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On the use of the finite born series in potential scattering

Об использовании кон ечного борцовского р яда в потенциальном р ассеянии

  • Published:
Il Nuovo Cimento B (1971-1996)

Summary

An analysis of the Born series in potential scattering is presented. Wellner’s formula for improving the convergence of a perturbation series is rederived and compared with some similar formulations. Convergence of the original perturbation expansion of the phase shift and its more convergent form obtained from the iterative scheme are studied. The numerical studies are made forS-wave scattering from an exponential potential. It is found that the first two terms of the Born series for the phase shift approximate the exact values quite closely over a fairly wide range of potential strength, except near threshold, where a further application of the iterative method improves the results significantly.

Riassunto

Si presenta un’analisi della serie di Born nello scattering di potenziale. Si deduce nuovamente la formula di Wellner per migliorarc la convergenza di una serie pertubativa e la si confronta con alcune formulazioni analogie. Si studiano la convergenza dello sviluppo perturbativo originale dello spostamento di fase e la sua forma più convergente ottenuta dallo schema iterativo. Si eseguono studi numerici per lo scattering in ondaS da un potenziale esponenziale. Si trova che i primi due termini della serie di Born per lo spostamento di fase si approssimano abbastanza da vicino ai valori esatti per un intervallo abbastanza ampio di intensità di potenziale, tranne presso la soglia dove un’ulteriore applicazione del metodo di iterazione migliora i risultati in modo significativo.

Резюме

Проводится анализ бо рцовского ряда в поте нциальном рассеянии. Заново выводится формула Ве ллнера для улучшения сходимости ряда теор ии возмущений, которая сравниваетс я с некоторыми другим и аналогичными форму лировками. Исследуются сходимо сть исходного разлож ения теории возмущен ий для фазового сдвига и более сходящ аяся форма разложени я, которая получается из итерационной схемы. Проводится чис ленное рассмотрение дляS-волнового рассе яния на экспоненциальном по тенциале. Получено, чт о первые два члена бор но-вского ряда для фазового сдвига х орошо аппроксимирую т точные значения в до вольно широкой области величин поте нциалов, за исключени ем области вблизи пор ога, где дополнительное прим енение итерационног о метода значительно улучшает полученные результаты.

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References

  1. This method was used to obtain the solution of the one-dimensional Schrödinger equation for the delta-function potential byI. R. Lapidus:Am. Journ. Phys.,37, 1064 (1969). See also ref. (2) for such a remark.

    Article  ADS  Google Scholar 

  2. R. H. Dalitz:Proc. Roy. Soc,206 A, 509 (1951).

    Article  MathSciNet  ADS  Google Scholar 

  3. B. L. Moiseiwitsch andT. J. O’Brien:Journ. Phys. B,3, 191 (1970), and references therein.

    Article  ADS  Google Scholar 

  4. See,e.g.,V. De Alfaro andT. Regge:Potential Scattering (New York, 1965).

  5. SeeJ. D. Bjorken andA. Goldberg:Nuovo Cimento,16, 539 (1960), for comparison of several approximation methods and original references.

    Article  MathSciNet  Google Scholar 

  6. M. Wellner:Phys. Rev.,132, 1848 (1963).

    Article  MathSciNet  ADS  Google Scholar 

  7. S. R. Singh:Journ. Phys. B,3, L 80 (1970), and references therein.

    Article  ADS  Google Scholar 

  8. B. L. Moiseiwitsch:Proc. Phys. Soc,78, 616 (1961).

    Article  ADS  Google Scholar 

  9. S. R. Singh andY. H. Smith jr.: to be published.

  10. S. R. Singh:Journ. Phys. B,3, 1216 (1970).

    Article  ADS  Google Scholar 

  11. P. M. Morse andH. Feshbach:Methods of Theoretical Physics, Part II, Chap. 9 (New York, 1953).

  12. P. M. Morse andH. Feshbach:Methods of Theoretical Physics, Part I (New York, 1953), p. 411.

  13. The method of Wellner, ref. (7), and the iterative method, ref. (8) (see alsoW. R. Conkie andS. R. Singh:Journ. Phys. B,2, 1293 (1969)), fall into this category.

    Article  ADS  Google Scholar 

  14. See,e.g.,G. A. Bakee jr.:Advances in Theoretical Physics, Vol. 1, edited byK. A. Beueckner (New York, 1965), p. 1.

  15. See,e.g.,M. Moe andD. S. Saxon:Phys. Rev.,111, 950 (1958), for such relationships (rom the usual calculations.

    Article  MathSciNet  ADS  Google Scholar 

  16. M. Alexanian andM. Wellner:Phys. Bev.,137, B 155 (1965);140, B 1079 (1965).

    MathSciNet  ADS  Google Scholar 

  17. See,e.g.,H. Morawitz:Journ. Math. Phys.,11, 649 (1970).

    Article  ADS  Google Scholar 

  18. H. A. Bethe andR. Bacher:Rev. Mod. Phys.,8, 111 (1936).

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Chemistry, Queen’s University, Kingston, Ont.

    S. R. Singh

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  1. S. R. Singh
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Supported, in part, by a grant from the National Research Council of Canada.

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Singh, S.R. On the use of the finite born series in potential scattering. Nuov Cim B 5, 1–10 (1971). https://doi.org/10.1007/BF02737704

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  • DOI: https://doi.org/10.1007/BF02737704

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