There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. S. 211-215

Incorporating recent data, a new set of recommended values of the basic constants and conversion factors of physics and chemistry has just been issued.

The fundamental physical constants

Chapter II. Baryon-Baryon Interactions and Single-Particle Aspects of Hypernuclei : Structure of Hypernuclei

We study a system of A identical interacting bosons trapped by an external field by solving ab initio the many-body Schroedinger equation. A complete solution by using, for example, the traditional hyperspherical harmonics (HH) basis develops serious practical problems due to the large degeneracy of HH basis. Symmetrization of the wave function, calculation of the matrix elements, etc., become an immensely formidable task as A increases. Instead of the HH basis, here we use a new basis, called 'potential harmonics' (PH) basis, which is a subset of HH basis. We assume that the contribution to the orbital and grand orbital [in 3(A-1)-dimensional space of the reduced motion] quantum numbers comes only from the interacting pair. This implies inclusion of two-body correlations only and disregard of all higher-body correlations. Such an assumption is ideally suited for the Bose-Einstein condensate (BEC), which is required, for experimental realization of BEC, to be extremely dilute. Hence three and higher-body collisions are almost totally absent. Unlike the (3A-4) hyperspherical variables in HH basis, the PH basis involves only three active variables, corresponding to three quantum numbers--the orbital l, azimuthal m, and the grand orbital 2K+l quantum numbers for any arbitrary A. It drastically reduces themore » number of coupled equations and calculation of the potential matrix becomes tremendously simplified, as it involves integrals over only three variables for any A. One can easily incorporate realistic atom-atom interactions in a straightforward manner. We study the ground and excited state properties of the condensate for both attractive and repulsive interactions for various particle number. The ground state properties are compared with those calculated from the Gross-Pitaevskii equation. We notice that our many-body results converge towards the mean field results as the particle number increases.« less

https://arxiv.org/pdf/quant-ph/0408061.pdf

Potential harmonics expansion method for trapped interacting bosons: Inclusion of two-body correlation

Hyperspherical harmonics expansion (HHE) method has been applied to study the structure and ΛΛ dynamics for the ground and first excited states of low and medium mass double-Λ hypernuclei in the framework of core+Λ+Λ three-body model. The ΛΛ potential is chosen phenomenologically while core-Λ potential is obtained by folding the phenomenological Λ N interaction into the density distribution of the core. The parameters of this effective Λ N potential is obtained by the condition that they reproduce the experimental (or empirical) data for core-Λ subsystem. The three-body (core+Λ+Λ) Schrodinger equation is solved by hyperspherical adiabatic approximation (HAA) to get the ground state energy and wave function. This ground state energy and wave function are used to construct a partner potential. The three-body Schrodinger equation is solved once again for this partner potential. According to supersymmetric quantum mechanics, the ground state energy of this potential is exactly the same as that of the first excited state of the potential used in the first step. In addition to the two-Λ separation energy for the ground and first excited state, some geometrical quantities for the ground state of double-Λ hypernuclei , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , are computed. These include the ΛΛ bond energy and various r.m.s. radii.

Study of the excited state of double-λ hypernuclei by hyperspherical supersymmetric approach

Calculation of the 2+ resonance state of 6He in a three-body model (4He + n + n) is done by adopting a novel theoretical technique. The effective three-body potential for the 2+ state of 6He is obtained using hyperspherical harmonics and it presents a shallow well followed by a low and wide barrier. Numerical diffculties present in the calculation of the resonance energy of such a shallow well–barrier combination are overcome by the construction of a one-parameter (λ) isospectral potential having a bound state in the continuum. This potential develops a deep well followed by a high barrier for small positive values of λ. This effectively traps the system in a sharp resonant state and facilitates calculation of the resonance energy accurately. We obtain the first 2+ resonance at 1.814 MeV with a width of 135 keV. We also get a second 2+ resonance at 4.9 MeV.

Computation of 2+ resonance in 6He: Bound state in the continuum

Ground state energies of exotic three-body atomic systems consisting two muons and a positively charged nucleus like: 1H+μ−μ−, 4He2+μ−μ−, 3He2+μ−μ−, 7Li3+μ−μ−, 6Li3+μ−μ−, 9Be4+μ−μ−, 12C6+μ−μ−, 16O8+μ−μ−, 20Ne10+μ−μ−, 28Si14+μ−μ− and 40Ar18+μ−μ− have been calculated using hyperspherical harmonics expansion (HHE) method. Calculation of matrix elements of two body interactions involved in the HHE method for a three body system is greatly simplified by expanding the bra- and ket- vector states in the hyperspherical harmonics basis states appropriate for the partition corresponding to the interacting pair. This involves the Raynal-Revai coefficients (RRC), which are the transformation coefficients between the hyperspherical harmonics bases corresponding to the two partitions. Use of these coefficients found to be very useful for the numerical solution of three-body Schrodinger equation where the two-body potentials are other than Coulomb or harmonic oscillator type. However, in this work the interaction potentials involved are purely Coulomb. The calculated energies have been compared with (i) those obtained by straight forward manner; and (ii) with those found in the literature (wherever available). The calculated binding energies agree within the computational error.

Hyperspherical three-body calculation for muonic atoms

We propose a novel theoretical technique for the calculation of resonances at low excitation energies in weakly bound systems. Starting from an effective potential, supersymmetric quantum mechanics can be successfully used to generate families of isospectral potentials having desirable and adjustable properties. For resonance states, for which there is no bound ground state of the same spin-parity, one can construct an isospectral potential with a bound state in the continuum (BIC). The potential looks quite different but is strictly isospectral with the original one. The quasi-bound state in the original shallow potential will be effectively trapped in the deep well of the isospectral family facilitating an easier and more accurate calculation of the resonance energy. Application to 6He, 6Be, and 6Li systems yields quite accurate results. The beauty of our technique: We get both the bound ground state and the resonances by a single technique and using the same potential.

Resonances in A=6 Nuclei: Use of Supersymmetric Quantum Mechanics

The hyperspherical harmonics expansion method is applied to a three-body model of two-neutron halo nuclei Convergence of the expansion has been ensured A short-range repulsive part is incorporated in the interaction between the core and the extra-core neutron, to simulate the Pauli principle Two neutron separation energy, rms radii, correlation factor and halo neutron density distributions have been calculated for  It is found that the convergence of the two-neutron separation energy is relatively slow, while other quantities reach convergence quickly

Md. Abdul Khan

Papers

Study of the excited state of double-λ hypernuclei by hyperspherical supersymmetric approach

Computation of 2+ resonance in 6He: Bound state in the continuum

Hyperspherical three-body calculation for muonic atoms

Resonances in A=6 Nuclei: Use of Supersymmetric Quantum Mechanics

Hyperspherical three-body calculation for neutron drip-line nuclei