Skip to main content
SpringerLink
Log in

Quantum states with continuous spectrum for a general time-dependent oscillator

  • Published:
Pramana Aims and scope Submit manuscript

Abstract

We investigated quantum states with continuous spectrum for a general time-dependent oscillator using invariant operator and unitary transformation methods together. The form of the transformed invariant operator by a unitary operator is the same as the Hamiltonian of the simple harmonic oscillator:I’ = p2/2 +ω 2 q 2/2. The fact thatω 2 of the transformed invariant operator is constant enabled us to investigate the system separately for three cases, whereω 2 > 0,ω 2 < 0, andω 2 = 0. The eigenstates of the system are discrete forω 2 > 0. On the other hand, forω 2 <− 0, the eigenstates are continuous. The time-dependent oscillators whose spectra of the wave function are continuous are not oscillatory. The wave function forω 2 < 0 is expressed in terms of the parabolic cylinder function. We applied our theory to the driven harmonic oscillator with strongly pulsating mass.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G E Giacaglia,Perturbation methods in nonlinear systems (Springer, Berlin, 1972), Applied Mathematical Science Series, Vol. 18

    Google Scholar 

  2. W T Van Horssen,SIAM J. Appl. Math. 59, 1444 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. L S Brown,Phys. Rev. Lett. 66, 527 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Fu-li Li, S J Wang, A Weiguny and D L Lin,J. Phys. A27, 985 (1994)

    ADS  MathSciNet  Google Scholar 

  5. C I Um, J R Choi and K H Yeon,J. Korean Phys. Soc. 38, 447 (2001)

    Google Scholar 

  6. J Y Ji, J K Kim and S P Kim,Phys. Rev. A51, 4268 (1995)

    ADS  Google Scholar 

  7. J R Choi,Pramana —J. Phys. 61, 7 (2003)

    Article  ADS  Google Scholar 

  8. C I Um, J R Choi, K H Yeon and T F George,J. Korean Phys. Soc. 40, 969 (2002)

    Google Scholar 

  9. R L Kobes and K L Kowalski,Phys. Rev. D34, 513 (1986)

    ADS  MathSciNet  Google Scholar 

  10. N P Landsman and Ch G van Weert,Phys. Rep. 145, 141 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  11. E Calzetta and B L Hu,Phys. Rev. D37, 2878 (1988)

    ADS  MathSciNet  Google Scholar 

  12. A O Caldeira and A J Leggett,Ann. Phys. 149, 374 (1983)

    Article  ADS  Google Scholar 

  13. S Baskoutas, A Jannussis and R Mignani,J. Phys. A27, 2189 (1994)

    ADS  MathSciNet  Google Scholar 

  14. A O Caldeira and A J Leggett,Phys. Rev. Lett. 46, 211 (1981)

    Article  ADS  Google Scholar 

  15. A Widom and T D Clark,Phys. Rev. Lett. 48, 63 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  16. B K Berger,Phys. Rev. D12, 368 (1975)

    ADS  Google Scholar 

  17. H Lamba, S McKee and R Simpson,J. Phys. A31, 7065 (1998)

    ADS  Google Scholar 

  18. H Dehmelt,Rev. Mod. Phys. 62, 525 (1990)

    Article  ADS  Google Scholar 

  19. W Paul,Rev. Mod. Phys. 62, 531 (1990)

    Article  ADS  Google Scholar 

  20. J B Maddox and E R Bittner,Phys. Rev. E65, 026143 (2002)

    ADS  Google Scholar 

  21. Z Qi,Phys. Rev. A53, 3805 (1996)

    ADS  Google Scholar 

  22. S Yngve and S Linnaeus,J. Phys. A19, 3017 (1986)

    ADS  MathSciNet  Google Scholar 

  23. S Guillouzic, I L’Heureux and A Longtin,Phys. Rev. E61, 4906 (2000)

    ADS  Google Scholar 

  24. K H Yeon, S S Kim, Y M Moon, S K Hong, C I Um and T F George,J. Phys. A34, 7719 (2001)

    ADS  MathSciNet  Google Scholar 

  25. J R Choi,Mod. Phys. Lett. B17, 1365 (2003)

    ADS  Google Scholar 

  26. M A Castagnino, H Giacomini and L Lara,Phys. Rev. D63, 044003 (2001)

    ADS  MathSciNet  Google Scholar 

  27. H Suzuki and M Tanaka,Phys. Rev. D49, 6692 (1994)

    ADS  Google Scholar 

  28. R Haydock,Phys. Rev. B63, 125413 (2001)

    ADS  Google Scholar 

  29. T Geisel, R Ketzmerick and G Petschel,Phys. Rev. Lett. 66, 1651 (1991)

    Article  ADS  Google Scholar 

  30. O Watanabe, K Ogura, T Cho and Md R Amin,Phys. Rev. E63, 056503 (2001)

    ADS  Google Scholar 

  31. H R Lewis Jr,Phys. Rev. Lett. 27, 510 (1967)

    Article  ADS  Google Scholar 

  32. H R Lewis Jr and W B Riesenfeld,J. Math. Phys. 10, 1458 (1969)

    Article  ADS  MathSciNet  Google Scholar 

  33. K-H Yeon, D-H Kim, C-I Um, T F George and L N Pandey,Phys. Rev. A55, 4023 (1997)

    ADS  MathSciNet  Google Scholar 

  34. D C Khandekar, S V Lawande and K V Bhagwat,Path-integral methods and their applications (World Scientific, Singapore, 1993) p. 10

    MATH  Google Scholar 

  35. A Erdély,Higher transcendental functions (McGraw-Hill, New York, 1953) Vol. II, pp. 116–196

    Google Scholar 

  36. A Erdély,Higher transcendental functions (McGraw-Hill, New York, 1953) Vol. I, pp. 255–267

    Google Scholar 

  37. I A Pedrosa,Phys. Rev. A55, 3219 (1997)

    ADS  Google Scholar 

  38. M S Abdalla and R K Colegrave,Phys. Rev. A32, 1958 (1985)

    ADS  MathSciNet  Google Scholar 

  39. R K Colegrave and M S Abdalla,J. Phys. A15, 1549 (1982)

    ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of New Material Science, Division of Natural Sciences, Sun Moon University, 336-708, Asan, Korea

    Jeong-Ryeol Choi

Authors
  1. Jeong-Ryeol Choi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jeong-Ryeol Choi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Choi, JR. Quantum states with continuous spectrum for a general time-dependent oscillator. Pramana - J Phys 65, 165–176 (2005). https://doi.org/10.1007/BF02898608

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02898608

Keywords

PACS Nos

Search

Navigation