Skip to main content

Dirac Equation with the Coulomb Potential

  • Chapter
Wave Equations in Higher Dimensions
  • 1574 Accesses

Abstract

The exact solutions of quantum system with a 1/r type potential are of importance in quantum mechanics. Due to the recent interest of the higher-dimensional field theory, many problems related to the Schrödinger equation and Klein-Gordon equation in (D+1) dimensions have been discussed. To fill in the gap between them, we have carried out the Dirac equation with this potential in (D+1) dimensions. In this Chapter, we shall study the exact solutions of the radial equations,the variations of energy difference and energy levels on the dimension D as well as the variations of energy levels on the potential strength. Finally, we deal with the Dirac equation with a Coulomb potential plus a scalar potential.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
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

Notes

  1. 1.

    Notice that Eq. (13.44) is a special case of the Tricomi equation as given in Eq. (13.14).

References

  1. Schiff, L.I.: Quantum Mechanics, 3rd edn. McGraw-Hill, New York (1955)

    MATH  Google Scholar 

  2. Landau, L.D., Lifshitz, E.M.: Quantum Mechanics-Nonrelativistic Theory, 3rd edn. Pergamon, New York (1977)

    Google Scholar 

  3. Erdélyi, A.: Higher Transcendental Functions, vol. 2, Bateman Manuscript Project, p. 232. McGraw-Hill, New York (1953)

    Google Scholar 

  4. Nieto, M.M.: Hydrogen atom and relativistic pi-mesic atom in N-space dimensions. Am. J. Phys. 47, 1067 (1979)

    Article  ADS  Google Scholar 

  5. Gu, X.Y., Ma, Z.Q., Dong, S.H.: Exact solutions to the Dirac equation for a Coulomb potential in D+1 dimensions. Int. J. Mod. Phys. E 11(4), 335–346 (2002)

    Article  ADS  Google Scholar 

  6. Dong, S.H., Gu, X.Y., Ma, Z.Q., Yu, J.: The Klein-Gordon equation with a Coulomb potential in D dimensions. Int. J. Mod. Phys. E 12, 555–565 (2003)

    Article  ADS  Google Scholar 

  7. Dong, S.H.: The Dirac equation with a Coulomb potential in D dimensions. J. Phys. A, Math. Gen. 36, 4977 (2003)

    Article  ADS  MATH  Google Scholar 

  8. Dong, S.H., Sun, G.H., Popov, D.: Group theory approach to the Dirac equation with a Coulomb plus scalar potential in D+1 dimensions. J. Math. Phys. 44, 4467–4479 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  9. Dong, S.H., Hou, X.W., Ma, Z.Q.: Relativistic Levinson theorem in two dimensions. Phys. Rev. A 58, 2160–2167 (1998)

    Article  ADS  Google Scholar 

  10. Spector, H.N., Lee, J.: Relativistic one-dimensional hydrogen atom. Am. J. Phys. 53, 248 (1985)

    Article  ADS  Google Scholar 

  11. Moss, R.E.: The hydrogen atom in one dimension. Am. J. Phys. 55, 397 (1987)

    Article  ADS  Google Scholar 

  12. Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals, Series, and Products, 5th edn. Pergamon, New York (1994)

    Google Scholar 

  13. Dong, S.H., Hou, X.W., Ma, Z.Q.: Levinson’s theorem for the Klein-Gordon equation in two dimensions. Phys. Rev. A 59, 995–1002 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  14. Dong, S.H., Hou, X.W., Ma, Z.Q.: Levinson’s theorem for non-local interactions in two dimensions. J. Phys. A, Math. Gen. 31, 7501 (1998)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  15. Dirac, P.A.: The Principles of Quantum Mechanics, 4th edn. Oxford University Press, London (1958)

    MATH  Google Scholar 

  16. Greiner, W.: Relativistic Quantum Mechanics-Wave Equations. Springer, Berlin (1990)

    MATH  Google Scholar 

  17. Vaidya, A.N., Souza, L.E.S.: Algebraic calculation of the S-matrix for the Dirac-Coulomb problem in 2+1 dimensions. Phys. Lett. A 293, 129–132 (2002)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  18. Greiner, W.: Quantum Mechanics—An Introduction. Springer, Berlin (1989)

    Google Scholar 

  19. Ikhadir, S.M., Mustafa, O., Sever, R.: Solution of the Dirac equation for vector and scalar potentials and some applications. Hadron. J. 16(1), 57–74 (1993)

    Google Scholar 

  20. Dong, S.H., Gu, X.Y., Ma, Z.Q.: Exact solutions of the Dirac equation with a Coulomb plus scalar potential in 2+1 dimensions. Int. J. Mod. Phys. E 11, 483–489 (2002)

    Article  ADS  Google Scholar 

  21. Dong, S.H., Lozada-Cassou, M.: On the analysis of the eigenvalues of the Dirac equation with a 1/r potential in D dimensions. Int. J. Mod. Phys. Lett. E 13, 917–931 (2004)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Shi-Hai Dong .

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Science+Business Media B.V.

About this chapter

Cite this chapter

Dong, SH. (2011). Dirac Equation with the Coulomb Potential. In: Wave Equations in Higher Dimensions. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-1917-0_13

Download citation

Publish with us

Policies and ethics