Elliptical cones

Symmetry 50. Intercepts 50. Asymptotes 50. Equations of Slope 51. Tangents 51. Equations of a Straight Line 52. Equations of a Circle 53. Equations of a Parabola 53. Equations of an Ellipse of Eccentricity e 54. Equations of a Hyperbola 55. Equations of Three-Dimensional Coordinate Systems 56. Equations of a Plane 56. Equations of a Line 57. Equations of Angles 57. Equation of a Sphere 57. Equation of an Ellipsoid 57. Equations of Hyperboloids and Paraboloids 58. Equation of an Elliptic Cone 59. Equation of an Elliptic Cylinder 59. 

Equation (standard form) of an Elliptic Cone (Figure 1-50)... 

The common cone has a circular section, whm the graerating lines always intersect a given circle. An elliptic cone is another special case, and square and rectangular cones are others. 

The hydrogen atom confined in semi-infinite space with an elliptical cone boundary... 

Rotations of asymmetric molecules in semi-infinite spaces with elliptical cone boundaries... 

Both works [2] and [3] show the separations of the eigenvalue equations for H and H, and H and H, in their respective spheroconal coordinates, into Lame differential equations in the individual elliptical cone angular coordinates. The corresponding solutions are Lam6 spheroconal polynomials included in the classic book of Whittaker and Watson [12]. In practice, the numerical evaluation of such Lame functions was not developed in an efficient manner so that the exact formulation of Ref. [2] did not prosper. Consequently, the analysis of rotations of asymmetric molecules took the route of perturbation theory using the familiar basis of spherical harmonics. 

Table 4.1 Sample of numerical values of alternative molecular asymmetry distribution and elliptical-cone parameter sets, from Eqs. (37 and 44)...

The construction of the Lam6 spheroconal harmonic polynomials involves Eq. (43) with matching parameters Eq. (44), matching species, and matching excitations of the respective elliptical cone coordinate degrees of freedom. The matching of species and kinds are fhe following ... 

The numerical results for the eigenenergies and eigenfunctions evaluated in Refs. [5] and [6] for molecules with different asymmetry distributions and states are accurate and consistent. The zeros of the individual Lame functions can be determined with high accuracy, and are illustrated in Figure 1 in Ref. [6]. They allow writing the Lame polynomial in product forms presented in Sections 2.1 and 2.2. They are also the key to implement the boundary condition for the rotations of molecules confined by elliptical cones as discussed in Section 3. 

A GUIDE TO ROTATIONS OF THE HYDROGEN ATOM AND ASYMMETRIC MOLECULES CONFINED BY ELLIPTICAL CONES... 

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. 

In fact, the impenetrable elliptical cone confinement imposes the boundary conditions... 

The Lame quasi-periodic functions are common eigenfunctions of fhe operators U and H as discussed in Sections 2.2 and 2.5. In Refs. [1] and [8], we chose the notation of G for the latter in order to emphasize that it represents the geometry of the spheroconal elliptical cone confinement, in contrast with its dynamical character for the rotations of asymmetric molecules. The eigenvalues and /i are numerically found to satisfy the relationships... 

Because the Hamiltonian of any central potential quantum system, H p/ commutes with the operators and H, they also have common eigenfunctions, including the situation of confinement by elliptical cones. Although Ref. [8] focused on the hydrogen atom. Ref. [1] included the examples of the free particle confined by elliptical cones with spherical caps, and the harmonic oscillator confined by elliptical cones. They all share the angular momentum eigenfunctions of Eqs. (98-101), which were evaluated in Ref. [8] and could be borrowed immediately. Their radial functions and their... 

Readers are invited to do their own reading of Ref. [1], including the effects of the confinement by elliptical cones on the energy spectra and eigenfunctions of the familiar free particle and harmonic oscillator. 

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). 

The rotations of asymmetric molecules confined by the two alternative families of spheroconal elliptical cone boundaries xi = Xio and X2 = X20 is a richer problem compared with the previous ones. In fact, the boundary... 

For the asymmetric molecules 0° < a < 60°, the eigenvalues and eigenfunctions associated with the respective elliptical-cone coordinates, interpolate between the above limiting cases. 

Table 4.5 First eigenvalue h] and matching /i of the respective Lam functions as functions of angular momentum values X for the successive families of elliptical cones 1 >kl>0...

Table 4.11 Table of energy eigenvalues and versus positions of the confining elliptical cones, sn(xio (T) and sn(x2ol< ) respectively, for the successive excited states of asymmetric molecules 60° > (T > 0°... 

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