Hydrogen atom, confinement

The hydrogen atom confined in semi-infinite spaces limited by conoidal boundaries... 

In particular, for the hydrogen atom, it was recognized that the confinements by elliptical cones and by dihedral angles were pending. Section 5.1 in the Preview of Ref. [9] formulated the problem of the hydrogen atom confined by a family of elliptical cones identified in its Eqs. (123 and 124), with the boundary condition that the wavefunctions vanish in such cones, Eqs. (125 and 126). The corresponding solution [8] is the subject of Section 3.3. 

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

Here we shall consider the simple case of a single hydrogen atom confined in a box. Such a system is akin to an octahedral interstitial... 

Table 1 The energies of hydrogen atom confined to a box of radius R, obtained from numerical calculations, model wave functions, and the simple expression in Equation (3.39). For each state and R, the first row is from numerical calculation, the second row in brackets is from model wave function, and the third row in brackets is from the simple expression in Equation (3.39). The last column is the dipole polarizability for the ground state...

The Hydrogen Atom Confined in Semi-infinite Spaces Limited by Conoidal Boundaries... 

Our contemporary article on the hydrogen atom confined by a conical boundary [22] also exhibits the properties commented in (ii)—(iv). In particular, the cone becomes the equatorial plane when the polar angle is ninety degrees, corresponding to Levine s 1965 pioneering model for an impurity atom on the surface of a solid [31]. 

While the hydrogen atom confined in spaces limited by closed boundaries does show a monotonic and unlimited increase of energy of its levels, as the boundary approaches the nucleus and the volume of confinement is reduced [2,14,17,18] its confinement by open conoidal boundaries is characterized by the monotonic increasing of energy of its levels only up to zero energy in the corresponding limit situations, with the consequent infinite degeneracy, as it was shown in [9,21,22] and commented on above. 

Ground state energy of the two-dimensional hydrogen atom confined with conical curves through the variational method... 

The comment on their work focuses on the limitations of their treatment of the hydrogen atom confined inside an angle. When we became aware of their incorrect results we decided to study the exact solutions for the confinement... 

For consistency in this subsection, it is also necessary to question the validity of the trial variational function and the results of the calculation for the ground-state energy of the hydrogen atom confined by a hyperbolic boundary in [3]. In fact, the corresponding function in their notation is... 

The conclusion of these comments is that the trial functions and the results reported in [3] do not provide a reliable description of the two-dimensional hydrogen atom ground-state energy for confinements by an angle and by a hyperbola. The solutions of the Schrodinger equation for the hydrogen atom confined by a hyperbola can also be constructed transparently and accurately using standard methods. 

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. 

For open boundaries, the energy and wave numbers of the type of this equation are not quantized, in contrast with the situation of the closed boundaries as in Equation (100). Correspondingly, the threshold value of the energy in Equation (104) is zero, validating it as the reference for the analysis of ionization of the hydrogen atom confined by an open boundary. 

This is the counterpart of Equation (100), which is recovered in the limit of vanishing / and infinite u, so that fu becomes r, the radial coordinate. Correspondingly, it is the reference for the analysis of the ionization of the hydrogen atom confined by prolate spheroids. 

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