Radial eigenfunctions

It has been found that these ratios do not depend on the precise shape of the eigenfunction, as for example, the ratios for a 5/-radial eigenfunctions are... 

Figure 7.1 Radial eigenfunctions Pn((r) = rR fr) for the electron in the hydrogen atom (in atomic units) where n is the principal quantum number,

Radial Eigenfunctions. After much grief, one gets the normalized nodeless radial Dirac eigenfunctions... 

The relevant equations for the derivative Numerov-Cooley (DNC) method closely follow Cooley s [111] presentation. Let R be the radial coordinate or bond displacement coordinate, P R) a radial eigenfunction, and U R) the potential function. The one-dimensional Schrodinger equation is then... 

Of course, the connection between Sections 4.1 and 4.2.2 is provided by Euler s formula = cos mtp isin mtp. Thus, the simultaneous raising and lowering actions of the linear momentum operators on the order of the angular momentum and radial eigenfunctions is established for cartesian, spherical, and spheroconal representations. 

Section 2.6 recognizes that for the hydrogen atom, its Hamiltonian also commutes with and H correspondingly, it also admits solutions with Lame spheroconal harmonics polynomial eigenfunctions. It also shares the same radial eigenfunction with the familiar solution with spherical harmonics, and additionally both can be obtained from a common generating function and both satisfy the addition theorem. 

The radial eigenfunctions of Equation (18) are also known [10], and here we write only their form without radial excitation np = 0 ... 

The analysis of the ionization limit, for vanishing E in Equation (33) or infinite nr in the eigenfunction of Equation (38), leads to the identification of the corresponding radial eigenfunctions in terms of Bessel functions ... 

A second linearly independent solution of Kummer s equation, conventionally denoted U(a, b, x), is not regular at the origin and can often be excluded on physical grounds. However, as we have shown recently [2-4], when a normal hydrogen atom is confined inside a finite sphere or spherical shell, its radial eigenfunctions may involve U(a,b, x) or a combination of two linearly independent solutions of Kummer s equation, chosen so as to satisfy the boundary conditions which reflect the particular confinement considered. We have felt it necessary to address this choice in Section 2 of the present work. 

It follows from a charge conjugation that for 0 < 7 < 1 the operator has the eigenvalues — n)- The corresponding radial eigenfunctions can be expressed in terms of hypergeometric functions ... 

This is seen in Figure 3.11 that illustrates radial eigenfunctions for the manganese atom. The distance from the nucleus at which the probability to find Is, 2s, 2p electrons is less than the Bohr radius. On the contrary, a 4s electron is located far off the nucleus because of shielding by inner electron shells. 

Figure 3.11 Radial eigenfunctions ripi (r) for the manganese atom (Z = 25). The extremum in the 3d is close to those of the 3s and 3p shells even though these are much...

Figure 11.1 Calculated eigenfunctions for the levels of atomic sodium plotted about two neighboring nuclei separated by a distance of 0.37 nm in bulk sodium. The solid curves are rip r) for the Is, 2s, 3s levels. The dashed curve is r times the radial eigenfunction for...

Because the Hulthen potential is spherically symmetric, the Green function can be expanded in terms of radial eigenfunctions and the spherical... 

---