Wave Functions In Closed Form Research Articles

Energetics and Efimov states of three interacting bosons and mass-imbalanced fermions in a three-dimensional spherical harmonic trap

We consider a system of three particles, either three identical bosons or two identical fermions plus an impurity, within a three-dimensional isotropic trap interacting via a contact interaction. Using two approaches, one using an infinite sum of basis states for the wavefunction and the other a closed form wavefunction, we calculate the allowable energy eigenstates of the system as a function of the interaction strength, including the strongly and weakly interacting limits. For the fermionic case this is done while maintaining generality regarding particle masses. We find that the two methods of calculating the spectrum are in excellent agreement in the strongly interacting limit. However the infinite sum approach is unable to uniquely specify the energy of Efimov states, but in the strongly interacting limit there is, to a high degree of accuracy, a correspondence between the three-body parameter required by the boundary condition of the closed form approach and the summation truncation order required by the summation approach. This specification of the energies and wavefunctions forms the basis with which thermodynamic variables such as the virial coefficients or Tan contacts, or dynamic phenomena like quench dynamics can be calculated.

Scattering of Anti-Plane (SH) Waves by a Shallow Semi-Elliptical Hill with a Concentric Elliptical Tunnel

A closed-form wave function analytic solution of two-dimensional scattering and diffraction of incident plane SH-waves by a shallow semi-elliptical hill with a concentric elliptical tunnel on an elastic half-space is presented. This research extends the previous solution by authors in 2013 which addressed only the semi-elliptical hill, to a hill-tunnel system. Ground surface displacement near the hill are studied and exhibited. It was observed from the results that the aspect ratio of hill and tunnel profiles, frequency of incident waves, and incident angles are the primary factors that may affect the surface ground motion amplitudes near the hill-tunnel system.

Soil-structure interaction on shallow rigid circular foundation: plane SH waves from far-field earthquakes

A closed-form wave function analytic solution of two-dimensional scattering and diffraction of incident plane SH-waves by a flexible wall on a rigid shallow circular foundation embedded in an elastic half-space is presented. This research generalizes the previous solution by Trifunac in 1972, which tackled only the semi-circular foundation, to arbitrary shallow circular-arc foundation cases, and is thus comparatively more realistic. Ground surface displacement spectra at higher frequencies are also obtained. As an analytical series solution, the accuracy and error analysis of the numerical results are also discussed. It was observed from the results that the rise-to-span ratio of the foundation profile, frequency of incident waves, and mass ratios of different media (foundation-structure-soil) are the three primary factors that may affect the surface ground motion amplitudes near the structure.

Soil-Structure Interaction on a Shallow Rigid Circular Foundation: SH Wave Source from Near-Field Excitations

A closed-form wave function analytic solution is presented in this article regarding the two-dimensional scattering and diffraction of a flexible wall sitting on a rigid shallow circular foundation embedded in an elastic half-space that is activated by a nearby anti-plane line source such as a blast caused by underground construction or mineral exploration or a near-field fault rupture, using similar methodology as the other paper in a series [Lee and Luo, 2013]. These wave propagation influences, although often treated as a transient process, may be simulated as linear combinations of steady-simple harmonic responses as studied in this article. Ground surface displacements spectra for wide-band of incident wave frequencies are calculated. Based on the spectra obtained, the dependence of near-field ground displacements are shown with respect to the rise-to-span ratio of foundation profile, frequency of incident waves, distance of source from the foundation, and mass ratios of various media (foundation-structure-soil). The screening effect of rigid foundation upon ground motions behind grazing incident waves is also presented.

Anti-plane foundationless soil–structure interaction

This paper presents a closed-form wave function analytic solution of two-dimensional scattering and diffraction of anti-plane SH-waves by a two-dimensional foundationless structure that corresponds to a shear wall on an elastic half-space. A wave-function expansion method is used to solve this model by first prescribing a set of wave functions with undetermined coefficients and then assembling them together based on the stress and displacement boundary conditions on the surface between the structure and half space. This results in a set of infinite equations to be solved by truncating to a finite set. The amplitudes and residuals of the displacement and stress distributions around the structure and nearby ground surface will be discussed carefully. While the solution is analytical, the computation of the numerical results involves the evaluation of complicated integrals. This analytic solution will be helpful to the understanding of propagation of seismic or other stress waves within the superstructure(s) undergoing earthquakes or other blast loads.

Antiplane (SH) Waves Diffraction by a Semicircular Cylindrical Hill Revisited: An Improved Analytic Wave Series Solution

An improved accurate closed-form wave function analytic solution of two-dimensional scattering and diffraction of antiplane SH waves by a semicircular cylindrical hill on an elastic half space is presented. In the previous solution, stress and displacement residual auxiliary functions were defined at the circular interface above and below the circular hill. The method of weighted residues (moment method) was used to solve for the unknown scattered and transmitted waves by requiring each term of Fourier series expansion of these auxiliary residual functions to vanish. It was found that the stress residual amplitudes on both (left and right) rims of the hill (ideally should be zero) are not numerically insignificant, irrespective of how many terms used. It was pointed out that the shear stress at the rim is infinite, and that the stress auxiliary function is discontinuous at both rims of the hill, exhibiting a problem for the numerical solution that is more complicated than Gibbs' phenomenon. The problem with the overshoot of the stress residual amplitudes at the rim was most likely numerical. In this paper, all displacement and stress waves were expressed as cosine functions, and the solution of the circular hill problem was reformulated in this paper, and, for the solution to be correct, the computed stress and displacement residual amplitudes were shown to be numerically negligible everywhere, including those at both rims of the hill. Displacements at higher frequencies are also computed.

Modified Riccati approach to partially solvable quantum Hamiltonians. III. Related families of Pöschl-Teller potentials.

Within the modified Riccati approach developed elsewhere, we construct quasiexactly solvable potentials related to the P\"oschl-Teller and modified P\"oschl-Teller potentials. This is achieved by considering closed-form wave functions whose logarithmic derivatives are rational in the mapping u(x)=cosh(x) or u(x)=cos(x). With the hyperbolic mapping we construct an interesting four-parameter quasiexactly solvable confining potential with one, two, or three minima upon the value of its coupling constants. We give explicit expressions for the bound-state eigenfunctions and corresponding energy levels for some particular cases. The analytic continuation x\ensuremath{\rightarrow}ix transforms the previous potential into a periodic confining well, or a Kronig-Penney-like potential with an interesting structure of minima in each cell and a finite number of gaps. The energy levels in the last case are some of the band edges of the spectrum. In any case a finite but arbitrarily large number of energy levels and corresponding eigenfunctions are determined from the diagonalization of a finite matrix.

Coordinate systems and analytic expansions for three-body atomic wavefunctions. II. Closed form wavefunction to second order in r

For pt.I, see ibid., vol.20, no.8, p.2043-75 (1987). Several coordinate systems for solving the few-electron Schrodinger equation are presented. Formal solutions corresponding to each coordinate system are given in terms of the Fock expansion and their interrelationships and general structure are examined. Attention is focused on the solutions obtained using spherical polar coordinates for a Coulomb potential of arbitrary symmetry. The wavefunction is obtained up to second order in the hyperradius r=(r21+r(sup)22)12/, and the special case of 1S states is then reduced to a closed form using classical techniques. The insight gained from this reduction suggests methods for solving the wavefunction to all orders. The results hint at the existence of closed form wavefunctions for few-body systems.

Erratum: Improved closed form wavefunction for H+2

Erratum: Improved closed form wavefunction for H+2

Improved closed form wavefunction for H+2

A possible closed form wavefunction for the hydrogen molecular ion H+2 is presented and the expectation values for the grand state energy are calculated by solving the wave equation using the closed form wave functions. (AIP)