Descriptions of operators in quantum mechanics

The problem of expressing a general dynamical variable in quantum mechanics as a function of a primitive set of operators is studied from several points of view. In the context of the Heisenberg commutation relation, the Weyl representation for operators and a new Fourier-Mellin representation are related to the Heisenberg group and the groupSL(2,R) respectively. The description of unitary transformations via generating functions is analysed in detail. The relation between functions and ordered functions of noncommuting operators is discussed, and results closely paralleling classical results are obtained.

Wigner distribution for angle coordinates in quantum mechanics

The method of Wigner distribution functions, and the Weyl correspondence between quantum and classical variables, are extended from the usual kind of canonically conjugate position and momentum operators to the case of an angle and angular momentum operator pair. The sense in which one has a description of quantum mechanics using classical phase‐space language is much clarified by this extension.

Analysis of continuum using boundary compatibility conditions of integrated force method

Static and vibration problems of an indeterminate continuum are traditionally analyzed by the stiffness method. The force method is more or less non-existent for such problems. This situation is primarily due to the incomplete state of development of the compatibility conditions which are essential for the analysis of indeterminate structures by the flexibility method. The understanding of the Compatibility Conditions (CC) has been substantially augmented. Based on the understanding of CC, a novel formulation termed the Integrated Force Method (IFM) has been established. In this paper IFM has been extended for the static and vibration analyses of a continuum. The IFM analysis is illustrated taking three examples: 1. (1) rectangular plate in flexure 2. (2) analysis of a cantilevered dam 3. (3) free vibration analysis of a beam. From the examples solved it is observed that the force response of an indeterminate continuum with mixed boundary conditions can be generated by IFM without any reference to displacements in the field or on the boundary. Displacements if required can be calculated by back substitution.

Molecular Mechanics Studies of Homopolymeric and Mixed Sequence Py.Pu*Pu Triple Helices,

DNA triple helices containing two purine strands and one pyrimidine strand (C.G*G and T.A*A) have been studied, using model building followed by energy minimisation, for different orientations of the third strand resulting from variation in the hydrogen bonding between the Watson-Crick duplex and the third strand and the glycosidic torsion angle in the third strand. Our results show that in the C.G*G case the structure with a parallel orientation of the third strand, resulting from Hoogsteen hydrogen bonds between the third strand and the Watson-Crick duplex, is energetically the most favourable while in the T.A*A case the antiparallel orientation of the third strand, resulting from reverse Hoogsteen hydrogen bonds, is energetically the most favourable. These studies when extended to the mixed sequence triplexes, in which the second strand is a mixture of G and A, correspondingly the third strand is a mixture of G and APT, show that though the parallel orientation is still energetically more favourable, the antiparallel orientation becomes energetically comparable with an increasing number of thymines in the third strand. Structurally, for the mixed triplexes containing G and T in the third strand, it is seen that the basepair non-isomorphism between the C.G*G and the T.A*T triplets can be overcome with some changes in the base pair parameters without much distortion of either the backbone or the hydrogen bonds.

Chiral polyhydroxylated tetrahydrothiophene derivatives: novel synthesis and structural elucidation by X-ray crystallography, NMR spectroscopy and molecular mechanics calculations

The dideoxygenation reaction of 1,3;4,6-di-O-alkylidene-2,5-di-S-methylthiocarbonyl-D-mannitol derivatives under Barton-McCombie reaction conditions gave the hexahydrodipyranothiophenes 4 and 7 instead of the expected 2,5-dideoxy products. Structural and conformational information on these novel derivatives has been obtained by NMR spectroscopy, single-crystal X-ray crystallography and molecular mechanics calculations.

Molecular expression for dielectric friction on a rotating dipole: Reduction to the continuum theory

Recently we presented a microscopic expression for dielectric friction on a rotating dipole. This expression has a rather curious structure, involving the contributions of the transverse polarization modes of the solvent and also of the molecular length scale processes. It is shown here that under proper limiting conditions, this expression reduces exactly to the classical continuum model expression of Nee and Zwanzig [J. Chem. Phys. 52, 6353 (1970)]. The derivation requires the use of the asymptotic form of the orientation‐dependent total pair correlation function, the neglect of the contributions of translational modes of the solvent, and also the use of the limit that the size of the solvent molecules goes to zero. Thus, the derivation can be important in understanding the validity of the continuum model and can also help in explaining the results of a recent computer simulation study of dielectric relaxation in a Brownian dipolar lattice.

The Reproducing Kernel DMS-FEM: 3D Shape Functions and Applications to Linear Solid Mechanics

We propose a family of 3D versions of a smooth finite element method (Sunilkumar and Roy 2010), wherein the globally smooth shape functions are derivable through the condition of polynomial reproduction with the tetrahedral B-splines (DMS-splines) or tensor-product forms of triangular B-splines and ID NURBS bases acting as the kernel functions. While the domain decomposition is accomplished through tetrahedral or triangular prism elements, an additional requirement here is an appropriate generation of knotclouds around the element vertices or corners. The possibility of sensitive dependence of numerical solutions to the placements of knotclouds is largely arrested by enforcing the condition of polynomial reproduction whilst deriving the shape functions. Nevertheless, given the higher complexity in forming the knotclouds for tetrahedral elements especially when higher demand is placed on the order of continuity of the shape functions across inter-element boundaries, we presently emphasize an exploration of the triangular prism based formulation in the context of several benchmark problems of interest in linear solid mechanics. In the absence of a more rigorous study on the convergence analyses, the numerical exercise, reported herein, helps establish the method as one of remarkable accuracy and robust performance against numerical ill-conditioning (such as locking of different kinds) vis-a-vis the conventional FEM.

Constitutive modeling of ice in the high strain rate regime

The objective of the present work is to propose a constitutive model for ice by considering the influence of important parameters such as strain rate dependence and pressure sensitivity on the response of the material. In this regard, the constitutive model proposed by Carney et al. (2006) is considered as a starting basis and subsequently modified to incorporate the effect of brittle cracking within a continuum damage mechanics framework. The damage is taken to occur in the form of distributed cracking within the material during impact which is consistent with experimental observations. At the point of failure, the material is assumed to be fluid-like with deviatoric stress almost dropping down to zero. The constitutive model is implemented in a general purpose finite element code using an explicit formulation. Several single element tests under uniaxial tension and compression, as well as biaxial loading are conducted in order to understand the performance of the model. Few large size simulations are also performed to understand the capability of the model to predict brittle damage evolution in un-notched and notched three point bend specimens. The proposed model predicts lower strength under tensile loading as compared to compressive loading which is in tune with experimental observations. Further the model also asserts the strain rate dependency of the strength behavior under both compressive as well as tensile loading, which also corroborates well with experimental results. (C) 2010 Elsevier Ltd. All rights reserved.

E. C. G. Sudarshan and Symmetry in Classical Dynamics, Optics and Quantum Mechanics

We review work initiated and inspired by Sudarshan in relativistic dynamics, beam optics, partial coherence theory, Wigner distribution methods, multimode quantum optical squeezing, and geometric phases. The 1963 No Interaction Theorem using Dirac's instant form and particle World Line Conditions is recalled. Later attempts to overcome this result exploiting constrained Hamiltonian theory, reformulation of the World Line Conditions and extending Dirac's formalism, are reviewed. Dirac's front form leads to a formulation of Fourier Optics for the Maxwell field, determining the actions of First Order Systems (corresponding to matrices of Sp(2,R) and Sp(4,R)) on polarization in a consistent manner. These groups also help characterize properties and propagation of partially coherent Gaussian Schell Model beams, leading to invariant quality parameters and the new Twist phase. The higher dimensional groups Sp(2n,R) appear in the theory of Wigner distributions and in quantum optics. Elegant criteria for a Gaussian phase space function to be a Wigner distribution, expressions for multimode uncertainty principles and squeezing are described. In geometric phase theory we highlight the use of invariance properties that lead to a kinematical formulation and the important role of Bargmann invariants. Special features of these phases arising from unitary Lie group representations, and a new formulation based on the idea of Null Phase Curves, are presented.

Prediction of nonlocal scaling parameter for armchair and zigzag single-walled carbon nanotubes based on molecular structural mechanics, nonlocal elasticity and wave propagation

Atomic vibration in the Carbon Nanotubes (CNTs) gives rise to non-local interactions. In this paper, an expression for the non-local scaling parameter is derived as a function of the geometric and electronic properties of the rolled graphene sheet in single-walled CNTs. A self-consistent method is developed for the linearization of the problem of ultrasonic wave propagation in CNTs. We show that (i) the general three-dimensional elastic problem leads to a single non-local scaling parameter (e(0)), (ii) e(0) is almost constant irrespective of chirality of CNT in the case of longitudinal wave propagation, (iii) e(0) is a linear function of diameter of CNT for the case of torsional mode of wave propagation, (iv) e(0) in the case of coupled longitudinal-torsional modes of wave propagation, is a function which exponentially converges to that of axial mode at large diameters and to torsional mode at smaller diameters. These results are valid in the long-wavelength limit. (C) 2011 Elsevier Ltd. All rights reserved.

The role of Compton and Raman scattering in the quasar continuum

There are three ways in which an electromagnetic wave can undergo scattering in a plasma: (i) when the scattering of radiation occurs by a single electron, it is called Compton Scattering (CS); (ii) if it occurs by a longitudinal electron plasma mode, it is called Stimulated Raman Scattering (SRS), and (iii) if it occurs by a highly damped electron plasma mode, it is called Stimulated Compton Scattering (SCS). The non-thermal continuum of quasars is believed to be produced through the combined action of synchrotron and inverse Compton processes, which are essentially single-particle processes. Here, we investigate the role of SRS and SCS in the generation of continuum radiation from these compact objects. It is shown as an example that the complete spectrum of 3C 273 can be reproduced by suitably combining SCS and SRS. The differential contributions of SCS and SRS under different values of the plasma parameters are also calculated.

The effect of boundaries on the plane Couette flow of granular materials: a bifurcation analysis

The tendency of granular materials in rapid shear flow to form non-uniform structures is well documented in the literature. Through a linear stability analysis of the solution of continuum equations for rapid shear flow of a uniform granular material, performed by Savage (1992) and others subsequently, it has been shown that an infinite plane shearing motion may be unstable in the Lyapunov sense, provided the mean volume fraction of particles is above a critical value. This instability leads to the formation of alternating layers of high and low particle concentrations oriented parallel to the plane of shear. Computer simulations, on the other hand, reveal that non-uniform structures are possible even when the mean volume fraction of particles is small. In the present study, we have examined the structure of fully developed layered solutions, by making use of numerical continuation techniques and bifurcation theory. It is shown that the continuum equations do predict the existence of layered solutions of high amplitude even when the uniform state is linearly stable. An analysis of the effect of bounding walls on the bifurcation structure reveals that the nature of the wall boundary conditions plays a pivotal role in selecting that branch of non-uniform solutions which emerges as the primary branch. This demonstrates unequivocally that the results on the stability of bounded shear how of granular materials presented previously by Wang et al. (1996) are, in general, based on erroneous base states.

Search for sub-mm, mm and radio continuum emission from extremely red objects

We present the results of sub-mm, mm (850 mum, 450 mum and 1250 mum) and radio (1.4 and 4.8 GHz) continuum observations of a sample of 27 K-selected Extremely Red Objects, or EROs, (14 of which form a complete sample with K < 20 and I - K > 5) aimed at detecting dusty starbursts, deriving the fraction of UltraLuminous Infrared Galaxies (ULIGs) in ERO samples, and constraining their redshifts using the radio-FIR correlation. One ERO was tentatively detected at 1250 mum and two were detected at 1.4 GHz, one of which has a less secure identification as an ERO counterpart. Limits on their redshifts and their star forming properties are derived and discussed. We stacked the observations of the undetected objects at 850 mum, 1250 mum and 4.8 GHz in order to search for possible statistical emission from the ERO population as a whole, but no significant detections were derived either for the whole sample or as a function of the average NIR colours. These results strongly suggest that the dominant population of EROs with K < 20 is not comprised of ULIGs like HR 10, but is probably made of radio-quiet ellipticals and weaker starburst galaxies with L < 10(12) L . and SFR < few hundred M. yr(-1).

Simulation of the infrared spectra of thioacetamide by the extended molecular mechanics method

The infrared spectrum of the matrix-isolated species of thioacetamide has been simulated using the extended molecular mechanics method. The equilibrium structure, vibrational frequencies, dipole moment and infrared absorption intensities of thioacetamide have been calculated in good agreement with the experiment. The vibrational frequencies and infrared absorption intensities for the isotopic molecules (CH2CSNH2)-C-13, (CH3CSNH2)-N-15 and (CH2CSND2)-C-13 have also been calculated consistent with the experiment. The infrared spectra of the matrix isolated species of N- and C- deuterated isotopomers of thioacetamide, CH3CSND2 and CD3CSNH2 have also been simulated in satisfactory agreement with the experimental spectra.