CALCULATION OF ENERGY EIGENVALUES AND EIGENVECTORS FOR THREE-BODY MOLECULES IN JACOBI AND HYPERSPHERICAL COORDINATES

2010 ◽  
Vol 19 (03) ◽  
pp. 419-435 ◽  
Author(s):  
M. R. ESKANDARI ◽  
H. KHAJEHAZAD

The Jacobi coordinates is used to eliminate center of mass motion of three-body systems. We write the results in hyperspherical coordinates and expand eigenfunction in a series of orthonormal complete set of Ykαi(Ωi) in partition i of Jacobi coordinates. The matrix elements of two-body interaction potential in hyperspherical harmonic approach are determined exactly using computed analytical form of Raynal–Revai coefficients to change the base set of Ykαi(Ωi) to other set such as Ykαj(Ωj). The generalized Laguerre functions are used to change the second order coupled differential equations to a non-differential matrix equation. This is solved to find energy eigenvalues and eigenfunctions of three-body molecules. The obtained results are in agreement with computational methods.

1874 ◽  
Vol 164 ◽  
pp. 173-182 ◽  

The tertiary deposits of the east coast of Patagonia, which yielded to the researches of Mr. Darwin and Admiral Sulivan such interesting and aberrant mammals as Macrauchenia , Nesodon , and Toxodon , have again disclosed a new and remarkable form of extinct animal life. The evidence upon which the existence of this new genus rests consists of a nearly complete set of teeth and some fragments of bone, discovered on the bank of the River Gallegos, by Dr. Robert O. Cunningham, Naturalist to H.M.S. ‘Nassau.’ during the voyage undertaken for the purpose of surveying in the Strait of Magellan and the west coast of Patagonia in the years 1866, 1867, 1868, and 1869. The spot was visited in conformity with instructions received before leaving England, “to insti­tute a search for a deposit of fossil bones discovered by Admiral Sulivan and the pre­sent Hydrographer of the Navy, Rear-Admiral G. H. Richards, about twenty years previously, and which Mr. Darwin, Professor Huxley, and other distinguished naturalists were anxious should be carefully examined”. The conditions under which the specimens were found will be best understood from the following additional extract from Dr. Cunningham’s narrative. “Accordingly, joined by the steamer, which again took us in tow, we proceeded onwards till we arrived opposite the first deposit of fallen blocks at the foot of the cliffs. The cutter was then anchored in the stream, while we pulled in towards the shore in the galley till she grounded, when we landed, armed with picks and geological hammers for our work. After examining the first accumulation of blocks, and finding in the soft yellow sandstone of which certain of them were composed some small fragments of bone, we proceeded to walk along the beach, carefully examining the surface of the cliffs and the piles of fragments which occurred here and there at their base. The height of the cliffs varied considerably, and the highest portions, averaging about 200 feet, extended for a distance of about ten miles, and were evidently undergoing a rapid process of disinte­gration, a perpetual shower of small pieces descending in many places, and numerous large masses being in process of detaching themselves from the parent bed. They were principally composed of strata of hard clay (sometimes almost homogeneous in its texture, and at others containing numerous rounded boulders) ; soft yellow sandstone ; sandstone abounding in hard concretions; and, lastly, a kind of conglomerate, resembling solidified, rather fine gravel. The lowermost strata, as a rule, were formed of the sand­ stone with concretions; the middle, of the soft yellow sandstone, which alone appeared to contain organic remains; and the upper, of the gravelly conglomerate and hard clay. Nearly the whole of the lower portion of the cliffs, as well as all the principal deposits of fallen blocks, were examined by us in the course of the walk, and we met with numerous small fragments of bone ; but very few specimens of any size or value occurred, and the generality of these were in such a state of decay as to crumble to pieces when we attempted, although with the utmost amount of care that we could bestow, to remove them from the surrounding mass. To add to this, the matrix in which they were imbedded was so exceedingly soft as not to permit of being split in any given direction. The first fossil of any size observed by us was a long bone, partially protruding from a mass, and dissolved into fragments in the course of my attempts to remove it. At some distance from this a portion of what appeared to be the scapula of a small quadruped, with some vertebrse, occurred; and further on one of the party (Mr. Vereker) directed my attention to a black piece of bone projecting from one side of a large block near its centre. This, which was carefully removed at the expense of a large amount of labour, with a considerable amount of the matrix surrounding it, by three of the officers, to whose zeal in rendering me most valuable assistance in my work I shall ever feel deeply indebted, afterwards proved to be a most valuable specimen for on carefully removing more of the matrix when we returned to the ship, I found that it was the cranium of a quadruped of considerable size, with the dentition of both upper and lower jaws nearly complete. As no other specimens of importance were discovered, we reembarked towards the close of the afternoon.


2009 ◽  
Vol 18 (05n06) ◽  
pp. 1166-1175
Author(s):  
SHASHIKANT C. PHATAK

The behavior of a nucleon in nuclear medium is discussed in Chiral Color Dielectric Model. It is assumed that the nucleons in nuclear medium produces a background dielectric field and the quark and dielectric field equations are solved self consistantly in presence of the dielectric field. A nucleon in nuclear medium is then constructed by means of standard procedure followed in chiral bag models. The corrections due to center of mass motion, color magnetic interaction and meson interaction are included. The calculations show that the nucleon becomes bigger in the medium but its mass does not change much. It is found that beyond a certian density, bound solutions in which quarks are bound in self-generated dielectric field are not possible. Thus, the calculations indicate that there is a critical density beyond which the matter consists of deconfined quarks.


1966 ◽  
Vol 44 (9) ◽  
pp. 2095-2110 ◽  
Author(s):  
Marcel Banville ◽  
P. D. Kunz

The three-body wave function for particles of equal mass is expanded in a systematic way by making use of a hyperspherical coordinate system. Apart from the center-of-mass coordinates, three of the variables are the usual Euler angles describing the orientation of the plane defined by the three particles. The other three variables, which describe the shape of the triangle, are represented in terms of a radial coordinate and two angular coordinates. The kinetic energy for these last three coordinates is separable and allows one to expand the three-body wave function in a complete set of orthogonal functions based upon the angular variables. The particular symmetry of the internal part of the wave function under permutations of the three particles is easily represented in terms of the set of functions for one of the angular variables. By choosing a particular set of radial functions one can then obtain the upper limit on the binding energy for the three-body system through the Rayleigh–Ritz variational procedure. The advantage of this particular coordinate system is that all but a few of the variational parameters occur linearly in the wave function, and the minimum energy can be obtained by diagonalizing a small number of the energy matrices. The method is applied to find the lower limit to a standard spin-independent potential of Gaussian shape.


1992 ◽  
Vol 46 (11) ◽  
pp. 7162-7178 ◽  
Author(s):  
W. Ren ◽  
J. D. Cresser ◽  
H. J. Carmichael

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Vivian Martins Gomes ◽  
Antonio Fernando Bertachini de Almeida Prado ◽  
Justyna Golebiewska

The present research studies the motion of a particle or a spacecraft that comes from an orbit around the Sun, which can be elliptic or hyperbolic, and that makes a passage close enough to the Earth such that it crosses its atmosphere. The idea is to measure the Sun-particle two-body energy before and after this passage in order to verify its variation as a function of the periapsis distance, angle of approach, and velocity at the periapsis of the particle. The full system is formed by the Sun, the Earth, and the particle or the spacecraft. The Sun and the Earth are in circular orbits around their center of mass and the motion is planar for all the bodies involved. The equations of motion consider the restricted circular planar three-body problem with the addition of the atmospheric drag. The initial conditions of the particle or spacecraft (position and velocity) are given at the periapsis of its trajectory around the Earth.


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