We present a utilitarian review of the family of matrix groups Sp(2n, ℛ), in a form suited to various applications both in optics and quantum mechanics. We contrast these groups and their geometry with the much more familiar Euclidean and unitary geometries. Both the properties of finite group elements and of the Lie algebra are studied, and special attention is paid to the so-called unitary metaplectic representation of Sp(2n, ℛ). Global decomposition theorems, interesting subgroups and their generators are described. Turning ton-mode quantum systems, we define and study their variance matrices in general states, the implications of the Heisenberg uncertainty principles, and develop a U(n)-invariant squeezing criterion. The particular properties of Wigner distributions and Gaussian pure state wavefunctions under Sp(2n, ℛ) action are delineated.

A rigorous derivation of the optical theorem (OT) from the conservation of probability flux (CPF) is presented for scattering on an arbitrary spherically symmetric potential inN-spatial dimensions (ND). The constructed expression for the OT is found to yield the corresponding well-known results for two- and three-dimensional cases in a rather natural way. The Aharonov-Bohm (AB) effect is considered as a scattering event of an electron by a magnetic field confined in an infinitely long shielded solenoid and a similar derivation is attempted for an appropriate optical theorem. Our current understanding of the scattering theory is found to be inadequate for the purpose. The reason for this is discussed in some detail.

BD-FRW universe filled with imperfect fluid having bulk viscosity is investigated under the framework of Israel-Stewart-Hiscock causal theory. The field equations have been solved by using the relationφ=KR^α whereK andα are constants, between the Brans-Dicke scalar fieldϕ and the scale factorR. This relation, in fact, leads to a constant deceleration parameterq. It is shown that the constancy of the deceleration parameter permits only two possibilities i.e. eitherH=constant withm=1 orm=(1+b −α)/(2(1+b) −α), irrespective of the value ofɛ.

Using Lexan polycarbonate plastic as the fission fragment track detectors, the fragment angular distributions have been measured in the cases of fission of²³²Th and²³⁸U induced by alpha particles of various energies ranging from 40 to 70MeV obtained from the 88″ variable energy cyclotron at Calcutta. The center-of-mass angular distributions have been calculated and fitted by a series of Legendre polynomials. TheW(10°)/W(90°) ratios (defined as anisotropy) were measured at several energies for both the targets. These data are utilized in calculation of the energy dependence ofK₀², the standard deviation of the distribution in the angular momentum projection on the nuclear symmetry axis at the saddle point. Values of Γ_f/Γ_η, i.e. the ratio of the fission width to neutron emission width have been determined for²³²Th and²³⁸U nuclei. The integral cross-section for alpha induced fission in each target was determined by numerical integration of the respective center-of-mass angular differential cross-sections. The results were compared with similar data available in the literature which served to resolve some of the discrepancies observed in earlier measurements. The results were also compared with theoretical cross-sections.

The first relativistic correction of orderα² to the dipole polarizability of a hydrogenic ion has been investigated by using mean excitation energy of the ion within the second-order perturbation theory. The density-dependent mean excitation energy is estimated via Bethe theory for the stopping cross section for a moving point charge interacting with the hydrogenic ion. In this approach only the unperturbed Dirac wavefunctions are required to evaluate the appropriate matrix elements. The first relativistic correction turns out to be − (13/12)(αZ)². This has the correct sign and is within 5% of the exact result which is −(28/27)(αZ)².

Line strengthS and radial matrix elementσ for the dipole allowed transitions withinn=2 complex of ions in the Be isoelectronic sequence have been fitted in the formsZ²S=A+B/(Z − C) andZσ=A′ + B′/(Z − C′). The constantsA, B, C andA′, B′, C′ have been calculated by employing a non-linear least square method. The relevant data forS andσ have been taken from calculations which includes correlation effects. It is shown that the fitted yalues ofA andA′ are in excellent accord with their hydrogenic values (Z →α) provided that we express the zeroth-order wavefunction of the ground state 1s²2s²¹S as a quantum-mechanical admixture of the Hartree-Fock (HF) state 1s²2s²¹S and the near-degenerate state 1s²2p²¹S.

Dynamics of a wavepacket on a model fractal surface has been studied by solving the pertinent time dependent Schrödinger equation. Spatial and temporal behaviour of charge and current densities and a local chemical potential for two different fractal lattices have been considered. Important insight into the dynamics has been obtained through time dependence of various quantities like macroscopic kinetic energy, global current, Shanon entropy, density correlation and global chemical potential. This study would be helpful in simulating adsorption and catalysis.

A self-consistent model for the description of the ion sound wave in a dusty plasma is given. We show that proper consideration of the ion attachment to the grains gives rise to the dissipation of the ion acoustic wave in a dusty plasma medium. Dissipation rate is proportional to the sum of the electron and twice the ion attachment frequencies.

This paper addresses itself to a practical problem encountered in using iterative learning rules for associative memory models. The performance of a learning rule based on linear programming which overcomes this problem is compared with that of a representative iterative rule by numerical simulation. Results indicate superior performance by the linear programming rule. An algorithm for computing radii of maximal hyperspheres around patterns in the state space of a model is presented. Fractional volumes of basins of attractions are computed for the representative iterative rule as well as the linear programming rule. With the radii of maximal hyperspheres as weight factors for corresponding patterns to be stored, the linear programming rule gives rise to the maximal utilisation of the state space.