The low-energy two-body scattering S- and P-wave parameters, together with a shape-dependent S-wave quantity, which appear in the well-known low-density expansions for the ground-state energy of many-boson and many-fermion systems, are accurately calculated for several central pair potentials for spin-polarized atoms, helium atoms, and nucleons. An integral-equation method allowing accurate determination of the coefficients of the above-mentioned parameters expanded in powers of the attraction strength of the pair potential is presented and applied. The results are needed for perturbation studies of ground-state energies of various quantum many-body systems.
The low-energy two-body scattering S- and P-wave parameters, together with a shape-dependent S-wave quantity, which appear in the well-known low-density expansions for the ground-state energy of many-boson and many-fermion systems, are accurately calculated for several central pair potentials for spin-polarized atoms, helium atoms, and nucleons. An integral-equation method allowing accurate determination of the coefficients of the above-mentioned parameters expanded in powers of the attraction strength of the pair potential is presented and applied. The results are needed for perturbation studies of ground-state energies of various quantum many-body systems.
At long distances, interactions between neutral ground-state atoms can be described by the van der Waals potential. In the ultracold regime, atom-atom scattering is dominated by s-waves phase shifts given by an effective range expansion in terms of the scattering length ${\ensuremath{\alpha}}_{0}$ and the effective range r0. We show that while the scattering length cannot be predicted for these potentials, the effective range is given by the universal low-energy theorem ${r}_{0}=A+B/{\ensuremath{\alpha}}_{0}+C/{\ensuremath{\alpha}}_{0}^{2}$, where A, B, and C depend on the dispersion coefficients Cn and the reduced diatom mass. We confront this formula to about 100 determinations of r0 and ${\ensuremath{\alpha}}_{0}$ and show why the result is dominated by the leading dispersion coefficient C6. Universality and scaling extend much beyond naive dimensional analysis estimates.
Low-energy properties of spin-polarized electron-helium (23S) and spin-polarized positronium-helium (23S) scattering are calculated using the modified confined variational method. To take the van der Waals interaction into consideration, the van der Waals coefficient between positronium and helium (23S) is calculated using the explicitly correlated Gaussian basis. Compared with the ground state noble gases, the larger scattering length and zero pickoff annihilation rate indicates that He(23S) may be a more suitable cooling gas. In the energy range we considered, no similarity is found for the S-wave cross sections between these two scatterings. A Ramsauer-Townsend minimum is observed in spin-polarized electron-helium(23S) scattering due to the large polarization of metastable helium. This may open a new way to find and study the Ramsauer-Townsend effect.
I am looking at the old problem of the ground-state energy of quantum many-Fermion systems. My tools will be a version of perturbation theory. It is plain that the ordinary, low-density rearrangement can not be used directly to compute the saturation energy and density of a system for which the two-body potential has a strong repulsive core and an attractive portion beyond the core, because the path of continuation from low-density to saturation density passes through the two-phase (vacuum, liquid) region. Likewise it is difficult to supply an expansion directly in lambda for fixed rho (density) because of the difficulty of the strong repulsive core and the tediousness of the calculations. My approach here is to expand the energy in a double series in k/sub F/, the Fermi momentum, and lambda, the strength of the attractive part of the potential. The plan is then to sum up, by the Pade method or a generalization of it, the k/sub F/ expansions of the coefficients of each power of lambda into a function of k/sub F/. The coefficient of lambda/sup 0/ is just the repulsive-core energy itself.
The low-energy scattering of three bosons or distinguishable particles with short-range interactions is characterized by a fundamental parameter, the three-body scattering hypervolume, which is responsible for the nonuniversal effects in dilute Bose-Einstein condensates. We derive an analytical formula of the three-body scattering hypervolumes for weak interactions. When the interaction supports two-body bound states, the three-body scattering hypervolume gains a negative imaginary part, which is directly related to the rate constant for three-body recombinations. We develop a numerical method to calculate the three-body scattering hypervolumes for some model potentials with variable strengths. For real atoms with van der Waals potential, the three-body scattering hypervolumes are much harder to compute, because the three-body wave function is highly oscillatory at smaller inter-atomic distances. However, they may be extractable from precision data such as the collective-mode frequencies of trapped Bose-Einstein condensates. In many numerical simulations, the system being simulated is put into a large but finite volume, such as a large periodic box of size L. To extract the low-energy scattering properties of two particles in infinite space from such simulations, L\\"uscher's formula must be used. In the second part of this thesis, we generalize L\\"uscher's formula to d spatial dimensions. We obtain its s-wave and p-wave approximations and the systematic expansions of the energies of the low-lying states. At a s-wave resonance in dge4 dimensions, we identify two low-lying states close to the threshold with nearly opposite energies, Esimpm1/Ld/2 when dge5, or Esimpm1/L2sqrtlnL when d=4. These calculations provide important insights to the physics of three particles at a three-body resonance in finite volumes in two or three dimensions. Three-body resonances are important phenomena not yet completely explored in the experiments on cold atoms.
A method by Post, which has been used to calculate lower bounds on the energies of the lowest spatially symmetric or antisymmetric states of many-identical-particle systems, is shown to apply quite generally to the ground states of systems of bosons or fermions for any pair interactions whatever. In particular, the N -nucleon system is considered. No ordinary central pair potential was found to give the empirical binding energies of 2 H, 3 H and 4 He. For the exponential and square-well potentials the calculated bounds on the energies are closer than 1% over wide ranges of the potential parameters. The method justifies and generalizes certain `equivalent two-body' calculations originated by Wigner.
Quantum corrections to the properties of a homogeneous interacting Bose gas at zero temperature can be calculated as a low-density expansion in powers of ρa3, where ρ is the number density and a is the S-wave scattering length. We calculate the ground state energy density to second order in ρa3. The coefficient of the ρa3 correction has a logarithmic term that was calculated in 1959. We present the first calculation of the constant under the logarithm. The constant depends not only on a, but also on an extra parameter that describes the low energy 3→3 scattering of the bosons. In the case of alkali atoms, we argue that the second order quantum correction is dominated by the logarithmic term, where the argument of the logarithm is ρaℓV2, and ℓV is the length scale set by the van der Waals potential.