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Atomic Physics. Minimum Calculations

Part of the Lecture Notes in Physics book series (LNP, volume 747)

Let us consider an electron state in a hydrogen atom characterized by quantum numbers n ≫1 and l = n - 1, which corresponds, as it is known, to circular orbits. Is the radial wave function Rn,n-1(r) of this state semiclassical? At first sight, no. Indeed, the radial quantum number nr = n - l - 1 is in this case in no way large, but is equal to zero, so that the wave function Rn,n-1(r) has no nodes at all.

Keywords

Wave Function Angular Momentum Total Angular Momentum Atomic Physic Bohr Radius 
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References

  1. We follow in this section the work by I.B. Khriplovich, D.V. Matvienko (2007).Google Scholar
  2. The analysis of numerical values of transition probabilities (H. Bethe, E. Salpeter, 1957) has demonstrated that even for n and l comparable with unity, i.e., in a nonclassical situation, radiation with Δl = 1 is considerably more probable than radiation with Δl = 1.Google Scholar
  3. See: L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields . §70.Google Scholar
  4. See: L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields . §75.Google Scholar
  5. See: L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields . §70.Google Scholar
  6. See: L.D. Landau and E.M. Lifshitz, Quantum Mechanics. §36.Google Scholar
  7. From now on in this section we use atomic units.Google Scholar
  8. We note that the magnetic dipole transition 1S 0 3P 1 is forbidden. As is known, the magnetic moment operator is proportional to Lz + 2Sz. Obviously, neither Lz nor Sz can transform 1S 0 into 3P, as well as 1D2 into 3P. Thus, in the given electron configuration p2 only the following M1 transitions are possible: 3P 2 3P 1 3P 1 3P 0.Google Scholar
  9. It was given by I.B. Khriplovich and G.Yu. Ruban (2004).Google Scholar
  10. See: I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series and products. New York, Academic Press, 1994.We use formula 9.327.1 from this book for μ = 1/2, omit in this formula the overall factor 1/G (1/2 +μ − λ) inessential for our purpose (the notations here are the same as in the quoted book) and rewrite the series at ln z in a compact form, as a confluent hypergeometric function.Google Scholar
  11. See for instance: L.D. Landau and E.M. Lifshitz, Quantum Mechanics. §112, Problem 1.Google Scholar
  12. See: I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series and products. New York, Academic Press, 1994.Google Scholar
  13. See for instance: L.D. Landau and E.M. Lifshitz, Quantum Mechanics. §112, Problem 1.Google Scholar

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© Springer 2008

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