Institute of Physical Chemistry

Using a rapidly convergent composite basis of Frankowski-Pekeris and Frankowski functions, we have accurately calculated the nodal surfaces of low-lying excited states of the helium atom to investigate Bressanini and Reynolds' conjecture ͓D. Bressanini and P. J. Reynolds, Phys. Rev. Lett. 95, 110201 ͑2005͔͒ that these nodal surfaces are rigorously independent of the interelectronic angle 12 . We find that in fact there is a slight dependence of the nodal surfaces on 12 , but it is so small that the assumption of strict independence may well yield extremely useful approximations in fixed-node quantum Monte Carlo calculations. We explain how Kato's cusp conditions determine the qualitative features of these nodal surfaces, which can accurately be modeled using the familiar ansatz of a symmetric or antisymmetric linear combination of products of hydrogenic orbitals, with some adjustments of the parameters. We explain why a similar near independence of the nodal surfaces on the angular variables can be expected for the ground and singly excited states of the lithium atom, but generally not for larger atoms.

We apply a physical and historical analysis to a passage by the medieval scholar Michael Scot concerning
multiple rainbows, a meteorological phenomenon whose existence has only been acknowledged in recent
history. We survey various types of physical models to best decipher Scot’s description of four parallel
rainbows as well as a linguistic analysis of Scot’s special etymology. The conclusions have implications on
Scot’s whereabouts at the turn of the 13th century.

We generalize the 1 + 1-dimensional gravity formalism of Ohta and Mann to 3 + 1 dimensions by developing the canonical reduction of a proposed formalism applied to a system coupled with a set of point particles. This is done via the Arnowitt-Deser-Misner method and by eliminating the resulting constraints and imposing coordinate conditions. The reduced Hamiltonian is completely determined in terms of the particles’ canonical variables (coordinates, dilaton field and momenta). It is found that the equation governing the dilaton field under suitable gauge and coordinate conditions, including the absence of transverse-traceless metric components, is a logarithmic Schrödinger equation. Thus, although different, the 3 + 1 formalism retains some essential features of the earlier 1 + 1 formalism, in particular the means of obtaining a quantum theory for dilatonic gravity.

Data clustering is a vital tool for data analysis. This work shows that some existing useful methods in data clustering are actually based on quantum mechanics and can be assembled into a~powerful and accurate data clustering method where the efficiency of computational quantum chemistry eigenvalue methods is therefore applicable. These methods can be applied to scientific data, engineering data and even text.

Herein, we use Hardy’s notion of the “false derivative” to obtain exact multiple roots in closed form of the transcendental-algebraic equations representing the generalized LambertW function. In this fashion, we flesh out the generalized Lambert W function by complementing previous developments to produce a more complete and integrated body of work. Finally, we demonstrate the usefulness of this special function with some applications.

The nonlinear logarithmic Schrodinger equation (log SE) appears in many branches of fundamental physics, ranging from macroscopic superfluids to quantum gravity. We consider here a model problem, in which the log SE includes an attractive Coulomb interaction. We derive an analytical solution for the ground state energy and wave function as a function of the strength of the logarithmic interaction. We develop an iterative finite method to solve the Coulombic log SE for the spherically symmetric states. The ground state results agree with the exact solution to better than one part in 10^10. The excited states are converged to better than one part in 10^8. We also construct a remarkably simple variational wave function, consisting of a sum of Gaussons with n free parameters. One can obtain an approximation to the energy and wave function that is in good agreement with the finite element results. Although the Coulomb problem is interesting in its own right, the iterative finite element method and the variational Gausson basis approach can be applied to any central force Hamiltonian.

Experimental data suggests that, at temperatures below 1 K, the pressure in liquid helium has a cubic dependence on density. Thus the speed of sound scales as a cubic root of pressure. Near a critical pressure point, this speed approaches zero whereby the critical pressure is negative, thus indicating a cavitation instability regime. We demonstrate that to explain this dependence, one has to view liquid helium as a mixture of three quantum Bose liquids: dilute (Gross-Pitaevskii-type) Bose-Einstein condensate, Ginzburg-Sobyanin-type fluid, and logarithmic superfluid. Therefore, the dynamics of such a mixture is described by a quantum wave equation, which contains not only the polynomial (Gross-Pitaevskii and Ginzburg-Sobyanin) nonlinearities with respect to a condensate wavefunction, but also a non-polynomial logarithmic nonlinearity. We derive an equation of state and speed of sound in our model, and show their agreement with experiment.

A general analysis of the two-body Dirac equation is presented for the case of equal masses interacting via a static Coulomb potential. Radial equations are derived and their analytical structure is discussed. Standard analytical and perturbative methods have failed to provide solutions to the radial equations due to the presence of the singularity on the negative radial axis at roughly the distance of the classical electron radius. The exact radial equations are solved using finite-element analysis, and the low-lying bound states are obtained to an accuracy of one part in 10'. The effect of the singularity is clearly seen in the structure of the finite-element radial components.

Some form of the time-independent logarithmic Schrödinger equation (log SE) arises in almost every branch of physics. Nevertheless, little progress has been made in obtaining analytical or numerical solutions due to the nonlinearity of the logarithmic term in the Hamiltonian. Even for a central potential, the Hamiltonian does not commute with L^2 or L z the Hamiltonian is invariant under the parity operation only if the wave function is an eigenstate of the parity operator. We show that the solutions with well-defined parity can be expressed as a linear combination of eigenstates of L 2 and L z where the parity restrictions on l and m determine the nodal structure of the wave function. The dominant contribution in the sum is designated as l-tilde and m-tilde; these serve as approximate quantum numbers. Using an iterative finite element approach, we also carry out fully converged numerical calculations in 1D, 2D and 3D for the special case of a Coulomb potential. The nodal structure of the wave functions and the expectation values < L^2 > and  < Lz^2> are consistent with the analytical predictions. Values for the energy and expectations values are tabulated for the low-lying states. The methods developed for this problem are applicable across many areas of physics.

We have analyzed and reduced a general (quantum-mechanical)
expression for the atom-atom exchange energy formulated as a five-dimensional
surface integral, which arises in studying the charge exchange processes
in diatomic molecules. It is shown that this five-dimensional surface integral can
be decoupled into a three-dimensional integral and a two-dimensional angular
integral which can be solved analytically using a special decomposition. Exact
solutions of the two-dimensional angular integrals are presented and generalized.
Algebraic aspects, invariance properties and exact solutions of integrals
involving Legendre and Chebyshev polynomials are also discussed.

Herein, we present analytical solutions for the electronic energy eigenvalues of the hydrogen molecular ion H ¡ , namely the one-electron two-fixed-center problem. These are given for the homonuclear case for the countable infinity of discrete states when the magnetic quantum number © is zero i.e. for ¡ states. In this case, these solutions are the roots of a set of two coupled three-term recurrence relations. The eigensolutions are obtained from an application of experimental mathematics using Computer Algebra as its principal tool and are vindicated by numerical and algebraic demonstrations. Finally, the mathematical nature of the eigenenergies is identified.

We present a general procedure, based on the Holstein-Herring method, for calculating exactly the leading term in the exponentially small exchange energy splitting between two asymptotically degenerate states of a diatomic molecule or molecular ion. The general formulas we have derived are shown to reduce correctly to the previously known exact results for the specific cases of the lowest Σ and Π states of H + 2. We then apply our general formulas to calculate the exchange energy splittings between the lowest states of the diatomic alkali cations K + 2 , Rb + 2 , and Cs + 2 , which are isovalent to H + 2. Our results are found to be in very good agreement with the best available experimental data and ab initio calculations.

We formulate an efficient exact method of propagating optical wave packets (and cw beams) in isotropic and nonisotropic dispersive media. The method does not make the slowly varying envelope approximation in time or space and treats dispersion and diffraction exactly to all orders, even in the near field. It can also be used to determine the partial differential wave equation for pulses (and beams) to any order as a power series in the partial derivatives with respect to time and space. The method can treat extremely focused pulses and beams, e.g., from near-field scanning optical microscopy sources whose transverse spatial extent in smaller than a wavelength.