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The short-wavelength approximation to the Schrödinger equation

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Il Nuovo Cimento (1955-1965)

Summary

We study the short-wavelength approximation to the Schrödinger equation in the presence of one and two turning points. Two methods are presented which allow the evaluation of the higher corrections to the elementary W.K.B. approximation. The first method, the « real method », employs higher transcendental functions, the second one uses complex integration techniques.

Riassunto

Si studia l’approssimazione di piccole lunghezze d’onda per l’equazione di Schrödinger con uno o due punti di transizione. Vengono presentati due metodi che permettono la valutazione delle correzioni alla semplice approssimazione W.K.B. Il primo metodo, il « metodo real » fa uso di funzioni speciali, mentre il secondo si serve di tecniche di integrazione complessa.

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References

  1. This technique was first applied to a Schördinger-like equation byH. Jeffreys:Proc. Math. Soc. London,23, 428 (1923) and to quantum mechanics byL. Brillouin:Compt. Rend.,183, 24 (1926);G. Wentzel (ref. (3)) andH. A. Kramers:Zeits. f. Phys.,39, 828 (1926). For a mathematical discussion of the argument seeA. Erdelyi:Asymptotic Expansions (New York, 1956), andF. Tricomi:Equazioni differenziali, 3rd ed. (Torino, 1960). In particular the paper we discuss in the following is byR. E. Langer:Trans. Amer. Math. Soc.,67, 461 (1949).

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Author notes

  1. G. Furlan

    Present address: Istituto di Fisica dell’Università, Trieste

Authors and Affiliations

  1. Istituto di Fisica dell’Università, Bologna

    L. Bertocchi

  2. Istituto di Fisica dell’Università, Torino

    S. Fubini

  3. CERN, Geneva

    S. Fubini & G. Furlan

Authors
  1. L. Bertocchi
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  2. S. Fubini
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  3. G. Furlan
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Additional information

The research reported in this document has been sponsored in part by the Air Force Office of Scientific Research OAR through the European Office, Aerospace Research, United States Air Force.

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Bertocchi, L., Fubini, S. & Furlan, G. The short-wavelength approximation to the Schrödinger equation. Nuovo Cim 35, 599–632 (1965). https://doi.org/10.1007/BF02735340

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