Hydrogen, confined circular cone

This is the title of Chapter 3 in Ref. [9], Advances in Quantum Chemistry, Vol. 57, dedicated to confined quantum systems. The conoidal boundaries involve spheres, circular cones, dihedral angles, confocal paraboloids, con-focal prolate spheroids, and confocal hyperboloids as natural boundaries of confinement for the hydrogen atom. In fact, such boundaries are associated with the respective coordinates in which the Schrodinger equation is separable and the boundary conditions for confinement are easily implemented. While spheres and spheroids model the confinement in finite volumes, the other surfaces correspond to the confinement in semi-infite spaces. 

Our experience with the hydrogen atom confined by a circular cone [39] involves going from Legendre polynomials P cos9) to Legendre functions... 

The early antecedent of the investigation presented in this section is Sommerfeld and Hartmann s work on the rotor constrained in a circular cone [43]. That work and also the hydrogen atom in the same situation of confinement [39] involve eigenfunctions described by Legendre functions vanishing at the circular cone boundaries... 

The same eigenfunctions in spherical coordinates used in the previous subsection and introduced in 3.1 are the basis for the analysis of the hydrogen atom confined by a circular cone defined by a fixed value of the polar angle 9 = 9q. The boundary condition requiring the vanishing of the wave function at such an angle must be satisfied by the hypergeometric function in Equation (36),... 

We invite the reader to follow the changes of the energy levels of the hydrogen atoms confined by circular cones, starting infinitesimally close to the south pole, passing the equator and approaching the north pole, by referring them to [22], At the end we will get back to the south pole itself to clarify the difference between "almost-free" and free. 

The free electron confined by a circular cone shares the same harmonic angular functions with the hydrogen atom, and its radial function is the adaptation of Equation (99) with a spherical Bessel function of order X and wave number determined by its energy ... 

Both sets of families span the shapes from circular cones, for ki = 1 and eccentricity zero, to dihedral angles, for ki = 0 and eccentricity 1, keeping in mind the complementarity between their parameters, Equation (50). While in [37] these parameters are related to the asymmetry of the molecules, here they determine the shape of the confining elliptical cones. The spheroconal harmonics borrowed from [37] in Section 3.2 are some of the solutions for the hydrogen atom confined by an elliptical cone, when the latter coincides with one of the elliptical-cone nodes in Equation (57). Of course, just as already discussed for the other conoidal boundaries in Section 4, here there are other solutions for the same elliptical-cone boundary, and their evaluation is discussed next. 

The confinement of the two-dimensional hydrogen atom by angles and hyperbolas [3,10] and 2.3, has its counterparts with circular cones [22] and 4.2, and hyperboloids [9] and 4.5 in the three-dimensional case. The "almost-free" hydrogen atom limit happens at the other end of the domain of the... 

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