arXiv subjects

Sigurd Yves Larsen

Publications and source records attributed to Sigurd Yves Larsen.

5 recordsLinked to original sources

arXiv2020

Analytical Expressions for a Hyperspherical Adiabatic Basis Suitable for a particular Three Particle Problem in 2 Dimensions

For a particular case of three-body scattering in two dimensions, and matching analytical expressions at a transition point, we obtain accurate solutions for the hyperspherical adiabatic basis and potential. We find analytical expressions for the respective, asymptotic, inverse logarithmic and inverse power potential behaviors, that arise as functions of the radial coordinate. The model that we consider is that of two particles interacting with a repulsive step potential, a third particle acting as a spectator. The model is simple but gives insight, as the 2-body interaction is long ranged in hyperspherical coordinates. The fully interacting 3-body problem is known, numerically, to yield similar behaviors that we can now begin to understand.That, clearly, is the ultimate aim.

quant-ph↗

arXiv2016

A Generalized Uhlenbeck and Beth Formula for the Third Cluster Coefficient

Relatively recently (A. Amaya-Tapia, S. Y. Larsen and M. Lassaut. Ann. Phys., vol. 306 (2011) 406), we presented a formula for the evaluation of the third Bose fugacity coefficient - leading to the third virial coefficient - in terms of three-body eigenphase shifts, for particles subject to repulsive forces. An analytical calculation for a 1-dim. model, for which the result is known, confirmed the validity of this approach. We now extend the formalism to particles with attractive forces, and therefore must allow for the possibility that the particles have bound states. We thus obtain a true generalization of the famous formula of Uhlenbeck and Beth (G.E. Uhlenbeck and E. Beth. Physica, vol. 3 (1936) 729; E. Beth and G.E. Uhlenbeck. ibid, vol.4 (1937) 915) (and of Gropper (L. Gropper. Phys. Rev. vol. 50 (1936) 963; ibid vol. 51 (1937) 1108)) for the second virial. We illustrate our formalism by a calculation, in an adiabatic approximation, of the third cluster in one dimension, using McGuire's model as in our previous paper, but with attractive forces. The inclusion of three-body bound states is trivial; taking into account states having asymptotically two particles bound, and one free, is not.

physics.chem-ph↗

arXiv2001

Hyperspherical Adiabatic Formalism of the Boltzmann Third Virial

First, we show that, if there are no bound states, we can express the q.m. third cluster - involving 3 and fewer particles in Statistical Mechanics - as a formula involving adiabatic eigenphase shifts. This is for Boltzmann statistics. From this q.m. formulation, in the case of purely repulsive forces, we recover, as h goes to 0, the classical expressions for the cluster. We then discuss difficulties which arise in the presence of 2-body bound states and present a tentative formula involving eigenphase shifts and the 2 and 3 body bound state energies. We emphasize that important difficulties have not been resolved.

physics.chem-ph↗

arXiv2001

Potential Harmonic Calculations of Helium Triplet States

The hyperspherical harmonics (HH) provide a complete basis for the expansion of atomic wave functions, but even for two particles the number of harmonics for a given order is not trivial and, as the number of electrons increases, this degeneracy becomes absolutely prohibitive. We modify the method by selecting a subset of the basis that, we feel, will yield the physically most important part of the wave function, and test the idea on simple systems. In a previous work (with M. Haftel) of the singlet ground and first excited states of the helium atom we found that the error in the binding energy of the ground state was of the order of 1 part in 10,000 and that it decreased for the first excited state. We now have applied our method to the equivalent triplet states. We report on this work, and our results, and hope to draw attention to the interesting accuracy that we obtain with the relatively simple wave functions of our formulation.

physics.atom-ph↗

arXiv2001

Analytical Expressions for a Hyperspherical Adiabatic Basis Three Particles in 2 Dimensions

For a particular case of three-body scattering in 2 dimensions, we demonstrate analytically that the behaviour of the adiabatic potential is different from that of the hyperspherical coupling matrix elements, thereby leading to a phase shift that tends to zero, as the energy goes to zero, instead of to a constant. We consider two particles interacting with binary repulsive step potentials, one acting as a spectator, and solve analytically for the adiabatic eigenvalues and eigenvectors, for all values of rho. We are thereby able to obtain the leading terms in the long range behaviour of the effective potentials, and confirm its inverse logarithmic nature, suggested from a previous numerical study.

physics.chem-ph↗