Hyperboloid hydrogen confinement

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. 

This contribution as a chapter in the special volume of ADVANCES IN QUANTUM CHEMISTRY on Confined Quantum Systems is focussed on (i) the hydrogen atom, (ii) confinement by conoidal boundaries, and (iii) semi-infinite spaces however, some of its discussions may extend their validity to other physical systems and to confinement in closed volumes. The limitations in the title are given as a point of reference, and also take into account that several of the other chapters deal with confinement in finite volumes. A semantic parenthesis is also appropriate and self-explanatory Compare conical curves (circles, ellipses, parabolas, hyperbolas and their radial asymptotes) with conoidal surfaces (spheres, spheroids, paraboloids, hyperboloids and their radial asymptotic cones). 

The author became interested in the models of confinement of the hydrogen atom inside finite volumes [2,14,17,18] in connection with the measurements of the hyperfine structure of atomic hydrogen trapped in a-quartz [19,20]. Ten years later, he extended his interests to confinement in semi-infinite spaces limited by a paraboloid [21], a hyperboloid [9] and a cone [22] in connection with the exoelectron emission by compressed rocks [23,24], Jaskolski s report [1] cited several of the above-mentioned works [9,14,17, 18,21], each one of which had formulated and constructed exact solutions for new types of confinement for the hydrogen atom. This subsection is focussed on his citation of our article [9] ... 

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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