Summary
SU
2⊗SU
2, being the universal covering group ofSO
4, is equally well suited to be the dynamical invariance group of the hydrogen atom. A representation
of the corresponding group ring is used to classify completely the eigenfunctions and to represent every operator commuting with the Hamiltonian as a unique linear combination of some special ring elements. An obvious extension of these ring elements in question yields a special basis of the linear operator space ℒ(ℋ’) containing the ring of bounded operators ℒ(ℋ) and
as subspaces. This operator basis will serve for the definition of a further basis of ℒ(ℋ’) whose elements are irreducible tensor operators with respect toSU
2⊗SU
2.
Riassunto
Il gruppoSU
2⊗SU
2, essendo il gruppo universale di ricoprimento diSO
4, si presta bene a rappresentare il gruppo di invarianza dinamica dell’atomo di idrogeno. Si usa una rappresentazione
del corrispondente anello di gruppi per classificare completamente le autofunzioni e rappresentare ciascuno degli operatori che commutano con l’hamiltoniano come unica combinazione lineare di certi elementi particolari dell’anello. Con un’estensione ovvia di questi elementi dell’anello si ottiene una base speciale dello spazio degli operatori limitati ℒ(ℋ’) che contiene come sottospazi gli anelli di operatori limitati ℒ(ℋ) e
. Per mezzo di questa base di operatori si definisce un’ulteriore base di ℒ(ℋ’), i cui elementi sono operatori tensoriali non riducibili nei confronti diSU
2⊗SU
2.
Резюме
SU
2⊗SU
2, которая является универсальной оболоченой группой дляSO
4, представляет динамическую инвариантную группу для атома водорода. Представление
соответствующего группового кольца используется для классификации собственных функций и представления любого оператора, коммутирующего с Гамильтонианом, в виде единственной линейной комбинации некоторых специальных кольцевых элементов. Очевидное расширение рассматриваемых элементов кольца дает специальный базис для пространства линейных операторов ℒ(ℋ’), содержазего кольцо ограниченных оператороб ℒ(ℋ) и
в качестве подпространств. Этот операторный базис служит для определения дополнительного базиса ℒ(ℋ’), чьи элементы представляют неприводимые тензорные операторы относительноSU
2⊗SU
2.
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Dirl, R. The group ring of the dynamical invariance group of the hydrogen atom.—I. Nuovo Cim B 23, 417–440 (1974). https://doi.org/10.1007/BF02723648
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DOI: https://doi.org/10.1007/BF02723648