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Generalized Dirac identities and explicit relations between the permutational symmetry and the spin operators for systems of identical particles
Author(s) -
Katriel J.,
Paldus J.,
Pauncz R.
Publication year - 1985
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.560280203
Subject(s) - eigenvalues and eigenvectors , operator (biology) , quantum mechanics , mathematical physics , mathematics , symmetry in quantum mechanics , irreducible representation , spin (aerodynamics) , physics , pure mathematics , theoretical physics , algebra over a field , relativistic quantum mechanics , quantum , chemistry , biochemistry , repressor , transcription factor , quantum dynamics , gene , thermodynamics
The well known one‐to‐one correspondence between the eigenstates of the total spin for a system of spin‐½ particles and irreducible representations of the symmetric group with up to two rows in the Young shape is the basis of interesting formal developments in quantum chemistry and in the theory of magnetism. As an explicit manifestation of this correspondence the class operators of the symmetric group are demonstrated to be expressible in terms of the total spin operator. This correspondence does not hold for higher elementary spins. The extension to arbitrary spin is investigated using Schrödinger's generalization of the Dirac identity, which expresses the transpositions in terms of two‐particle spin operators. It is shown that additional operators, which for σ = ½ reduce to the total spin operator, are needed for a complete classification. Some aspects of the formalism are developed in detail for σ = 1. In this case a classification identical with that provided by the irreducible representations of the symmetric group is obtained in terms of the eigenstates of two commuting operators, one of which is the total spin operator.