Nazakat Ullah

INVARIANCE HYPOTHESIS AND HIGHER CORRELATIONS OF HAMILTONIAN MATRIX ELEMENTS

Abstract A method is developed for calculating the averages of the components of a randomly oriented unit vector in an N dimensional orthogonal, unitary and symplectic space. The method is extended to calculate the averages of the components of two orthogonal N dimensional randomly oriented unit vectors. Using the invariance hypothesis it is shown that the off-diagonal matrix elements are distributed symmetrically about zero mean and that there are no correlations between an odd power of the off-diagonal matrix element and any power of diagonal or another off-diagonal matrix element. An expression is given for the correlation coefficient of the squares of diagonal matrix elements.

On a Generalized Distribution of the Poles of the Unitary Collision Matrix

An expression for the joint distribution of the complex poles of the unitary collision matrix is derived for the single‐channel case, which is valid for all values of the ratio of the width to the spacing. The derivation uses the statistical distribution of the parameters of the real R‐matrix theory. We find that unitarity gives rise to the statistical correlations between the width and the spacing of the collision matrix. It is shown that the distribution of the poles of the unitary collision matrix using Feshbachs unified theory of nuclear reactions is the same as the one obtained using R‐matrix theory, provided we make a particular choice of the arbitrary boundary condition in the latter theory. A remark is made about the use of the random complex orthogonal matrix in the study of the parameters of the statistical collision matrix.

Exact Distributions of the Reduced‐Width Amplitude

The invariance hypothesis is used to derive the various multivariate distributions of the reduced‐width amplitude. Simple expressions are given for the multi‐level and multi‐channel distributions valid for all dimensions of the random orthogonal matrix.

Asymptotic Reduced‐Width Amplitude Distributions

The method of moments is used to derive the reduced‐width amplitude distributions. The explicit dependence of the distribution function on the dimension N of the random orthogonal matrix for large values of N is obtained. It is shown that in the limit N → ∞, the distribution is the same as the one obtained using the explicit assumption of level independence.

On the Reduction of the Generalized RPA Eigenvalue Problem

The 2n‐dimensional eigenvalue problem, which arises when the random phase approximation (RPA) matrix is not real, is reduced to an n‐dimensional eigenvalue problem. Some properties of the reduced eigenvalue problem are studied. A numerical example is considered for illustrative purposes.

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(a2 −r2)3/2,K, a being constants. Two broad features of this distribution are mentioned.

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→∞.

On the expansion of the single eigenvalue probability density function

The probability density function of the single eigenvalue is expanded in terms of the reciprocal of the dimension of the matrix using Bessel functons. It is shown that for the new matrix ensembles this expansion gives Wigners semicircle centered at the mean value of the matrix elements plus terms of the order ofN−1, whereN is the dimension of the matrix.

Parity mixing in deformed hartree-fock calculations

Abstract It is shown that parity non-conserving single-particle orbitals can result from a self-consistent calculation. Different types of single-particle orbitals are examined for the simple case of two levels of opposite parities to determine the best variational solution. Results of detailed numerical calculations are reported for some nitrogen and oxygen isotopes.

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.