Harmonic confinement

Lo, J.M.H. and Mobukowski, M. (2007) Relativistic calculations on the ground and excited states of AgH and AuH in cylindrical harmonic confinement Theoretical Chemistry Accounts, 118, 607-622. 

Low-Lying Excited States of the Hydrogen Molecule in Cylindrical Harmonic Confinement... 

Connection of Harmonic Confining Force Constants and Frequencies... 

The motion of electrons in a magnetic field in a situation in which inhomogeneity of some kind exists remains of considerable interest at the time of writing. Therefore, in this Appendix, we shall first summarize some results of Freeman and March [49] for the current density in a simple model of independent harmonically confined electrons in a constant magnetic field. Then we shall go on to discuss the semiclassical theory of current density in atoms. 

The trapped condensate in a magnetic and/or optical trap is not bound, but rather produced and kept in an external harmonic confining potential Vext = [14, 106], which acts on each atom of mass m and where... 

J.M.H. Lo, M. Klobukowski, G.H.F. Diercksen, Low-lying excited states of the hydrogen molecule in cylindrical harmonic confinement, Adv Quantum. Chem. 48 (2005) 59-89. 

In Section 10.3.4, the dynamics were treated of two harmonically-confined ions having small amplitudes of oscillation around the equilibrium positions of the ions. More specifically, in deriving Equations 10.10 and 10.13 (for z = zjit was assumed that the change of the ion distance is small compared to the equilibrium distance Az, such that the Coulomb interaction energy can be approximated by a harmonic potential. For large changes in the ion-ion distance, additional terms in the Coulomb energy lead to an anharmonic interaction and, hence, to an amplitude-dependent oscillation frequency. 

The analysis above can be generalized to describe collisions of molecules on an optical lattice with harmonic confinement in one dimension. The Hamiltonian of the collision system can be written as... 

As noted before, for z = 0 the repulsive dipole-dipole interaction dominates over the attractive van der Waals potential at distances r r given by Equation 12.22. In addition, for cox > 0 the harmonic potential confines the particle s motion in the z-direction. Thus, the combination of the dipole-dipole interaction and the harmonic confinement yields a repulsive potential that provides for a three-dimensional barrier separating the long-distance from the short-distance regime. If the collision energy is much smaller than this barrier, the particles motion is confined to the long-distance region, where the potential is purely repulsive. 

So far, we have studied separately the ordering and splitting of positive-parity excitations and the corresponding pattern of the negative-parity states. We now address the following question which is the lowest excitation the positive-parity radial excitation l) = [56,0 ] or the negative-parity orbital excitation 2)s[70,1 ] This question is motivated by the anomalously low location of the Roper resonance in the excitation spectrum of N and A. In the case of harmonic confinement, the radial... 

Restricting the quadrupole deformation to a mixing of closest unperturbed states is exact for harmonic confinement, but leads us to underestimate (2n by 20% for a smooth potential j8 = 0.1. The latter model, with the parameters (11.19), gives = 0.004 fm. ... 

For a gas in a harmonic confining potential, the phase space density can be directly related to the density and temperature, such that... 

To provide quantitative predictions about how to detect the superfluid-insulator transition in these experiments, Kashurnikov, Prokofev and Svistunov performed quantum Monte Carlo simulations of the singleparticle density matrix p,y = (I I /). They used the Bose-Hubbard model with harmonic confining potential and carried out world-line Monte Carlo simulations with the continuous-time Worm algorithm. The diagonal elements of the density matrix provide the real-space particle density, and... 

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