On formulas in closed form for Van Vleck expansions

Abstract

AbstractsCanonical transformations have been widely used to simplify Hamiltonians and other operators. In molecular and in solid state theory, the so-called Van Vleck expansion is usually employed for this purpose while in theories of particles interacting with fields a combination of canonical transformations in closed form with Van Vleck type expansions has been found effective. For some of the transformations used in applications formulas in closed form are well known. I t will be shown here that such formulas can be derived whenever the transformation function is bilinear in the canonical variables, and further that the use of matrix operators makes it possible to simplify these derivations substantially. 'The Cayley-Hamilton theorem is then used to express the expansions for the matrix operators in closed form. The number of separate operator terms appearing in the formulas thus obtained is the same as the rank of the matrices used. To calculate the coefficients of these operator terms a new type of special functions is introduced. The resulting linear canonical transformations include generalized rotations in both ordinary and phase-space. Explicit results have been obtained for several two-to .four-dimensional problems.Des transformations canoniques ont Ct C employtes frkquemment pour simplifier 1'Hamiltonien et d'autres optrateurs. Dans les thtories moltculaire et de l'ttat solide le dCveloppement de Van Vleck est gtneralement utilisC B cette fin, tandis que dans les thtories des particules intbragissant avec des champs on a prtfCrt une combinaison de transformations canoniques dans une forme close avec un type de dtveloppement de Van Vleck. Pour quelques-unes des transformations employkes dans les applications des formules en forme close sont bien connues. I1 est dtmontrC ici, que de telles formules peuvent Ctre obtenues quand la fonction de transformation est bilintaire dans les variables canoniques, et aussi qu'B l'aide des optrateurs matriciels il est possible de simplifier considCrablement les dkductions correspondantes. Avec le thCoreme de Cayley-Hamilton on peut exprimer les dtveloppements pour les optrateurs matriciels dans une forme close. Le nombre de termes d'optrateurs distincts ainsi obtenu est le mCme que le rang des matrices utilistes. Pour calculer les coefficients de ces termes d'opCrateurs on introduit un nouveau type de fonctions spkciales. Les transformations canoniques lintaires, qui en rCsultent comprennent des rotations gtntralistes dans l'espace ordinaire et dans l'espace des phases. Des resultats explicites ont t t t obtenus pour plusieurs problemes deux jusqu'8 quatre dimensions.Kanonische Transformationen sind oft verwendet worden um Hamilton-und andere Operatoren zu vereinfachen. In Molekul-und Festkorpertheorie wird gewohnlich die SOgenannte Van Vleck-Entwicklung zu diesem Zweck angewandt, wahrend in Theorien

Novel methodology for systematically constructing global effective models from ab initio-based surfaces: A new insight into high-resolution molecular spectra analysis

“…The oldest approach is known as the Rayleigh-Schrödinger perturbation method 74 which has been generalized to tackle the case of quasi-degenerate states 75,76 . Another approach was suggested by Van Vleck 77 and is known as canonical perturbation theory or contact transformation (CT) method [78][79][80][81][82][83][84][85][86][87] falling in the domain of the Lie algebra 88 . The basic idea is to a apply a series of unitary transformations T CT to the nuclear motion Hamiltonian in order to transform it to a new, block-diagonal representation.…”

Mnemonic symbols in commutator perturbation method

“…CPM is the Lie‐algebraic version of the Van Vleck–Primas perturbation method. Van Vleck 6 introduced the use of the contact transformations in quantum mechanics 7–25. This method is a quantum analog of the classical Lie‐algebraic perturbation theory 26.…”