform Semiclassical (WKB) Solutions for Nonseparable Problems

다차원 비분리계의 균일준고전적 해법

  • Byung C. Eu (Department of Chemistry, McGill University)
  • 유병찬 (카나다 McGill 대학교 화학과)
  • Published : 1978.08.30
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Abstract

Uniform semiclassical (WKB) solutions are obtained for nonseparable systems without using a close coupling formalism and are given explicitly in terms of well known analytic functions for various physically interesting and realistic cases. They do not become singular at turning points or surfaces and when taken in their asymptotic forms, they reduce to the usual WKB solutions that could be obtained if the Stokes phenomenon was properly taken care of for solutions. In obtaining such uniform solutions, the Schroedinger equations for nonseparable systems are suitably "renormalized" to solvable "normal" forms through certain transformations. Ehrenfest's adiabatic principle plays an important guiding role for obtaining such "renormalized" uniform solutions for nonseparable systems. The eigenvalues of the Hamiltonian can be calculated from the extended Bohr-Sommerfeld quantization rules when appropriate classical trajectories are obtained. An application is made to many-electron systems and for one of the simplest examples to show the utility of the method the approximate wavefunction is calculated of the ground state helium atom.

본 논문에서는 비분리계의 균일준고전(WKB)해를 close coupling formalism을 쓰지 않고 구하였으며, 여러가지 물리적으로 흥미있고 현실적인 경우들에, 구해진 해들이 잘 알려져 있는 해석 함수들로서 주어졌다. 전환점이나 전환면이 구해진 해들의 특이점이 되지 않으며, 그들의 점근형을 취했을 때, 그 해들은 보통의 WKB해들로 복귀된다. 그러한 균일해들을 얻기 위해서는 비분리계의 Schroedinger 방정식들을 변형하여 풀 수 있는 형으로 적절히 "재규격화"하였다. Ehrenfest의 단열 원리가 그러한 "재규격화"된 균일해들을 도출하는데 중요한 역할을 한다. 적절한 고전적 궤적들이 얻어지면 Hamiltonian의 고유치들은 확장된 Bohr-Sommerfeld 양자화 규칙으로 계산된다. 다전자계에 대한 응용이 시사되었고, 현 방법의 유용성을 보여주기 위한 가장 간단한 예의 하나로서 helium 원자의 바닥상태의 파동함수를 근사적으로 계산하였다.

Keywords

References

  1. Proc. Nat. Acad. Acad. Sci. (U. S. A.) v.14 J.H. Van Vleck
  2. Principles of Quantum Mechanics P.A.M. Dirac
  3. Phys. Rev. v.81 C. Morette
  4. Phys. Rev. v.125 R. Schiller
  5. Phys. Rev. v.181 P. Pechukas
  6. J. Chem. Phys. v.53 W.H. Miller
  7. J. Chem. Phys. v.54 R.A. Marcus
  8. J. Chem. Phys. v.56 R.A. Marcus
  9. J. Chem. Phys. v.57 R.A. Marcus
  10. J. Chem. Phys. v.57 B.C. Eu
  11. Methods of Theoretical Physics v.1 P.M. Morse;H. Feshbach
  12. Introduction to Phase-Integral Methods J. Heading
  13. The JWKB Approximation N. Froman;P.O. Froman
  14. Phys. Rev. v.157 H.M. Van Horn;E.E. Salpeter
  15. Ann. Phys. (N.Y.) v.4 J.B. Keller
  16. Ann. Phys. (N. Y.) v.9 J.B. Keller;I. Rubinow
  17. Dis. Farady Soc. v.55 R.A. Marcus
  18. J. Chem. Phys. v.61 W. Eastes;R.A. Marcus
  19. J. Chem. Phys. v.62 D.W. Noid;R.A. Marcus
  20. Mole. Phys. v.31 I.C. Percival;N. Pomphrey
  21. Bull. Amer. Math. Soc. v.49 R.E. Langer
  22. Trans. Amer. Math. Soc. v.67 R.E. Langer
  23. J. Math. Phys. v.1 A. Erdelyi
  24. Phil. Trans. Roy. Soc.(London) v.A247 F. W. J. Olver
  25. Phil. Trans. Roy. Soc.(London) v.A249 F. W. J. Olver
  26. Phil. Trans. Roy. Soc.(London) v.A250 F. W. J. Olver
  27. J. Chem. Phys. v.55 B.C. Eu
  28. J. Chem. Phys. v.56 B.C. Eu
  29. J. Chem. Phys. v.58 B.C. Eu
  30. J. Chem. Phys. v.59 B.C. Eu
  31. Phys. Rev. v.A7 B.C. Eu;T.P. Tsien
  32. Chem. Phys. Letts. v.31 U.I. Cho;B.C. Eu
  33. J. Chem. Phys. U.I. Cho;B.C. Eu
  34. Mol. Phys. v.32 no.1 U.I. Cho;B.C. Eu
  35. Phys. Rev. v.91 S. C. Miller, Jr.;R.H. Good, Jr.
  36. Comm. Pure Appl. Math. v.19 D. Ludwing
  37. Comm. Pure Appl. Math. v.20 D. Ludwing
  38. Comm. Pure Appl. Math. v.20 R.M. Lewis;N. Bleistein;D. Ludwig
  39. Proc. Acad. Amsterdam v.19 P. Ehrenfest
  40. Ann. Phys. v.51 P. Ehernfest
  41. Phil. Mag. v.33 P. Ehrenfest
  42. Proc. Acad. Amsterdam Paul Ehrenfest M.J. Klein
  43. Atomic Structure and Spectral Lines A. Sommerfeld
  44. Principles of Statistical Mechanics R.C. Tolman
  45. Can. J. Phys. v.52 B. C. Eu;H. Guerin
  46. J. Math. Phys. v.12 M.C. Gutzwiller
  47. J. Chem. Phys. v.63 W.H. Miller
  48. Handbook of Mathematical Functions M. Abramowitz;I.A. Stegun
  49. Analyticla Dynamics of Particles and Rigid Bodies E.T. Whittaker
  50. Quantum Chemistry H. Eyring;J. Walter;G.E. Kimball
  51. Roy. Astron, Soc. Month. Notice v.74 J. W. Nicholson
  52. J. Chem. Phys. v.56 W.H. Miller
  53. Phys. Rev. v.91 S. Chandrasekhar;D. Elbert;G. Herzberg
  54. Phys. Rev. v.98 S. Chandrasekhar;D. Elbert;G. Herzberg
  55. J. Korean. Chem. Soc. v.18 U.I. Cho;B. C. Eu
  56. J. Chem. Phys. v.63 S. Chapman;B.C. Garret;W.H. Miller