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Some comments on the treatment of symmetry properties in perturbation theory
Author(s) -
LÖwdin PerOlov
Publication year - 2009
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560020715
Subject(s) - linear subspace , perturbation (astronomy) , subspace topology , hamiltonian (control theory) , hilbert space , mathematics , mathematical physics , equations of motion , formalism (music) , physics , quantum mechanics , classical mechanics , mathematical analysis , pure mathematics , art , mathematical optimization , musical , visual arts
The treatment of constants of motion and symmetry properties in partitioning technique and perturbation theory is briefly discussed. A constant of motion may be characterized by a set of projection operators { Q Q } forming a resolution of the identity and leading to a splitting of the Hilbert space into orthogonal and non‐interacting subspaces. It is shown that, in the partitioning technique, it is sufficient to consider only one such subspace at a time, which is a considerable simplification. The treatment may be extended to perturbation theory ℋ = ℋ 0 + V , and it is shown that the unperturbed Hamiltonian ℋ 0 enters the formalism only in the form of the reduced resolvents occurring in ordinary perturbation theory. It is emphasized that, even if the infinite‐order expressions are unique, the finite‐order expansions in the perturbation V are ambiguous, and the relation between various forms is studied.