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Schrodinger's equation for the hydrogen molecule ion and the Helmholtz equation are separable in prolate and oblate spheroidal coordinates respectively. They share the same form of the angular equation. The first task in deriving the ground state energy of the hydrogen molecule ion, and in obtaining finite solutions of the Helmholtz equation, is to obtain the physically allowed values of the separation of variables parameter. The separation parameter is not known analytically, and since it can only have certain values, it is an important parameter to quantify. Chapter 2 of this thesis investigates an exact method of obtaining the separation parameter. By showing that the angular equation is solvable in terms of confluent Heun functions, a new method to obtain the separation parameter was obtained. We showed that the physically allowed values of the separation of variables parameter are given by the zeros of the Wronskian of two linearly dependent solutions to the angular equation. Since the Heun functions are implemented in Maple, this new method allows the separation parameter to be calculated to unlimited precision. As Schrodinger's equation for the hydrogen molecule ion is related to Helmholtz's equation, this warranted investigation of scalar beams. Tightly focused optical and quantum particle beams are described by exact solutions of the Helmholtz equation. In Chapter 3 of this thesis we investigate the applicability of the separable spheroidal solutions of the scalar Helmholtz equation as physical beam solutions. By requiring a scalar beam solution to satisfy certain physical constraints, we showed that the oblate spheroidal wave functions can only represent nonparaxial scalar beams when the angular function is odd, in terms of the angular variable. This condition ensures the convergence of integrals of physical quantities over a cross-section of the beam and allows for the physically necessary discontinuity in phase at z = 0 on the ellipsoidal surfaces of otherwise constant phase. However, these solutions were shown to have a discontinuous longitudinal derivative. Finally, we investigated the scattering of scalar waves by oblate and prolate spheroids whose symmetry axis is coincident with the direction of the incident plane wave. We developed a phase shift formulation of scattering by oblate and prolate spheroids, in parallel with the partial wave theory of scattering by spherical obstacles. The crucial step was application of a finite Legendre transform to the Helmholtz equation in spheroidal coordinates. Analytical results were readily obtained for scattering of Schrodinger particle waves by impenetrable spheroids and for scattering of sound waves by acoustically soft spheroids. The advantage of this theory is that it enables all that can be done for scattering by spherical obstacles to be carried over to the scattering by spheroids, provided the radial eigenfunctions are known.

The Theory of the Hydrogen Molecule Ion, Scalar Beams, and Scattering by Spheroids