Indian Journal of Physics

, Volume 91, Issue 4, pp 371–376 | Cite as

Use of quantum self-friction potentials and forces in standard convention for study of harmonic oscillator

Original Paper
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Abstract

In this paper, the physical nature of quantum usual and self-friction (SF) harmonic oscillators is presented. The procedure for studying these harmonic oscillators is identical; therefore, we can benefit from the theory of the usual harmonic oscillator. To study the SF harmonic oscillator, using analytical formulae for the \(L^{{(p_{l}^{ * } )}}\)-SF Laguerre polynomials (\(L^{{(p_{l}^{ * } )}}\)-SFLPs) and \(L^{{(\alpha^{*} )}}\)-modified SFLPs (\(L^{{(\alpha^{*} )}}\)-MSFLPs) in standard convention, the \(V^{{(p_{l}^{ * } )}}\)-SF potentials (\(V^{{(p_{l}^{ * } )}}\)-SFPs), \(V^{{(\alpha^{*} )}}\)-modified SFPs (\(V^{{(\alpha^{*} )}}\)-MSFPs), \(F^{{(p_{l}^{ * } )}}\)-SF forces (\(F^{{(p_{l}^{ * } )}}\)-SFFs) and \(F^{{(\alpha^{*} )}}\)-modified SFFs (\(F^{{(\alpha^{*} )}}\)-MSFFs) are investigated, where \(p_{l}^{ * } = 2l + 2 - \alpha^{*}\) and \(\,\alpha^{*}\) is the integer (\(\alpha^{*} = \alpha\), \(\, - \infty < \alpha \le 2)\) or non-integer (\(\alpha^{*} \ne \alpha\), \(\, - \infty < \alpha < 3)\) SF quantum number. We note that the potentials (\(V^{{(p_{l}^{ * } )}}\)-SFPs and \(V^{{(\alpha^{*} )}}\)-MSFPs), and forces (\(F^{{(p_{l}^{ * } )}}\)-SFFs and \(F^{{(\alpha^{*} )}}\)-MSFFs), respectively, are independent functions. It is shown that the numerical values of these independent functions are the same, i.e., \(V_{num}^{{(p_{l}^{ * } )}} = V_{num}^{{(\alpha^{*} )}}\) and \(F_{num}^{{(p_{l}^{ * } )}} = F_{num}^{{(\alpha^{*} )}}\). The dependence of the SF harmonic oscillator as a function of the distance is analyzed. The presented relationships are valid for arbitrary values of parameters.

Keywords

Harmonic oscillator Standard convention Laguerre polynomials Self-friction quantum numbers 

PACS Numbers

31.10.+z 31.15.a- 

References

  1. [1]
    L D Landau and E M Lifshitz Quantum Mechanics (New York: Pergamon) Vol 3, p. 245 (1987)Google Scholar
  2. [2]
    A S Davyidov Quantum Mechanics (New York: Pergamon) p. 157 (1965)Google Scholar
  3. [3]
    S H Dong Factorization Method in Quantum Mechanics (Dordrecht: Springer) p. 73 (2007)CrossRefGoogle Scholar
  4. [4]
    H Hassanabadi, E Maghsoodi, A N Ikot and S Zarrinkamar Adv. High Energy Phys. 2014 831938 (2014)Google Scholar
  5. [5]
    A N Ikot, B H Yazarloo, A D Antia and H Hassanabadi Indian J. Phys. 87 913 (2013)ADSCrossRefGoogle Scholar
  6. [6]
    H A Lorentz The Theory of Electrons (New York: Dover) p. 223 (1953)Google Scholar
  7. [7]
    W Heitler The Quantum Theory of Radiation (London: Oxford University) p. 57 (1950)MATHGoogle Scholar
  8. [8]
    L D Landau and E M Lifshitz The Classical Theory of Fields (New York: Pergamon) p. 132 (1987)Google Scholar
  9. [9]
    B A Mamedov Int. J. Quantum Chem. 114 361 (2014)CrossRefGoogle Scholar
  10. [10]
    I I Guseinov AIP Conf. Proceed. 899 65 (2007)ADSCrossRefGoogle Scholar
  11. [11]
    I I Guseinov Bull. Chem. Soc. Jpn. 85 1306 (2012)CrossRefGoogle Scholar
  12. [12]
    I I Guseinov Few-Body Syst. 54 1773 (2013)ADSCrossRefGoogle Scholar
  13. [13]
    W Magnus, F Oberhettinger, R P Soni Formulas and Theorems for the Special Functions of Mathematical Physics (New York: Springer) p. 324 (1966)Google Scholar

Copyright information

© Indian Association for the Cultivation of Science 2016

Authors and Affiliations

  1. 1.Department of Physics, Faculty of Arts and SciencesOnsekiz Mart UniversityÇanakkaleTurkey
  2. 2.Department of Physics, Faculty of Arts and SciencesGaziosmanpaşa UniversityTokatTurkey
  3. 3.Department of PhysicsBaku State UniversityBakuAzerbaijan

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