Hydrogenic wave functions, bound-state

Using either so(4, l)orso(4, 2) we can find infinite dimensional unirrepsfor which all bound-state scaled hydrogenic wave functions form a basis. The Lie algebra so(4, 2) is more suitable for our purposes since we have the simple expressions r = T3 — Tur = B — A. We can then calculate matrix elements of... 

In order to obtain the so(4,2) representation corresponding to the hydrogenic case we must specify the action of the 15 generators on the scaled bound-state hydrogenic wave functions nZm>. We have already done this for the generators L, A of so(4) [cf. Eqs. (177) and (178)]. For TUT2, T3 it follows... 

The unique properties of dilute metal-ammonia solutions depend not upon the nature of the metal species, but upon the solvated electron common to all these solutions. Thus, the electron-in-a-cavity model (17, 19, 21) seems best suited to describe the species present in these solutions since the model is independent of the type of cation present. Jortner and his associates (15, 16) have extended this model by assuming that the cavity arises from polarization of the medium by the electron. The energy levels of the bound electrons are obtained by using a potential function containing the static and optical dielectric constants of the bulk medium as parameters. Using one-parameter hydrogen-like wave functions for the first two bound states of the electron, the total energy of the ith state is expressed as... 

The usual Hilbert-space requirement of continuous gradients is not appropriate to Coulombic point-singularities of the potential function u(r) [ 196]. This is illustrated by the cusp behavior of hydrogenic bound-state wave functions, for which the Hamiltonian operator is... 

In this section we shall discuss in some detail the formalism needed to apply the so(4, 2) algebraic methods to problems whose unperturbed Hamiltonian is hydrogenic. First a scaling transformation is applied to obtain a new Hamiltonian whose unperturbed part is just the so(2, 1) generator T3, which has a purely discrete spectrum. Next we use the scaled hydrogenic eigenfunctions of T3 as a basis for the expansion of the exact wave function. This discrete basis is complete with respect to the expansion of bound-state wave functions whereas the usual bound-state eigenfunctions do not form a complete set continuum functions must also be included to ensure completeness (cf. Section VI,A)-... 

It divides into two categories. For E < 0 we obtain an infinite discrete sequence of bounded states for > 0 we obtain a continuous spectrum. Due to the similarity of (6.1.22) to the Z = 0 radial Hamiltonian of the hydrogen atom, the wave functions and energies can be computed analytically. For the bound state wave functions we obtain... 

We now consider the bound states of the hydrogen atom, with < 0. In this case, the quantity in parentheses in (6.66) is positive. Since we want the wave functions to remain finite as r goes to infinity, we prefer the minus sign in (6.66), and in order to get a two-term recursion relation, we make the substitution... 

Degeneracy. Are the hydrogen-atom energy levels degenerate For the bound states, the energy (6.94) depends only on n. However, the wave function (6.61) depends on all three quantum numbers n, I, and m, whose allowed values are [Eqs. (6.91), (6.92), (5.108), and (5.109)]... 

THE BOUND-STATE HYDROGEN-ATOM WAVE FUNCTIONS... 

Section 6.6 The Bound-State Hydrogen-Atom Wave Functions 143... 

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