Lie algebra projectors and the kinematics of collective motions

Gulshani, P.

doi:10.1088/0305-4470/14/1/010

Cited by 7 publications

(15 citation statements)

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“…The derivation: (i) bypasses the usual method of canonical transformation of the space-fixed-frame particle coordinates to intrinsic and angular coordinates and, in this way, it avoids resorting to the use of redundant coordinates, (ii) is a natural generalization of the transformation of the motion of a single-particle system to radial and angular motions, where there is no requirement, on either the single or multi-particle system, that the intrinsic state be deformed but only that the intrinsic state be a zero eigenstate of the angular momentum operator, (iii) chooses an Euler angle for which the Coriolis-coupling term in the derived Schrödinger equation vanishes exactly without imposing any constraints on the particle coordinates or the intrinsic wavefunction, and (iv) determines the kinematic moment of inertia in the derived Schrödinger equation using the Euler angle-angular momentum operator commutation relation. These are the features of the microscopic model that differ from those used previously by other authors [15,53,59,60,62,66,67,[70][71][72][73][74][75][76][77][78]. Restricting to a rotational motion about a single axis simplifies the analysis of the rotational problem and presentation of the results, provides physical insight into the nature of the rotational motion of a multi-particle system in its simplest form, and facilitates future generalization to rotational motion in three dimensions.…”

Section: Introductionmentioning

confidence: 92%

“…At the coordinate-system origin, Eq. (73) shows that V rot s in Eq. (72) is singular, i.e., the quantum flow has a free-line vortex at the origin, with a quantized circulation given by (compare Eq.…”

Section: Uni-axial Rotational Motion Of Single-particle Quantum Fluidmentioning

confidence: 98%

“…(32) and hence would result in a new constraining equation for Φ. In other words, any choice of θ different from the rigid flow (for example, a rigid-plus-irrotational flow) would not decouple the intrinsic and rotational motions and hence would generate rotation-distortion in the intrinsic system (similar to that investigated in [23,[68][69][70][71][72][73][74]). Therefore, such as a choice would not be consistent with the rotational model prescription in Eq.…”

Section: Condition For Zero Coriolis-coupling Term or Adiabaticitymentioning

confidence: 99%

“…V rot s =h l Mr 2 (−z e y + y e z ) = ω s × r (72) where the single-particle angular velocity ω s in the x direction is defined by: (73) ω s is observed to be quantized in units ofh Mr 2 . 9 The expressions in Eqs.…”

Section: Uni-axial Rotational Motion Of Single-particle Quantum Fluidmentioning

confidence: 99%

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Derivation of microscopic uni-axial unified adiabatic Bohr–Mottelson rotational model

Gulshani

2010

Nuclear Physics A

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Section: Introductionmentioning

confidence: 92%

Section: Uni-axial Rotational Motion Of Single-particle Quantum Fluidmentioning

confidence: 98%

Section: Condition For Zero Coriolis-coupling Term or Adiabaticitymentioning

confidence: 99%

Section: Uni-axial Rotational Motion Of Single-particle Quantum Fluidmentioning

confidence: 99%

See 2 more Smart Citations

Derivation of microscopic uni-axial unified adiabatic Bohr–Mottelson rotational model

Gulshani

2010

Nuclear Physics A

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“…In this transformation, the coordinates x were constrained to make the quadrupole tensor diagonal in the principal-axis frame. This approach was generalized to microscopically describe rotational and vibrational motions of the quadrupole degrees of freedom, and introduced additional constraints on the coordinates x [9][10][11][12][13][14][15][16]. The constraints on the coordinates x imply that the coordinates x are not independent of each other, i.e., some of them are redundant.…”

Section: Introductionmentioning

confidence: 99%

A semiclassical, microscopic model for nuclear collective rotation

Gulshani

2006

Can. J. Phys.

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In this article, a semiclassical, microscopic model (dubbed SMRM) is derived to describe collective rotation in deformed nuclei. The SMRM is derived by transforming the time-dependent, multiparticle Schrödinger equation to a rotating frame whose axes are chosen to coincide with the principal axes of the expectation value of an arbitrary, secondrank, symmetric, tensor (nuclear shape) operatorˆ . This transformation circumvents the difficulty associated with the introduction of redundant particle coordinates in the Villars' transformation. The SMRM Schrödinger equation, which resembles the cranking model (CM) equation, is a time-dependent, time-reversal-invariant, nonlinear integro-differential equation. In this equation, the angular velocity is determined by the wave function and deformationrotation shear operators, and this introduces the nonlinearity in the equation. A variational method is proposed and justified to obtain: a stationary solution of the SMRM Schrödinger equation in the Rayleigh-Ritz Hartree-Fock particle-hole formalism, the rotational energy increment, and the associated moment of inertia. When exchange interaction terms are neglected or a separable interaction is used, the SMRM moment of inertia is shown to reduce to that given by the CM provided that a certain relationship exists between the moment of inertia and the expectation value ofˆ . However, the SMRM and CM wave functions are not the same (SMRM preserves and CM violates time-reversal invariance) implying that the calculated values of other parameters, including the moment of inertia at higher values of the angular momentum, may not be the same in the two models. In any case, the SMRM derives the CM moment of inertia from a microscopic, time-reversal invariant, nonlinear theory. PACS Nos.: 21.60.Ev, 21.60.Fw, 21.60.Jz Résumé : Nous développons ici un modèle microscopique semi-classique (SMRM) pour décrire les rotations collectives dans les noyaux déformés. Ce modèle est obtenu en transformant l'équation de Schrödinger à N corps dépendante du temps dans un référentiel tournant dont les axes sont les valeurs propres d'un opérateur tensoriel symétrique du deuxième ordreˆ créé pour décrire la forme du noyau. Cette façon de faire évite la difficulté associée à l'introduction de coordonnées redondantes de particules dans la transformation de Villars. L'équation SMRM de Schrödinger, qui ressemble au modèle du cranking (CM), est une équation intégro-différentielle dépendante du temps et invariante sous renversement du temps. Dans cette équation la vitesse angulaire est déterminée par la fonction d'onde et les opérateurs de déformation-stress sous rotation, ce qui introduit une non-linéarité. Nous proposons et et justifions une méthode variationnelle pour obtenir les éléments suivants : une solution stationnaire de l'équation SMRM de Schrödinger dans le formalisme particule-trou

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A microscopic derivation of nuclear collective rotation-vibration model and its application to nuclei

Gulshani¹

2016

AIP Conference Proceedings

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We develop in this article a microscopic version of the successful phenomenological hydrodynamic Bohr-Davydov-Faessler-Greiner (BDFG) model for the collective rotationvibration motion of a deformed nucleus. The model derivation is not limited to small oscillation amplitudes. The model generalizes the author's previous model to include interaction between collective oscillations in each pair of spatial directions, and to remove many of the previousmodel approximations. To derive the model, the nuclear Schrodinger equation is canonically transformed to collective coordinates and then linearized using a constrained variational method. The associated transformation constraints are imposed on the wavefunction and not on the particle co-ordinates. This approach yields four self-consistent, time-reversal invariant, cranking-type Schrodinger equations for the rotation-vibration and intrinsic motions, and a selfconsistency equation. To facilitate comparison with the BDFG model, simplify the solution of the equations, and gain physical insight, we restrict in this article the collective oscillations to only two space dimensions. For harmonic oscillator mean-field potentials, the equations are then solved in closed forms and applied to the ground-state rotational bands in some even-even light and rare-earth nuclei. The computed ground-state rotational band excitation energy, quadrupole moment and electric quadrupole transition probabilities are found to agree favourably with measured data and the results from mean-field, Sp(3,R), and SU(3) models. quantum numbers) arising from a full 3-D vibration model. This 2-D submodel of the microscopic collective rotation-vibration-intrinsic model is derived in Section 2. In section 3, we solve in closed forms the model governing equations, obtain expressions for the model coupling parameters and the excitation energy for the members of the rotational bands, and solve the algebraic equations for the coupling parameters. In Section 4, we discuss how the model coupling parameters affect the nuclear energy, deformation, and collective and intrinsic motions, and generate self-consistency and feedback mechanisms among these motions. In Section 5, we compare the model with MVI, BDFG, and mean-field models including the cranking models. In Section 6, we use the model to predict the excitation energy, moment of inertia, and quadrupole moment, and transition probability rates for the members of the ground-state rotational band in light and rare-earth nuclei. The article is concluded in Section 7. Derivation of microscopic collective-intrinsic modelThe model is derived by transforming the nuclear Schrodinger equation to collective Euler angles and vibration co-ordinates with constraints imposed on the wavefunction. The transformation is performed in two steps.In the first step, we use the microscopic rotational model derived in [40,41] and described briefly here. We use the rotational-model product wavefunction:where the coefficients A, B, C, etc. and the moment of inertia J are functions of int...

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Lie algebra projectors and the kinematics of collective motions

Cited by 7 publications

References 15 publications

Derivation of microscopic uni-axial unified adiabatic Bohr–Mottelson rotational model

Derivation of microscopic uni-axial unified adiabatic Bohr–Mottelson rotational model

A semiclassical, microscopic model for nuclear collective rotation

A microscopic derivation of nuclear collective rotation-vibration model and its application to nuclei

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