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Canadian Journal of Physics

Transcription de la quasi-particule dans l'espace ideal et la méthode des bosons

P. A. Simard

Published Jan 1, 1969• 6 citations
Publisher

Abstract

A transcription of the quasi-particle into the ideal space is written in such a way that all the terms which contribute for the three quasi-particles system in QTD approximation can be found. Canonical transformations between the ideal quasi-particles are studied. An application to one shell is done with the result that the boson method gives exactly the same spectra as the Kuo method.

DOI
10.1139/p69-013

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I) i) Description dynamique de systèmes quantiques par la méthode fonctionnelle et la méthode du potentiel effectif, ii) Résolution d'une classe de potentiels quasi-exactement solubles

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Outre l'etude d'une classe de potentiels quasi-exactement soluble, ce travail contient des resultats originaux obtenus dans le cadre des integrales de parcours : 1) le propagateur relatif a une particule confinee dans un secteur a ete obtenu dans le formalisme de Feynman et dans l'espace des phases, la particule etant soumise a un potentiel harmonique et a un potentiel coulombien. Il a ete etabli que pour certaines valeurs de l'angle d'ouverture, le propagateur est la somme de termes correspondant aux contributions de tous les parcours classiques. La singularite coulombienne engendrant l'effondrement du propagateur, a pu etre rejetee a l'infini moyennant une transformation spatio-temporelle, ramenant ainsi le probleme a celui, bien connu, du potentiel de Poschel-Teller modifie. 2. La ressemblance formelle existant entre les integrales de parcours et les integrales de Wiener, a permis de resoudre de nombreux problemes de la physique statistique. Nous avons ainsi remplace dans nos calculs, le potentiel quantique de depart, par un potentiel effectif classique harmonique. Cette approximation nous a permis d'etudier, dans divers systemes de coordonnees, l'energie libre et la densite de particules d'un systeme soumis a des familles de potentiels de type coulombien perturbe et oscillateur anharmonique, en tenant compte de l'isotropie, ou de l'anisotropie de la pulsation du potentiel effectif classique. 3. Le propagateur relatif au lagrangien quadratique en vitesse a pu etre obtenu par la methode polygonale, en exploitant la forme doublement quadratique du lagrangien, qui a permis d'ecrire l'action sous forme matricielle, et d'obtenir, par integrations successives, la forme habituelle du propagateur

•Jan 1, 1996•0 citations
A transcription of the quasi-particle into the ideal space is written in such a way that all the terms which contribute for the three quasi-particles system in QTD approximation can be found. Canonical transformations between the ideal quasi-particles are studied. An application to one shell is done with the result that the boson method gives exactly the same spectra as the Kuo method.
The transcription of the quasi-particle in the ideal space has been studied in such a way that all terms in the Hamiltonian in the QTD approximation for the system of four quasi-particles can be found. It is found that an infinity of solutions exists which verify the commutator {α, α+} while yielding correct Hamiltonian matrix elements. Finally, particular solutions which do not verify this commutator are found. Their particular invariance properties under canonical transformations make them relatively easy to obtain. Nonphysical states are eliminated by these transcriptions and all the physical states appear properly antisymmetrized.
An antisymmetric basis has been set up in the ideal space as a combination of one boson and one ideal quasi-particle state; in this way a three quasi-particle system may be studied. The knowledge of solutions of linearized equations for two quasi-particles is a requisite for the setting up of this basis. The exclusion principle is thus introduced correctly. The spurious states coming from the non-conservation of the number of particles are separated out and can be rejected easily. The method can be used only in QTD approximation. The one-shell case is discussed as a verification of the method.
A new method treating the interplay of pairing and intrinsic modes of excitation is proposed. In this method, the pairing mode associated with the J = 0-coupled nucleon pairs is represented by pairing bosons and the intrinsic mode characterized by the seniority quantum number is explicitly treated by ideal quasi-particle operators. We obtain a closed expression of the single-nucleon operator in terms of pairing bosons and ideal quasi-particles. As a simple illustration, the superconducting system is treated by introducing the coherent state of pairing bosons and the relation to the Bogoliubov transformation is discussed. The relation between this method and the canonical transformation method with auxiliary variables is also clarified.
Outre l'etude d'une classe de potentiels quasi-exactement soluble, ce travail contient des resultats originaux obtenus dans le cadre des integrales de parcours : 1) le propagateur relatif a une particule confinee dans un secteur a ete obtenu dans le formalisme de Feynman et dans l'espace des phases, la particule etant soumise a un potentiel harmonique et a un potentiel coulombien. Il a ete etabli que pour certaines valeurs de l'angle d'ouverture, le propagateur est la somme de termes correspondant aux contributions de tous les parcours classiques. La singularite coulombienne engendrant l'effondrement du propagateur, a pu etre rejetee a l'infini moyennant une transformation spatio-temporelle, ramenant ainsi le probleme a celui, bien connu, du potentiel de Poschel-Teller modifie. 2. La ressemblance formelle existant entre les integrales de parcours et les integrales de Wiener, a permis de resoudre de nombreux problemes de la physique statistique. Nous avons ainsi remplace dans nos calculs, le potentiel quantique de depart, par un potentiel effectif classique harmonique. Cette approximation nous a permis d'etudier, dans divers systemes de coordonnees, l'energie libre et la densite de particules d'un systeme soumis a des familles de potentiels de type coulombien perturbe et oscillateur anharmonique, en tenant compte de l'isotropie, ou de l'anisotropie de la pulsation du potentiel effectif classique. 3. Le propagateur relatif au lagrangien quadratique en vitesse a pu etre obtenu par la methode polygonale, en exploitant la forme doublement quadratique du lagrangien, qui a permis d'ecrire l'action sous forme matricielle, et d'obtenir, par integrations successives, la forme habituelle du propagateur