Mathematics of Complexity and Dynamical Systems

2011 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Perturbation Theory in Quantum Mechanics

  • Luigi E. Picasso
  • Luciano Bracci
  • Emilio d'Emilio
Reference work entry
DOI: https://doi.org/10.1007/978-1-4614-1806-1_85

Article Outline

Glossary

Definition of the Subject

Introduction

Presentation of the Problem and an Example

Perturbation of Point Spectra: Nondegenerate Case

Perturbation of Point Spectra: Degenerate Case

The Brillouin–Wigner Method

Symmetry and Degeneracy

Problems with the Perturbation Series

Perturbation of the Continuous Spectrum

Time Dependent Perturbations

Future Directions

Bibliography

Keywords

Wave Function Perturbation Theory Irreducible Representation Point Spectrum Asymptotic Series 
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.
This is a preview of subscription content, log in to check access.

Bibliography

Primary Literature

  1. 1.
    Baker GA, Graves‐Morris P (1996) Padé approximants. Cambridge Univ. Press, CambridgeGoogle Scholar
  2. 2.
    Bargmann V (1964) Note on Wigner's theorem on symmetry operations. J Math Phys 5:862–868MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bender CM, Wu TT (1969) Anharmonic oscillator. Phys Rev 184:1231–1260MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bloch C (1958) Sur la théorie des perturbations des états liées. Nucl Phys 6:329–347CrossRefGoogle Scholar
  5. 5.
    Böhm A (1993) Quantum mechanics, foundations and applications. Springer, New York, pp 208–215Google Scholar
  6. 6.
    Borel E (1899) Mémoires sur le séries divergentes. Ann Sci École Norm Sup 16:9–136MathSciNetMATHGoogle Scholar
  7. 7.
    Born M, Heisenberg W, Jordan P (1926) Zur Quantenmechanik, II. Z Phys 35:557–615MATHCrossRefGoogle Scholar
  8. 8.
    Born M (1926) Quantenmechanik der Stossvorgänge. Z Phys 38:803–827CrossRefGoogle Scholar
  9. 9.
    Brillouin L (1932) Perturbation problem and self consistent field. J Phys Radium 3:373–389CrossRefGoogle Scholar
  10. 10.
    Courant R, Hilbert D (1989) Methods of mathematical physics, vol I. Wiley, New York, pp 343–350CrossRefGoogle Scholar
  11. 11.
    Dirac PAM (1926) On the theory of quantum mechanics. Proc Roy Soc A112:661–677MATHCrossRefGoogle Scholar
  12. 12.
    Dyson FJ (1952) Divergence of perturbation theory in quantum electrodynamics. Phys Rev 85:631–632MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Dyson FJ (1949) The radiation theories of Tomonaga, Schwinger and Feynman. Phys Rev 75:486–502MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Dyson FJ (1949) The S‑matrix in quantum electrodynamics. Phys Rev 75:1736–1755MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Epstein ST (1954) Note on perturbation theory. Amer J Phys 22:613–614MATHCrossRefGoogle Scholar
  16. 16.
    Epstein ST (1968) Uniqueness of the energy in perturbation theory. Amer J Phys 36:165–166CrossRefGoogle Scholar
  17. 17.
    Feynman RP (1939) Forces in molecules. Phys Rev 56:340–343MATHCrossRefGoogle Scholar
  18. 18.
    Fock VA (1935) Zur Theorie des Wasserstoffatoms. Z Phys 98:145–154CrossRefGoogle Scholar
  19. 19.
    Graffi S, Grecchi V, Simon B (1970) Borel summability: Application to the anharmonic oscillator. Phys Lett B32:631–634MathSciNetCrossRefGoogle Scholar
  20. 20.
    Grossman A (1961) Schrödinger scattering amplitude I. J Math Phys 3:710–713CrossRefGoogle Scholar
  21. 21.
    Hamermesh M (1989) Group theory and its application to physical problems. Dover, New York, pp 32–114Google Scholar
  22. 22.
    Hannabuss K (1997) Introduction to quantum theory. Clarendon, Oxford, pp 131–136MATHGoogle Scholar
  23. 23.
    Hellmann H (1937) Einführung in die Quantenchemie. Deuticke, LeipzigGoogle Scholar
  24. 24.
    Joachain J (1983) Quantum collision theory. North–Holland, AmsterdamGoogle Scholar
  25. 25.
    Kato T (1966) Perturbation theory for linear operators. Springer, New YorkMATHGoogle Scholar
  26. 26.
    Kato T (1949) On the convergence of the perturbation method I. Progr Theor Phys 4:514–523CrossRefGoogle Scholar
  27. 27.
    Kramers HA (1957) Quantum mechanics. North-Holland, Amsterdam, pp 198–202MATHGoogle Scholar
  28. 28.
    Krieger JB (1968) Asymptotic properties of perturbation theory. J Math Phys 9:432–435MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Landau LD, Lifshitz EM (1960) Mechanics. Pergamon Press, Oxford, p 129MATHGoogle Scholar
  30. 30.
    Lippmann BA, Schwinger J (1950) Variational principles for scattering processes I. Phys Rev 79:469–480MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Loeffel J, Martin A, Wightman A, Simon B (1969) Padé approximants and the anharmonic oscillator. Phys Lett B 30:656–658CrossRefGoogle Scholar
  32. 32.
    Padé H (1899) Sur la représentation approchée d'une fonction pour des fonctions rationelles. Ann Sci Éco Norm Sup Suppl 9(3):1–93Google Scholar
  33. 33.
    Rayleigh JW (1894–1896) The theory of sound, vol I. Macmillan, London, pp 115–118Google Scholar
  34. 35.
    Reed M, Simon B (1975) Methods of modern mathematical physics, vol II. Academic Press, New York, pp 282–283Google Scholar
  35. 34.
    Reed M, Simon B (1978) Methods of modern mathematical physics, vol IV. Academic Press, New York, pp 10–44Google Scholar
  36. 36.
    Rellich F (1937) Störungstheorie der Spektralzerlegung I. Math Ann 113:600–619MathSciNetCrossRefGoogle Scholar
  37. 37.
    Rellich F (1937) Störungstheorie der Spektralzerlegung II. Math Ann 113:677–685MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Rellich F (1939) Störungstheorie der Spektralzerlegung III. Math Ann 116:555–570MathSciNetCrossRefGoogle Scholar
  39. 39.
    Rellich F (1940) Störungstheorie der Spektralzerlegung IV. Math Ann 117:356–382MathSciNetCrossRefGoogle Scholar
  40. 40.
    Riesz F, Sz.-Nagy B (1968) Leçons d'analyse fonctionelle. Gauthier-Villars, Paris, pp 143–188Google Scholar
  41. 41.
    Rollnik H (1956) Streumaxima und gebundene Zustände. Z Phys 145:639–653MATHCrossRefGoogle Scholar
  42. 42.
    Sakurai JJ (1967) Advanced quantum mechanics. Addison-Wesley, Reading, pp 39–40Google Scholar
  43. 43.
    Scadron M, Weinberg S, Wright J (1964) Functional analysis and scattering theory. Phys Rev 135:B202-B207MathSciNetCrossRefGoogle Scholar
  44. 44.
    Schrödinger E (1926) Quantisierung als Eigenwertproblem. Ann Phys 80:437–490Google Scholar
  45. 45.
    Schwartz J (1960) Some non-self‐adjoint operators. Comm Pure Appl Math 13:609–639MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Simon B (1970) Coupling constant analyticity for the anharmonic oscillator. Ann Phys 58:76–136CrossRefGoogle Scholar
  47. 47.
    Simon B (1991) Fifty years of eigenvalue perturbation theory. Bull Am Math Soc 24:303–319MATHCrossRefGoogle Scholar
  48. 48.
    Thirring W (2002) Quantum mathematical physics. Springer, Berlin, p 177Google Scholar
  49. 49.
    von Neumann J, Wigner E (1929) Über das Verhalten von Eigenwerten bei adiabatischen Prozessen. Phys Z 30:467–470MATHGoogle Scholar
  50. 50.
    Watson G (1912) A theory of asymptotic series. Philos Trans Roy Soc Lon Ser A 211:279–313CrossRefGoogle Scholar
  51. 51.
    Weyl H (1931) The theory of groups and quantum mechanics. Dover, New YorkMATHGoogle Scholar
  52. 52.
    Wigner EP (1935) On a modification of the Rayleigh–Schrödinger perturbation theory. Math Natur Anz (Budapest) 53:477–482Google Scholar
  53. 53.
    Wigner EP (1959) Group theory and its application to the quantum mechanics of atomic spectra. Academic Press, New YorkMATHGoogle Scholar
  54. 54.
    Yosida K (1991) Lectures on differential and integral equations. Dover, New York, pp 115–131MATHGoogle Scholar
  55. 55.
    Yukalov VI, Yukalova EP (2007) Methods of self similar factor approximants. Phys Lett A 368:341–347CrossRefGoogle Scholar
  56. 56.
    Zemach C, Klein A (1958) The Born expansion in non‐relativistic quantum theory I. Nuovo Cimento 10:1078–1087MathSciNetMATHCrossRefGoogle Scholar

Books and Reviews

  1. 57.
    Hirschfelder JO, Byers Brown W, Epstein ST (1964) Recent developments in perturbation theory. In: Advances in quantum chemistry, vol 1. Academic Press, New York, pp 255–374Google Scholar
  2. 58.
    Killingbeck J (1977) Quantum‐mechanical perturbation theory. Rep Progr Phys 40:963–1031CrossRefGoogle Scholar
  3. 59.
    Mayer I (2003) Simple theorems, proofs and derivations in quantum chemistry. Kluwer Academic/Plenum Publishers, New York, pp 69–120Google Scholar
  4. 60.
    Morse PM, Feshbach H (1953) Methods of theoretical physics, part 2. McGraw-Hill, New York, pp 1001–1106Google Scholar
  5. 61.
    Wilcox CH (1966) Perturbation theory and its applications in quantum mechanics. Wiley, New YorkMATHGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Luigi E. Picasso
    • 1
    • 2
  • Luciano Bracci
    • 1
    • 2
  • Emilio d'Emilio
    • 1
    • 2
  1. 1.Dipartimento di FisicaUniversità di PisaPisaItaly
  2. 2.Istituto Nazionale di Fisica NucleareSezione di PisaPisaItaly