• Nazakat Ullah

Articles written in Pramana – Journal of Physics

• A study of the variable moment of inertia models

The variable moment of inertia (VMI) model proposed by Holmberg and Lipas has been shown to be a special case of the VMI model of Mariscottiet al. The solution of Mariscotti’s model is expressed in terms of hypergeometric functions, which directly give the rotational energies or their expansions in terms of the quantityF(F+1), whereF is the total angular momentum. The present way of looking at the VMI model also tells us how to write the general dependence of the vibrational energy and the moment of inertia on the energyEJ.

• Differential equation approach for the energy average of the scattering function

An exact differential equation is given to evaluate the energy average of the scattering function. The advantage of the differential equation as compared to the earlier methods based on series expansion is that one has to evaluate only single sums over the complex poles of the S-matrix. Using Wigner’s semicircle law for the distribution of the real parts of the poles of the scattering matrix, the earlier expression for the energy average of the scattering function is rederived.

• On the joint eigenvalue distribution for the matrix ensembles with non zero mean

Exact distributions are given for the two-dimensional case when the mean of the off-diagonal element is non-zero. The joint eigenvalue distribution for theN dimensional case, derived using the volume element in the space ofN ×N orthogonal matrices, is checked by rederiving the exact results forN=2. The smooth nature of theN-dimensional joint distribution supports the claim of the method of moments that the single eigenvalue distribution is a smooth function of the ratio of mean-to-mean square deviation.

• Approximate angular momentum projection from an intrinsic random phase approximation Hartree-Fock state

An approximation procedure is described to calculate the projected energies from an intrinsicrpa hf wave function. The method of moments is used to find the relevant parameters. A model calculation is carried out for illustrative purposes.

• On an integral over the surface ofN-dimensional unit sphere

An integral which occurs in the new matrix ensembles and the width fluctuation factor is evaluated using a transformation which changes a Gaussian into an exponential. It is expressed in the form of a series whose terms are found using a simple recursion relation. It is shown that the series can be summed in closed form for the two-dimensional case.

• Modified perturbation series for the anharmonic oscillator using linearization technique

The linearization technique of random phase approximation is applied to the anharmonic oscillator to find a modified perturbation series. It is shown that for the anharmonic termλx4, the ground state energyE0 upto the second order of perturbation is given byE0=(35/48) (3/4)1/3λ1/3 asλ→∞.

• Fourier transform of single eigenvalue probability density function using ensemble-averaged traces of the Hamiltonian

A determinantal identity is used to calculate the ensemble-averaged traces of the Hamiltonian. Using these averages a general expression is obtained for the Fourier transform of the single eigenvalue probability density function for all the three Gaussian ensembles for the two-dimensional case. It is shown how one can use the familiar step-up operators for the representation of a determinant. The ensemble-averaged traces are also used to derive the Fourier transform of the non-zero mean ensemble.

• Density of nucleons in heavy nuclei

A shell model description of heavy nuclei is used to show that the density of nucleons in heavy nuclei is of the formρ(r) =K(a2r2)3/2,K, a being constants. Two broad features of this distribution are mentioned.

• A note on the mean-internucleon distance in the central region of heavy nuclei

The density distribution of nucleons in a heavy nucleus is used to show that the mean-internucleon distance in the central region of heavy nuclei is 1.99 fm.

• Exact generator coordinate wave functions for some simple hamiltonians

A Gaussian form of the generator coordinate wave function is used to find the exact weight function for the ground state of H-atom using HWG integral equation. Exact pairs of GC wave functions and weight functions are then constructed for other simple Hamiltonians using a simple integral which converts an exponential into a Gaussian. A discussion as to how a GC wave function can be used as a trial variational wave function is also presented.

• Density of quarks in heavy spherical nuclei using NRQSM

A nonrelativistic quark shell model (NRQSM) is used to derive an expression for the density of quarks in heavy spherical nuclei. It is shown that quark density is related in a simple way with the probability of finding a nucleon in a nucleus. The quark density is used to determine the ratio of average distance between two quarks to the average distance between two nucleons.

• Reply to the comment on ‘Modified perturbation series for the anharmonic oscillator using linearization technique’

It is pointed out that the value of the perturbed energy calculated in the comment by Rath and Pattnayak, is incorrect. When the coupling parameter has the value unity, the energy up to second order of perturbation theory is 0.801 compared to its exact value of 0.804. The suggested split of the Hamiltonian into an unperturbed and perturbed part does not seem to be of any use as the unperturbed part contains anharmonic terms.

• Momentum space distribution of electrons in an atom using hydrogenic wave functions

A formulation is developed to derive exact analytic expressions for electron-electron correlation and density of electrons in momentum space using hydrogenic wave functions. It is shown that for large atoms the expression for density of electrons has a simple form.

• Quark distribution in light nuclei

The distribution of quarks in light nuclei is given using the quark cluster wave function. An analytic expression for the nucleus4He is obtained. The distribution so obtained is compared with the one obtained using a different theoretical formulation called mapping.

• Single nucleon distribution based on shell-model and matrix ensemble theory

An expression for the density of single nucleons in a heavy spherical nucleus is derived using the shell-model and matrix ensembles. It is shown that the theoretical expression gives an excellent fit to the density of nucleons for the nucleus197Au obtained usinge-scattering data.

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019