Abstract
In the previous chapters, we thoroughly investigated the simplest atom of all—the hydrogen atom. We have found that, owing to the existence of a sufficient number of integrals of motion, one can solve its energy spectrum exactly. Unfortunately, one cannot determine exactly the spectrum of helium nor of any of the heavier atoms. Nevertheless, we know from Chap. 2 that by means of the variational method we may approximate the solution to any desirable accuracy. We will show that the antisymmetry of the wave function with respect to the exchange of the electrons leads to a so-called exchange interaction. Accounting for this interaction subsequently leads to a qualitatively correct result even when only one two-electron configuration considered. This estimate can be further systematically improved by the inclusion of additional electron configurations. We will see that the symmetries of the helium atom, i.e., the existence of operators commuting with the Hamiltonian, substantially decrease the amount of configurations one needs to include in the calculations.
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Notes
- 1.
Precisely speaking, we need to consider all other internal characteristics of the particle as well. Although electrons possess no other internal properties but their spin, one would have to include also the color when discussing quarks of a given flavor, for example.
- 2.
As we demonstrated in the previous chapter, a sum of two integer angular momenta gives rise to another integer angular momentum. We also showed there that both the magnetic quantum number m and the orbital quantum number l can evaluate only to integer numbers.
- 3.
The derivation presented henceforth is inspired by the one in [7].
- 4.
It holds true for the δ-function that ∫δ (3)(r)d3r = 1. After transformation of the differential to spherical coordinates, we have d3r = r 2drdϑsinϑdφ. Since we have ∫δ(x)dx = 1 for the one-dimensional integrals, it must obviously be \(\delta ^{(3)}(\mathbf{r}) ={ 1 \over r^{2}} \delta (r){1 \over \sin \vartheta } \delta (\vartheta )\delta (\varphi )\).
- 5.
- 6.
- 7.
- 8.
So simple that the even authors managed it.
- 9.
Confront with the fine structure of positronium, see Exercise 10 in Sect. 4.4.9.
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Zamastil, J., Benda, J. (2017). The Helium Atom. In: Quantum Mechanics and Electrodynamics. Springer, Cham. https://doi.org/10.1007/978-3-319-65780-6_5
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DOI: https://doi.org/10.1007/978-3-319-65780-6_5
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