Il Nuovo Cimento A (1965-1970)

, Volume 52, Issue 1, pp 274–285 | Cite as

On the reduced-mass corrections to the fine structure of π-mesic atoms

  • M. Ciafaloni


The problem of mass corrections to the fine structure formula is examined for a spin 0-spin 1/2 atom with arbitrary masses of the bound particles. An approximate, purely electrodynamic, Bethe-Salpeter equation is discussed in this case, and from it a three-dimensional equation is derived which gives correctly all terms of order α2 ryd in the binding energy. It is found that, except forS-waves, the reduced mass fine structure formula is exactly true for the spinless particle, while for the spin-1/2 particle it is modified by terms of second order in the mass ratio.


Mass Correction Coulomb Gauge Spinless Particle Ladder Approximation Arbitrary Mass 
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О поправках приведенной массы к тонкой структуры п-меэоатомов


Исследуется проблема массовых поправок к формуле тонкой структуры для атома со спином 0-спином 1/2 и с произвольными массами связанных частиц. В этом случае обсуждается приближенное чисто электродинамическое уравнение Бете-Салпетера, и из него выводится трехмерное уравнение, которое корректно определяет все члены, порядка α2 ryd в энергии свяэи. Найдено, что за исключениемS-волн приведенная массовая формула тонкой структуры является точной для бесспиновой частицы, тогда как для частици со спином 1/2 эта формула видоизменяется за счет членов второго порядка по отношению масс.


Si esamina qui il problema delle correzioni di massa alla struttura fine di atomi costituiti da una particella di spin 0 e da una di spin 1/2 aventi massa qualsiasi. Viene discussa in questo caso una equazione di Bethe-Salpeter approssimata di puro tipo elettrodinamico e se ne ricava una equazione tridimensionale che dà correttamente tutti i termini della energia di legame dell’ordine α2 ryd. Si trova infine che la formula di struttura fine con massa ridotta è esatta per la particella di spin 0, eccetto che per l’ondaS, mentre quella per la particella di spin 1/2 è modificata da termini del secondo ordine nel rapporto delle masse.


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  1. (1).
    Corrections of this order of magnitude have been considered in calculating the theoretical level shift in a precise pion-mass measurement byR. E. Shafer, K. M. Crowe andD. A. Jenkins:Phys. Rev. Lett.,14, 923 (1965). We are grateful to Prof.E. Segrè for correspondence on this subject.ADSCrossRefGoogle Scholar
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    For computational convenience we shall separate the c.m. co-ordinates in the unconventional wayp 1=p, p2=P−p. Therefore, the singularities or ψ(P,p 0) in thep 0-plane are displaced by an amountmE/(m+M) with respect to those given byG. C. Wick:Phys. Rev.,96, 1125 (1954). However, no Wick’s rotation will be performed here. In the following we shall also assume that the nucleus has unit charge, because theZ-dependence of the fine structure formula is obvious.ADSMathSciNetCrossRefGoogle Scholar
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    The symmetrization is not necessary, but it makes clearer the computation.Google Scholar
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    We notice that we are expanding inv/c and not inm/M. This expansion is only formal in character and is correctly used in first-order perturbation theory for the binding energy. As for the wave function, eq. (15) must be handled more carefully (4,6),E. E. Salpeter:Phys. Rev.,87, 328 (1952),H. A. Bethe andE. E. Salpeter:Quantum Mechanics of One- and Two- Electron Systems, inEncyclopedia of Physics, Sect.20α and42.Google Scholar
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    Cf.H. A. Bethe andE. E. Salpeter: ref. (6). Sect.42.ADSMathSciNetCrossRefGoogle Scholar
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    For the spin-1/2 case there are also some numerically small corrections of order α(m/M) f.s. (4), (8)R. Karplus andA. Klein:Phys. Rev.,87, 848 (1952).ADSCrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica 1967

Authors and Affiliations

  • M. Ciafaloni
    • 1
    • 2
  1. 1.Scuola Normale SuperiorePisaItalia
  2. 2.Sezione di PisaIstituto Nazionale di Fisica NuclearePisaItalia

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