Paraboloid 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. 

Another property of interesf in the confined hydrogen afom is the pressure distribution on the walls of the confining surface. Reference [8] adapted the method introduced for the case of confinement by paraboloids [41] using Hirschfelder s work on the mechanical properties of quantum systems [42], for the specific case of confinement by elliptical cones. Table V and Figure 3 in Ref. [8] illustrate the variations of the pressure in different radial positions r and angular positions sn(x2 (r) on the walls of some confining cones sn(xiok). 

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] ... 

We consider the confinement of the hydrogen atom by a paraboloid defined by a fixed value off = fo- Its wave functions have the form of Equation (74), and the condition that it vanishes at the position of the confining boundary is expressed as... 

Our early work on the hydrogen atom confined inside prolate spheroidal boxes also dealt with the molecular hydrogen H+ and molecular HeH++ ions [18]. The investigation of the hydrogen molecular ion was extended recently for confinement in boxes with the same shape with penetrable walls [40]. On the other hand, more than ten years ago, we investigated the ground state of the helium atom confined in a semi-infinite space [46] and inside boxes [47] with paraboloidal boundaries. 

As a specific experience, we can point out that our works on the helium atom confined by paraboloids [46,47] were preceded by variational calculations for the hydrogen atom in the respective confinement situations [49] and by the construction of the paraboloidal harmonic expansion of the Coulomb potential [50]. The last reference also includes the corresponding expansions in prolate and oblate spheroidal coordinates. 

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