Theoretical Kaleidoscope pp 23-44 | Cite as

# Atomic Physics. Minimum Calculations

Chapter

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 *R**n,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 *R**n,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

- We follow in this section the work by
*I.B. Khriplovich, D.V. Matvienko*(2007).Google Scholar - 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 - From now on in this section we use atomic units.Google Scholar
- We note that the magnetic dipole transition
^{1}*S*_{0}*→*^{3}*P*_{1}is forbidden. As is known, the magnetic moment operator is proportional to*L*_{z}+^{2}*S*_{z}. Obviously, neither*L*_{z}nor*S*_{z}can transform^{1}*S*_{0}into^{3}*P*, as well as^{1}*D*_{2}into^{3}*P*. Thus, in the given electron configuration*p*^{2}only the following*M*1 transitions are possible:^{3}*P*_{2}*→*^{3}*P*_{1}^{3}*P*_{1}*→*^{3}*P*_{0}.Google Scholar - 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 - See: I.S. Gradshteyn and I.M. Ryzhik,
*Table of integrals, series and products*. New York, Academic Press, 1994.Google Scholar

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