Abstract
This chapter is dedicated to introduce the formalism of tensorial algebra which is used throughout the book. It also contains some mathematical proofs and further complementary material needed for deepening some of the topics covered in the book.
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- 1.
This law is not universally attributed to Gilbert. In fact, this law was discovered experimentally by Coulomb himself and could therefore be rightly called the “second Coulomb’s law”. William Gilbert (1564–1603) was an English physician who lived well before Coulomb. He is remembered for his studies on the terrestrial magnetism and by the fact that he realised that the magnetic force should increase with decreasing distance.
- 2.
These experiments were made possible thanks to the discovery of the electric battery by Alessandro Volta.
- 3.
With the introduction of capacity and inductance, together with their units, the Farad (F) and the Henry (H), the units in which μ 0 and ϵ 0 are expressed are, respectively, H m−1 and F m−1.
- 4.
This convention is not universally accepted. Some authors prefer to assume γ=1 also in the International System. In this case Ampère’s principle of equivalence is written as μ=iσ n while in Gilbert’s law the factor μ 0 is to appear in the numerator rather than in the denominator.
- 5.
The symbol [M L ,M S ] means the sum of the diagonal matrix elements of the Hamiltonian of the spin-orbit interaction over all the states Ψ A for which M L and M S are the eigenvalues of L z and S z , respectively. Similarly, the notation \((m_{1}^{\pm},m_{2}^{\pm}) \) is used to denote the diagonal matrix element of the same Hamiltonian on the state where the electron 1 has the magnetic quantum number m 1 and spin quantum number +1/2 or −1/2 and, similarly, the electron 2 has the magnetic quantum number m 2 and spin quantum number +1/2 or −1/2.
- 6.
We note that even if there are electrons in the closed subshells they do not produce any contribution to the equation.
- 7.
The quantity (1+,1−,0+,−1+), relative to the configuration p 4, is obtained from the corresponding quantity (0+,1+), relative to the configuration p 2, taking the “complementary” of the latter, i.e. (−1+,−1−,0−,1−), and then changing sign to all the values of m and m s .
- 8.
We note that in Chap. 11 we only introduced the diagonal elements of the density matrix, denoted for simplicity by the symbol ρ α instead of ρ αα .
- 9.
The dipole approximation is appropriate when considering the interaction between radiation and electrons that are bound in an atom. For free electrons, described by eigenfunctions of the type of a plane wave, the approximation cannot be applied.
- 10.
Recall that the first average, 〈PP ∗〉, has a similar meaning with respect to the spin states of the electron.
- 11.
\(\sigma= {2 \pi^{5} k_{\mathrm{B}}^{4} \over15 h^{3} c^{2}}\), \(a={ 4 \sigma\over c} = {8 \pi^{5} k_{\mathrm{B}}^{4} \over15 h^{3} c^{3}}\).
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© 2014 Springer-Verlag Italia
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Landi Degl’Innocenti, E. (2014). Appendix. In: Atomic Spectroscopy and Radiative Processes. UNITEXT for Physics. Springer, Milano. https://doi.org/10.1007/978-88-470-2808-1_16
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DOI: https://doi.org/10.1007/978-88-470-2808-1_16
Publisher Name: Springer, Milano
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