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International Journal of Quantum Chemistry

Theoretical and computational aspects of extended wave functions

Ramon Carbó‐Dorca, Jacek Karwowski

Published Jan 1, 2001• 4 citations
Publisher

Abstract

Abstract Relations between the extended wave functions defined by Carbó‐Dorca in connection with studies on quantum similarity problems and the wave functions of Schrödinger, Lévy‐Leblond, and Dirac equations are investigated. In particular, it is pointed out that solutions of the Lévy‐Leblond equation are equal to the appropriate extensions of the corresponding Schrödinger wave functions. The quantum mechanical continuity equation is applied to analyze the physical meaning of the extended wave functions. Finally, usefulness of the concept of the extended wave function in designing relativistic variational calculations in kinetically balanced basis sets is briefly discussed. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem 84: 331–337, 2001

DOI
10.1002/qua.1088

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Abstract Relations between the extended wave functions defined by Carbó‐Dorca in connection with studies on quantum similarity problems and the wave functions of Schrödinger, Lévy‐Leblond, and Dirac equations are investigated. In particular, it is pointed out that solutions of the Lévy‐Leblond equation are equal to the appropriate extensions of the corresponding Schrödinger wave functions. The quantum mechanical continuity equation is applied to analyze the physical meaning of the extended wave functions. Finally, usefulness of the concept of the extended wave function in designing relativistic variational calculations in kinetically balanced basis sets is briefly discussed. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem 84: 331–337, 2001
Abstract Attempts to put classical and quantum mechanics on an equal footing gave rise to establishing of many known phase‐space representations of quantum mechanics. The class of self‐dual representations occupies a distinguished position among them, as they could be endowed with a Hilbert space structure closely resembling the standard picture of quantum mechanics. Using a phase‐space variational principle and the concept of a phase‐space wave function, we established a phase‐space version of the density functional theory parallel to the original Hohenberg–Kohn formalism and the constrained‐search approach by Levy and Lieb. © 1995 John Wiley & Sons, Inc.
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New fundamental (Galilean invariant) dynamical equations for four- component spinor wavefunctions are derived by using a different phase function than the Schrodinger phase function. The derived equations are called the generalized Levy-Leblond and generalized Schrodinger equa- tions because these equations reduce to the standard Levy-Leblond and Schrodinger equations when the Schrodinger phase function is used. The generalized and standard equations describe the same free elementary particles with spin 1/2, however, masses of the particles described by these equations are not the same as the generalized equations automati- cally account for particles with larger masses. This has important phys- ical implications as it shows that elementary particles with the same physical properties but different masses can result from using phase functions different than the Schrodinger phase function. Therefore, the main conclusion of this paper is that the existence of the three currently known families of elementary particles in the Standard Model of particle physics can be theoretically accounted for by choosing different phase functions.
Abstract The methods of quantum chemistry and solid state theory to solve the many‐body problem are reviewed. We start with the definitions of reduced density matrices, their properties (contraction sum rules, spectral resolutions, cumulant expansion, and N‐representability), and their determining equations (contracted Schrödinger equations), and we summarize recent extensions and generalizations of the traditional quantum chemical methods, of the density functional theory, and of the quasiparticle theory: from finite to extended systems (incremental method), from density‐to‐density matrix (density matrix functional theory), from weak to strong correlation (dynamical mean field theory), from homogeneous (Kimball–Overhauser approach) to inhomogeneous and finite systems. Measures of the correlation strength are discussed. The cumulant two‐body reduced density matrix proves to be a key quantity. Its spectral resolution contains geminals, possibly the solutions of an approximate effective two‐body equation, and we sketch the idea of how its contraction sum rule can be used for a variational treatment. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004