Skip to main content
SpringerLink
Log in

On Supersymmetric Quantum Mechanics

  • Chapter
Fundamental World of Quantum Chemistry

  • 483 Accesses

Abstract

This paper constitutes a review on N = 2 fractional supersym-metric Quantum Mechanics of order k. The presentation is based on the introduction of a generalized Weyl-Heisenberg algebra W k . It is shown how a general Hamiltonian can be associated with the algebra W k . This general Hamiltonian covers various supersymmetrical versions of dynamical systems (Morse system, Pöschl-Teller system, fractional supersymmetric oscillator of order k, etc.). The case of ordinary supersymmetric Quantum Mechanics corresponds to k = 2. A connection between fractional supersymmetric Quantum Mechanics and ordinary supersymmetric Quantum Mechanics is briefly described. A realization of the algebra W k , of the N = 2 supercharges and of the corresponding Hamiltonian is given in terms of deformed-bosons and k-fermions as well as in terms of differential operators.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 299.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 379.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 379.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. Coleman and J. Mandula, Phys. Rev. 159, 1251 (1967).

    Article  CAS  Google Scholar 

  2. R. Haag, J.T. Lopuszanski and M. Sohnius, Nucl. Phys. B88, 257 (1975).

    Article  Google Scholar 

  3. P. Fayet, Phys. Lett. B64, 159 (1976).

    Article  Google Scholar 

  4. J. Wess and B. Zumino, Phys. Lett. B49, 52 (1974).

    Article  Google Scholar 

  5. F. Wilczek, Fractional Statistics and Any on Superconductivity (World Scientific, Singapore, 1990).

    Google Scholar 

  6. A. Metz, J. Jolie, G. Graw, R. Hertenberger, J. Gröger, C. Günther, N. Warr and Y. Eisermann, Phys. Rev. Lett. 83, 1542 (1999). See also: F. Iachello, Nucl. Phys. News 10, 12 (2000).

    Article  CAS  Google Scholar 

  7. ALEPH Collaboration, Phys. Lett. B495, 1 (2000). L3 Collaboration, Phys. Lett. B495, 18 (2000). OPAL Collaboration, Phys. Lett. B499, 38 (2001).

    Article  Google Scholar 

  8. E. Witten, Nucl. Phys. B138, 513 (1981).

    Article  Google Scholar 

  9. J.M. Leinaas and J. Myrheim, Nuovo Cimento B37, 1 (1977).

    Google Scholar 

  10. G.A. Goldin, R. Menikoff and D.H. Sharp, J. Math. Phys. 21, 650 (1980);

    Article  Google Scholar 

  11. G.A. Goldin, R. Menikoff and D.H. Sharp, J. Math. Phys. 22, 1664 (1981).

    Article  Google Scholar 

  12. G.A. Goldin and D.H. Sharp, Phys. Rev. Lett. 76, 1183 (1996).

    Article  CAS  Google Scholar 

  13. M. Frau, A. Lerda and S. Sciuto, in Proc. Int. School of Physics ‘E. Fermi’, Course CXXVII, eds. L. Castellani and J. Wess (North-Holland, Amsterdam, 1996);

    Google Scholar 

  14. L. Frappat, A. Sciarrino, S. Sciuto and P. Sorba, Phys. Lett. B369, 313 (1996);

    Article  CAS  Google Scholar 

  15. L. Frappat, A. Sciarrino, S. Sciuto and P. Sorba, J. Phys. A: Math. Gen. 30, 903 (1997).

    Article  CAS  Google Scholar 

  16. H.S. Green, Phys. Rev. 90, 270 (1953).

    Article  Google Scholar 

  17. O.W. Greenberg and A.M.L. Messiah, Phys. Rev. B138, 1155 (1965).

    Article  CAS  Google Scholar 

  18. Y. Ohnuki and S. Kamefuchi, Quantum Field Theory and Parastatistics (Tokyo Univ., Tokyo, 1982).

    Book  Google Scholar 

  19. M.S. Plyushchay, Int. J. Mod. Phys. A15, 3679 (2000).

    CAS  Google Scholar 

  20. S. Klishevich and M.S. Plyushchay, Nucl. Phys. B606, 583 (2001).

    Article  Google Scholar 

  21. M. Daoud and M. Kibler, Phys. Lett. A206, 13 (1995).

    Article  CAS  Google Scholar 

  22. M.R. Kibler, J. Meyer and M. Daoud, in Symmetry and Structural Properties of Condensed Matter, eds. T. Lulek, W. Florek and B. Lulek (World Scientific, Singapore, 1997).

    Google Scholar 

  23. M. Daoud and Y. Hassouni, Helv. Phys. Acta 71, 599 (1998).

    CAS  Google Scholar 

  24. M. Rachidi, E.H. Saidi and J. Zerouaoui, preprint: hep-th/0103212.

    Google Scholar 

  25. M. Arik and D.D. Coon, J. Math. Phys. 17, 524 (1976).

    Article  Google Scholar 

  26. A.J. Macfarlane, J. Phys. A: Math. Gen. 22, 4581 (1989).

    Article  Google Scholar 

  27. L.C. Biedenharn, J. Phys. A: Math. Gen. 22, L873 (1989).

    Article  Google Scholar 

  28. C.-P. Sun and H.-C. Fu, J. Phys. A: Math. Gen. 22, L983 (1989).

    Article  Google Scholar 

  29. A.I. Solomon, Phys. Lett. A196, 29 (1994).

    Article  CAS  Google Scholar 

  30. M.R. Kibler, in Symmetry and Structural Properties of Condensed Matter, eds. W. Florek, D. Lipihski and T. Lulek (World Scientific, Singapore, 1993).

    Google Scholar 

  31. V.A. Rubakov and V.P. Spiridonov, Mod. Phys. Lett. A3, 1337 (1988).

    Article  Google Scholar 

  32. J. Beckers and N. Debergh, Mod. Phys. Lett. A4, 1209 (1989);

    Article  Google Scholar 

  33. J. Beckers and N. Debergh, Nucl. Phys. B340, 767 (1990).

    Article  Google Scholar 

  34. N. Debergh, J. Math. Phys. 34, 1270 (1993);

    Article  Google Scholar 

  35. N. Debergh, J. Phys. A: Math. Gen. 26, 7219 (1993);

    Article  Google Scholar 

  36. N. Debergh, J. Phys. A: Math. Gen. 27, L213 (1994).

    Article  Google Scholar 

  37. A. Khare, J. Phys. A: Math. Gen. 25, L749 (1992);

    Article  Google Scholar 

  38. A. Khare, J. Math. Phys. 34, 1277 (1993).

    Article  Google Scholar 

  39. A.T. Filippov, A.P. Isaev and A.B. Kurdikov, Mod. Phys. Lett. A7, 2129 (1992);

    Article  Google Scholar 

  40. A.T. Filippov, A.P. Isaev and A.B. KurdikovInt. J. Mod. Phys. A8, 4973 (1993).

    Article  Google Scholar 

  41. S. Durand, Mod. Phys. Lett. A7, 2905 (1992);

    Article  Google Scholar 

  42. S. Durand, Phys. Lett. B312, 115 (1993);

    Article  CAS  Google Scholar 

  43. S. Durand, Mod. Phys. Lett. A8, 2323 (1993).

    Article  Google Scholar 

  44. M. Daoud and Y. Hassouni, Prog. Theor. Phys. 97, 1033 (1997).

    Article  CAS  Google Scholar 

  45. S. Klishevich and M.S. Plyushchay, Mod. Phys. Lett. A14, 2379 (1999).

    Article  Google Scholar 

  46. M. Daoud, Y. Hassouni and M. Kibler, in Symmetries in Science X, eds. B. Gruber and M. Ramek (Plenum, New York, 1998);

    Google Scholar 

  47. M. Daoud, Y. Hassouni and M. Kibler, Yad. Fiz. 61, 1935 (1998).

    CAS  Google Scholar 

  48. H.-Y. Pan, Z.S. Zhao, Phys. Lett. A 312 (2003) 1.

    Article  CAS  Google Scholar 

  49. M. Daoud and M.R. Kibler, in Symmetry and Structural Properties of Condensed Matter, eds. T. Lulek, B. Lulek and A. Wal (World Scientific, Singapore, 2001).

    Google Scholar 

  50. M. Kibler and M. Daoud, Recent Res. Devel. Quantum Chem. 2, 91 (2001).

    CAS  Google Scholar 

  51. M. Daoud and M. Kibler, Recent Res. Devel. Quantum Chem. 2, 101 (2001).

    CAS  Google Scholar 

  52. M. Daoud and M. Kibler, in Proceedings of the Sixth International Wigner Symposium (Bogazici Univ. Press, Istanbul, Turkey, 2002).

    Google Scholar 

  53. M.S. Plyushchay, Ann. Phys. (N.Y.) 245, 339 (1996);

    Article  CAS  Google Scholar 

  54. M.S. Plyushchay, Mod. Phys. Lett. All, 397 (1996);

    Google Scholar 

  55. M.S. Plyushchay, Mod. Phys. Lett. A12, 1153 (1997);

    Article  Google Scholar 

  56. M.S. Plyushchay, Nucl. Phys. B491, 619 (1997).

    Article  Google Scholar 

  57. J. Gamboa, M.S. Plyushchay and J. Zanelli, Nucl. Phys. B543, 447 (1999).

    Article  Google Scholar 

  58. C. Quesne and N. Vansteenkiste, Phys. Lett. A240, 21 (1998).

    Article  CAS  Google Scholar 

  59. C. Ahn, D. Bernard and A. LeClair, Nucl. Phys. B346, 409 (1990).

    Article  Google Scholar 

  60. A. LeClair and C. Vafa, Nucl. Phys. B401, 413 (1993).

    Article  Google Scholar 

  61. R. Kerner, J. Math. Phys. 33, 403 (1992).

    Article  Google Scholar 

  62. J.L. Matheus-Valle and M.A. R. Monteiro, Mod. Phys. Lett. A7, 3023 (1992);

    Article  Google Scholar 

  63. J.L. Matheus-Valle and M.A. R. Monteiro, Phys. Lett. B300, 66 (1993).

    Article  CAS  Google Scholar 

  64. L.P. Collatto and J.L. Matheus-Valle, J. Math. Phys. 37, 6121 (1996).

    Article  Google Scholar 

  65. E.H. Saidi, M.B. Sedra and J. Zerouaoui, Class, and Quant. Gravity 7, 1567 (1995).

    Article  Google Scholar 

  66. N. Mohammedi, Mod. Phys. Lett. A10, 1287 (1995).

    Article  Google Scholar 

  67. J.A. de Azcârraga and A.J. Macfarlane, J. Math. Phys. 37, 1115 (1996).

    Article  Google Scholar 

  68. A. Perez, M. Rausch de Traubenberg and P. Simon, Nucl. Phys. B482, 325 (1996).

    Article  Google Scholar 

  69. N. Fleury and M. Rausch de Traubenberg, Mod. Phys. Lett. A11, 899 (1996).

    Article  CAS  Google Scholar 

  70. M. Rausch de Traubenberg and M.J. Slupinski, Mod. Phys. Lett. A12, 3051 (1997).

    Article  Google Scholar 

  71. M. Rausch de Traubenberg and P. Simon, Nucl. Phys. B517, 485 (1998).

    Article  Google Scholar 

  72. M. Rausch de Traubenberg and M.J. Slupinski, J. Math. Phys. 41, 4556 (2000).

    Article  Google Scholar 

  73. A. Mostafazadeh, Int. J. Mod. Phys. A11, 1057, 2941, 2957 (1996).

    CAS  Google Scholar 

  74. K. Aghababaei Samani and A. Mostafazadeh, Nucl. Phys. B595, 467 (2001).

    Article  Google Scholar 

  75. H. Ahmedov and Ö.F. Dayi, J. Phys. A: Math. Gen. 32, 6247 (1999).

    Article  Google Scholar 

  76. D. Bonatsos, P. Kolokotronis and C. Daskaloyannis, Mod. Phys. Lett. A10, 2197 (1995).

    Article  Google Scholar 

  77. J. Katriel and C. Quesne, J. Math. Phys. 37, 1650 (1996).

    Article  Google Scholar 

  78. F. Calogero, J. Math. Phys. 10, 2191, 2197 (1969).

    Article  Google Scholar 

  79. M.A. Vasiliev, Int. J. Mod. Phys. A6, 1115 (1991);

    Article  Google Scholar 

  80. F. Calogero, Nucl. Phys. Proc. Suppl. b56, 241 (1997).

    Article  Google Scholar 

  81. J.-P. Gazeau, in Quantization, Coherent States, and Complex Structures, eds. J.P. Antoine, S.T. Ali, W. Lisiecki, I.M. Mladenov and A. Odzijewicz (Plenum, New York, 1996).

    Google Scholar 

  82. C. De Concini and V.G. Kac, Progr. Math. 92, 471 (1990).

    Google Scholar 

  83. M. Daoud, M. Kibler, Phys. Part. Nuclei (Suppl. 1) 33 (2002) S43;

    Google Scholar 

  84. M. Daoud, M. Kibler, Int. J. Quantum Chem. 91 (2003) 551; Phys. Lett. Axx, xxx (2004).

    Article  CAS  Google Scholar 

  85. U.P. Sukhatme, C. Rasinariu, A. Khare, Phys. Lett. A234, 401 (1997).

    Article  CAS  Google Scholar 

  86. G. Junker, Supersymmetric Methods in Quantum and Statistical Physics (Springer, Berlin, 1996).

    Book  Google Scholar 

  87. Y. Bérubé-Lauzière and V. Hussin, J. Phys. A: Math. Gen. 26, 6271 (1993).

    Article  Google Scholar 

  88. R.S. Dunne, A.J. Macfarlane, J.A. de Azcárraga and J.C. Pérez Bueno, Phys. Lett. B387, 294 (1996);

    Article  CAS  Google Scholar 

  89. R.S. Dunne, A.J. Macfarlane, J.A. de Azcárraga and J.C. Pérez Bueno, Czech. J. Phys. 46, 1145 (1996);

    Article  Google Scholar 

  90. R.S. Dunne, A.J. Macfarlane, J.A. de Azcárraga and J.C. Pérez Bueno, Czech. J. Phys. 46, 1235 (1996).

    Article  Google Scholar 

  91. M. Mansour, M. Daoud and Y. Hassouni, Phys. Lett. B454, 281 (1999).

    Article  CAS  Google Scholar 

  92. M. Chaichian, D. Ellinas and P.P. Kulish, Phys. Rev. Lett. 65, 980 (1990).

    Article  Google Scholar 

  93. J. Katriel and A.I. Solomon, J. Phys. A: Math. Gen. 24, 2093 (1991);

    Article  Google Scholar 

  94. R.J. McDermott and A.I. Solomon, J. Phys. A: Math. Gen. 27, L15 (1994);

    Article  Google Scholar 

  95. R.J. McDermott and A.I. Solomon, J. Phys. A: Math. Gen. 27, 2037 (1994).

    Article  Google Scholar 

  96. F.B. Wang and L.M. Kuang, Phys. Lett. A169, 225 (1992);

    Article  Google Scholar 

  97. L.M. Kuang and F.B. Wang, Phys. Lett. A173, 221 (1993);

    Article  Google Scholar 

  98. Le-Man Kuang, Fa-Bo Wang and Gao-Jian Zeng, Phys. Lett. A176, 1 (1993).

    Google Scholar 

  99. V.I. Man’ko, G. Marmo, E.C.G. Sudarshan and F. Zaccaria, Phys. Scripta 55, 520 (1997).

    Article  Google Scholar 

  100. S. Majid and M.J. Rodriguez-Plaza, J. Math. Phys. 35, 3753 (1994).

    Article  Google Scholar 

  101. A. Vourdas, Phys. Rev. A41, 1653 (1990);

    Article  Google Scholar 

  102. A. Vourdas, Phys. Rev. A43, 1564 (1991).

    Article  Google Scholar 

  103. A. Vourdas and C. Bendjaballah, Phys. Rev. A47, 3523 (1993).

    Article  Google Scholar 

  104. A. Vourdas, J. Phys. A: Math. Gen. 29, 4275 (1996).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut de Physique Nucléaire de Lyon, IN2P3-CNRS et Université Claude Bernard, F-69622, Villeurbanne Cedex, France

    M. R. Kibler

  2. Département de Physique Faculté des Sciences, Université Ibnou Zohr, B.P. 28/S, Agadir, Morocco

    M. Daoud

Authors
  1. M. R. Kibler
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Daoud
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Uppsala University, Uppsala, Sweden

    Erkki J. Brändas

  2. Bogoliubov Institute for Theoretical Physics, Kiev, Ukraine

    Eugene S. Kryachko

  3. University of Leuven, Leuven, Belgium

    Eugene S. Kryachko

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Kibler, M.R., Daoud, M. (2004). On Supersymmetric Quantum Mechanics. In: Brändas, E.J., Kryachko, E.S. (eds) Fundamental World of Quantum Chemistry. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0448-9_5

Download citation

  • DOI: https://doi.org/10.1007/978-94-017-0448-9_5

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-6687-9

  • Online ISBN: 978-94-017-0448-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation