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Springer series on atomic, optical, and plasma physics

Irreducible Components of the Density Matrix

Karl Blum

Published Oct 4, 2011• 2 citations
Publisher
DOI
10.1007/978-3-642-20561-3_4

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CONCERNING THE SYMMETRY PROPERTIES OF THE REDUCED DENSITY MATRIX AND THE NATURAL SPIN ORBITALS AND SPIN GEMINALS (THE NATURAL ONE- AND TWO-ELECTRON FUNCTIONS)

Werner Kutzelnigg

The density operator (density matrix) of a quantum mechenical system can be decomposed into operators which transform as irreducible representations of the symmetry group in coordinate and spin space. Each of these components has a physical meaning connected with the expectation values of certain operators. The reduced density matrices can be decomposed in a completely analogous way. The symmetry properties of the total wave function give rise to degeneracies of the eigenvalues of the reduced density matrices. These degeneracies can be removed by requiring that the natural spin orbitals (NSO, defined as the eigenfunctions of the first-order density matrix), as well as the natural spin geminals (NSG, the eigenfunctions of the second-order density matrix) and their spinless counterparts transform as irreducible representations of the symmetry group and are eigenfunctions of S/sup 2/ and S/sub z/. In many cases this requirement is compatible with the original definition of the NSO, the NSG, etc., e.g., when there is no spatial degeneracy of the total wave function and when the Z- component of the total spin vanishes. When these conditions are not fulfilled an alternative definition of the NSO and the NSG is proposed. (auth)

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Relaxation and generation of hyperfine coherences in an optically pumped heavy-alkali-metal vapor undergoing double-resonance excitation

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An extension of a theory of the relaxation, generation, and detection of Zeeman components in the presence of a weak static magnetic field to experiments where hyperfine coherences are also involved is presented. Our study focuses on the geometrical aspects of the optical-microwave double-resonance experiments without considering the transition-profile related phenomena. We determine the coupled evolution equations of the Zeeman and hyperfine components of the atomic density matrix in the irreducible formalism and calculate the stationary solutions to these equations. \textcopyright{} 1996 The American Physical Society.

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Stephen J. Gustafson, Israel Michael Sigal

Universitext•Jan 1, 2003•0 citations
The density operator (density matrix) of a quantum mechenical system can be decomposed into operators which transform as irreducible representations of the symmetry group in coordinate and spin space. Each of these components has a physical meaning connected with the expectation values of certain operators. The reduced density matrices can be decomposed in a completely analogous way. The symmetry properties of the total wave function give rise to degeneracies of the eigenvalues of the reduced density matrices. These degeneracies can be removed by requiring that the natural spin orbitals (NSO, defined as the eigenfunctions of the first-order density matrix), as well as the natural spin geminals (NSG, the eigenfunctions of the second-order density matrix) and their spinless counterparts transform as irreducible representations of the symmetry group and are eigenfunctions of S/sup 2/ and S/sub z/. In many cases this requirement is compatible with the original definition of the NSO, the NSG, etc., e.g., when there is no spatial degeneracy of the total wave function and when the Z- component of the total spin vanishes. When these conditions are not fulfilled an alternative definition of the NSO and the NSG is proposed. (auth)
An extension of a theory of the relaxation, generation, and detection of Zeeman components in the presence of a weak static magnetic field to experiments where hyperfine coherences are also involved is presented. Our study focuses on the geometrical aspects of the optical-microwave double-resonance experiments without considering the transition-profile related phenomena. We determine the coupled evolution equations of the Zeeman and hyperfine components of the atomic density matrix in the irreducible formalism and calculate the stationary solutions to these equations. \textcopyright{} 1996 The American Physical Society.